Filtrations and Buildings

Christophe Cornut

En hommage à Alexander Grothendieck

2000 Mathematics Subject Classification. 14L15, 18D10, 20E42, 20G05, 51E24.

Key words and phrases. Filtrations, Buildings and the Tannakian formalism.

Abstract. We construct and study a scheme theoretical version of the Tits vectorial building, relate it to filtrations on fiber functors, and use them to clarify various constructions pertaining to affine Bruhat-Tits buildings, for which we also provide a Tannakian description.

Contents

Chapter 1. Introduction  5
Chapter 2. The group theoretical formalism  10
2.1. Γ-graduations on smooth affine groups  10
2.2. Γ-filtrations on reductive groups  13
2.3. Relative positions of Γ-filtrations  21
2.4. Interlude on the dominance partial orders  25
Chapter 3. The Tannakian formalism  34
3.1. Γ-graduations and Γ-filtrations on quasi-coherent sheaves  35
3.2. Γ-graduations and Γ-filtrations on fiber functors  36
3.3. The subcategories of rigid objects  39
3.4. Skalar extensions  42
3.5. The regular representation  44
3.6. Relating Rep(G)(S) and Rep^{∘}(G)(S)  47
3.7. The stabilizer of a Γ-filtration, I  51
3.8. Grothendieck groups  55
3.9. The stabilizer of a Γ-filtration, II  56
3.10. Splitting filtrations  58
3.11. Consequences  64
3.12. Ranks and relative positions  72
3.13. Appendix: pure subsheaves  73
Chapter 4. The vectorial Tits building F^{Γ}(G)  76
4.1. Combinatorial structures  76
4.2. Distances and angles  82
Chapter 5. Affine F(G)-buildings  90
5.1. The dominance order  90
5.2. Affine F(G)-spaces and buildings  92
5.3. Further axioms  100
5.4. Walls and tight buildings  104
5.5. Metric properties  108
5.6. The affine F(P/U)-space T_{P}^{∞}X(G)  112
5.7. Example: F(G) as a tight affine F(G)-building  116
5.8. Example: a symmetric space  119
Chapter 6. Bruhat-Tits buildings  124
6.1. The Bruhat-Tits building of GL(V)  124
6.2. The Bruhat-Tits building of G  126
6.3. Functoriality for Bruhat-Tits buildings  128

6.4. A Tannakian formalism for Bruhat-Tits buildings  132
Nomenclature  142
Bibliography  148

CHAPTER 1 Introduction

Spherical, affine and vectorial buildings are covered by apartments which are respectively spheres, affine and vector spaces. They interact with each other as spheres, affine and vector spaces do.

The combinatorial Tits building of a reductive group G over a field K reflects the incidence relations between the parabolic subgroups of G. The spherical Tits building is the geometric realization of the combinatorial one, obtained by gluing spheres along common spherical sectors. Both buildings were defined by Tits in [46], they only depend upon the adjoint group of G, and they are not functorial in G. The vectorial Tits building was defined by Rousseau in [40], and it does depend functorially upon G. For a semi-simple group G, it may by defined as the cone on the spherical building of G, obtained by gluing vector spaces along common sectors. For a torus G, it is the group of K-rational cocharacters of G, tensored with ℝ. This vectorial Tits building is the unifying theme of our somewhat eclectic paper.

When K is a non-archimedean local field, there is also an affine building attached to G: the (extended) Bruhat-Tits building of G, as defined in [9, 10]. It is obtained by gluing affine spaces along common alcoves and it reflects the incidence relations between bounded open subgroups of G. The combinatorial Tits building of G encodes the geometry of this affine Bruhat-Tits building at infinity, and the combinatorial Tits buildings of the maximal reductive quotients of the special fibers of various (non-necessarily reductive) integral models of G similarly encode the local geometry of the affine Bruhat-Tits building of G.

For classical groups, which come equipped with a standard faithful representation, a global construction of the Bruhat-Tits building is given in [11, 12], generalizing the pioneering work of Goldman and Iwahori in [22], which dealt with the case of a general linear group and served as a model for the development of the whole Bruhat-Tits theory. For these classical groups, the Bruhat-Tits building of G is cut out from the space of all non-archimedean K-norms on the standard representation. My initial intention was to expand such a global construction to arbitrary reductive groups, and to clarify and canonify the aforementioned relations between the Bruhat-Tits building of G and the Tits buildings of various related groups.

Let me try to explain how I came to be interested in these questions. Grassmannians, flag manifolds and their affine counterparts are essentially orbits of G acting on its related buildings, and they show up in many branches of mathematics. In particular, integral p-adic Hodge theory makes extensive use of filtrations and lattices, which are respectively parametrized by flag manifolds and affine Grassmannians, or by the corresponding larger ambiant spaces: the vectorial Tits building and the affine Bruhat-Tits building. Working on classical results in p-adic Hodge theory [16, 15, 14], I eventually realized that a natural transitive action of the former on the latter explained many features of the interplay between filtrations and lattices in this area of mathematics. For G = GL_{n}, this circle of ideas was already more or less implicit in various works, for instance in Laffaille [31] or Fontaine and Rapoport [19]. But for more general groups, sound foundations seemed to be lacking or scattered in the existing litterature: I needed a flexible and extensive dictionnary connecting building-theoretical, geometric and metric notions and tools to properties of the relevant objects in linear algebra, filtrations and lattices.

For instance the aforementioned action itself was most definitely well-known to authors working in metric geometry. Here, the cone C(∂X) on the visual boundary ∂X of a CAT (0)-metric space (X, d) (see [8] for these notions) acts on X by nonexpanding maps as follows: an element of the cone is a pair (ξ, ℓ) where ξ ∈ ∂X is an asymptotic class of unit speed geodesic rays in X and ℓ ≥ 0 is some length (or speed); it acts on a point x of X by moving it at distance ℓ along the unique geodesic ray in ξ emanating from x. Taking X to be the Bruhat-Tits building of G over K, it was also known since Bruhat and Tits that X can be equipped with a non-canonical metric d which turns it into a CAT (0)-space, and for which the visual boundary ∂X is a realization of the spherical building of G over K – this geometric statement was initially encapsulated in the group-theoretical notion of double Tits systems, see [9, 5.1.33] and [40, 11.10]. But there were no clear-cut identifications between filtrations for G and the cone C(∂X), or between lattices (and norms) for G and its Bruhat-Tits building X; moreover, having to rely on the artificial choice of a metric to define the action furthermore blurred its naturality. This paper provides proper definitions of these notions, all the required canonical identifications, an explicit formula for the action of filtrations (which are elements of the vectorial Tits building) on norms (which are elements of the Bruhat-Tits building), along with many properties of these objects and constructions.

There is a general Grothendieckian recipe to pass from G = GL_{n} to more general algebraic groups: replace the natural standard representation by the entire category of all algebraic representations. This has become a very common method in p-adic Hodge theory [30, 17], and as far as filtrations are concerned, it was already implemented for reductive groups over arbitrary fields in the foundational work on Tannakian categories, Saavedra Rivano’s thesis [41]. For norms and lattices, it was completed more recently in Wilson’s thesis [50] for split reductive groups over complete discrete valuation fields. This Tannakian formalism also has many advantages: it has build-in functorialities, it works for arbitrary affine groups over arbitrary base schemes, it provides a conceptual framework for many algebraic constructions, and it gives rise to various interesting representable sheaves.

In chapter 2, we thus actually start with a reductive group G over an arbitrary base scheme S. For a totally ordered commutative group Γ = (Γ, +, ≤), we introduce there our fundamental G-equivariant cartesian diagram of S-schemes

\xymatrix{𝔾^{Γ}(G) \ar[r]^{Fil} \ar[d]_{F} & 𝔽^{Γ}(G) \ar[r]^{t} \ar[d]_{F} & ℂ^{Γ}(G) \ar[d]_{F} \\ 𝕆ℙℙ(G) \ar[r]^{p_{1}} & ℙ(G) \ar[r]^{t} & 𝕆(G)}

where ℙ(G) and 𝕆ℙℙ(G) are respectively the S-schemes of parabolic subgroups P of G and pairs of opposed parabolic subgroups (P, P′) of G, 𝕆(G) is the S-scheme of G-orbits in ℙ(G) or 𝕆ℙℙ(G), 𝔾^{Γ}(G) = \underline{Hom}(𝔻_{S}(Γ), G) where 𝔻_{S}(Γ) is the diagonalized multiplicative group over S with character group Γ, while 𝔽^{Γ}(G) and ℂ^{Γ}(G) are suitable quotients of 𝔾^{Γ}(G). The facet morphisms F are surjective and locally constant in the étale topology on their base, the p_{1} and Fil morphisms are affine smooth surjective with geometrically connected fibers and the type morphisms t are projective smooth surjective with geometrically connected fibers. Since 𝕆(G) is finite étale over S, all of the above schemes are smooth, separated and surjective over S. We also equip ℂ^{Γ}(G) and 𝕆(G) with S-monoid structures, and the facet map F : ℂ^{Γ}(G) → 𝕆(G) is compatible with them. We finally define two related partial orders on the S-monoid ℂ^{Γ}(G), the weak and strong dominance orders.

For Γ = ℤ, 𝔻_{S}(Γ) = 𝔾_{m,S} and 𝔾^{Γ}(G) is the S-scheme of cocharacters of G, whose conjugacy classes are classified by ℂ^{Γ}(G). For Γ = ℝ, 𝔽^{Γ}(G) is a scheme theoretical version of the Tits vectorial building defined by Rousseau in [40] and ℂ^{Γ}(G) is a scheme theoretical version of a closed Weyl chamber. More general Γ’s, for instance the valuation groups of valuation fields of height > 1 may also be useful in connection with recent developements in p-adic geometry.

In chapter 3, we show that 𝔾^{Γ}(G) and 𝔽^{Γ}(G) represent functors respectively related to Γ-graduations and Γ-filtrations on a variety of fiber functors. The main difficulty here is to show that the Γ-filtrations split fpqc-locally on the base scheme. For Γ = ℤ, this was essentially established in the thesis of Saavedra Rivano [41], at least when S is the spectrum of a field. We strictly follow Saavedra’s proof (which he attributes to Deligne), adding a considerable amount of details and some patch when needed. We advise our reader to read both texts side by side, only switching to ours when he feels uncomfortable with the necessary generalizations of Saavedra’s arguments^{1}. Various constructions of chapter 2 have counterparts in this Tannakian framework, which are reviewed in section 3.11. In particular, we show that the first line of our fundamental diagram is functorial in the reductive group G over S. The weak dominance order on ℂ^{Γ}(G) is compatible with this functoriality, but we would like to already emphasize here that the monoid structure is not.

In chapter 4, we study the sections of our schemes over a local ring O. We first equip F^{Γ}(G) = 𝔽^{Γ}(G)(O) with a collection of apartments F^{Γ}(S) indexed by the maximal split subtori S of G, and with the collection of facets F^{−1}(P) indexed by the parabolic subgroups P of G. The key properties of the resulting combinatorial structure are well-known when O is a field and Γ = ℝ, in which case F^{Γ}(G) is the Tits vectorial building, but most of them carry over to this more general situation, thanks to the wonderful last chapter of SGA3 [21]. We describe the behavior of these auxiliary structures under specialization (when O is Henselian) or generization (when O is a valuation ring). When Γ is a subring of ℝ, we also attach to every finite free faithful representation τ of G a partially defined scalar product on F^{Γ}(G) and the corresponding distance and angle functions, and we study their basic properties. When O is a field, a theorem of Borel and Tits [9] implies that these functions are defined everywhere, and one thus retrieves the aforementioned non-canonical distances on the vectorial Tits building F(G) = F^{ℝ}(G).

1For Γ = ℤ, Ziegler recently established the fpqc-splitting of ℤ-filtrations on fiber functors on arbitrary Tannakian categories [51], thereby proving a conjecture which was left open after Saavedra’s thesis. In particular, the ℤ-filtrations we consider have fpqc-splittings even when G is not reductive, but defined over a field. In the reductive case, the final arguments in Ziegler’s proof simplify those of Saavedra’s, but rely more on the Saavedra-Deligne theorem that all fiber functors on Tannakian categories are fpqc-locally isomorphic [18]. According to D. Schäppi, it follows from his own work [42, 43] and Lurie’s note on Tannaka duality that the same result holds for any ⊗-functor Rep^{fp}(G)(S) → QCoh(T) where: S is affine, T is an S-scheme, G is affine flat over S, Rep^{fp}(G)(S) is the ⊗-category of algebraic representations of G on finitely presented O_{S}-modules, and G has the resolution property: any finitely presented algebraic representation of G is covered by another one which is locally free. It then seems likely that Ziegler’s proof could yield a common generalization of his result (Γ = ℤ, G affine over a field) and ours (Γ and S arbitrary, but G reductive) on the existence of fpqc-splittings of Γ-filtrations, using a hefty dose of the stack formalism. We have chosen to stick to the constructive, down-to-earth original proof of Saavedra/Deligne – and to reductive groups as well.

Over a field K and with Γ = ℝ, we next define a notion of affine F(G)-spaces, which interact with the vectorial Tits building F(G) as affine spaces do with their underlying vector space. Strongly influenced by the formalism set up by Rousseau in [40] and Parreau in [36], we introduce various axioms that these spaces may satisfy, leading to the more restricted class of affine F(G)-buildings. Most of the abstract definitions of affine buildings that have already been proposed [29, 37, 40] also add a euclidean metric into the picture, and involve a covering atlas of charts, which are isometries from a given fixed euclidean affine space onto subsets of the building (its apartments) subject to various conditions. Our definition also involves a covering by apartments, but their affine structure is inherited from a globally defined G(K)- equivariant transitive operation (x, F) ↦ x + F of the vectorial building F(G) on the given affine F(G)-space. It is therefore essentially a boundary-based formalism for affine buildings, as opposed to the more usual apartment-based formalism.

Eventhough our affine F(G)-buildings have no fixed metric, they are equipped with a canonical metrizable topology and a canonical vector valued convex distance d, taking values in C(G) = t(F(G)). The choice of a faithful representation τ of G eventually equips them with a convex distance d_{τ} = ∥d∥_{τ} in the usual sense, for which they often become CAT (0)-metric spaces as defined in [8]. But the finer and canonical vectorial distance d really is a key feature of our buildings: it retrieves and generalizes many classical invariants in various set-up (such as types of filtrations and relative positions of lattices or quadrics), and its formal properties imply most, if not all, of the known inequalities among these invariants.

Of course F(G) is itself an affine F(G)-building, with a distinguished point. When K is equipped with a non-trivial, non-archimedean absolute value, we show in chapter 6 that the (extended) affine building B^{e}(G) constructed by Bruhat and Tits [9, 10] is canonically equipped with a structure of affine F(G)-building in our sense. This is our precise formalization of the “combinatorial” assertion that the visual boundary of the Bruhat-Tits building is a geometric realization of the combinatorial Tits building. This being done, we may fix a base point ∘_{G} in B^{e}(G) and try to describe the whole building as a quotient of F(G) using the surjective map F(G) ∋ F ↦ ∘_{G} + F ∈ B^{e}(G). We do this in the last section, assuming that our base point ∘_{G} is hyperspecial, i.e. corresponds to a reductive group G over the valuation ring O of K, which we also assume to be Henselian. Note that the existence of an hyperspecial point amounts to an assumption on G_{K} [47, 2.4].

More precisely, we first define a space of K-norms on the fiber functor

ω_{G}^{∘} : Rep^{∘}(G)(O) → Vect(K)

where Rep^{∘}(G)(O) is the category of algebraic representations of G on finite free Omodules. This space is equipped with a G(K)-action, an explicit G(K)-equivariant operation of F(G_{K}) and a base point α_{G} fixed by G(O). We show that the map ∘_{G} + F ↦ α_{G} + F is well-defined, injective, G(K)-equivariant and compatible with the operations of F(G_{K}). It thus defines an isomorphism α of affine F(G_{K})- buildings from B^{e}(G_{K}) to a set B(ω_{G}^{∘}, K) = α_{G} + F(G_{K}) of K-norms on ω_{G}^{∘}.

This Tannakian description of the extended Bruhat-Tits building immediately implies that the assignment G ↦ B^{e}(G_{K}) is functorial in the reductive group G over O. Such a functoriality was already established by Landvogt [32], with fewer assumptions on G_{K} but more assumptions on K. It also suggests a possible definition of Bruhat-Tits buildings for reductive groups over valuation rings of height greater than 1, as well as a similar Tannakian description of symmetric spaces (in the archimedean case, see 5.8). It is related to previous constructions as follows.

Our canonical isomorphism α : B^{e}(G_{K}) → B(ω_{G}^{∘}, K) assigns to a point x in B^{e}(G_{K}) and to any algebraic representation τ of G on a flat O-module V (τ) a K-norm α(x)(τ) on V_{K}(τ) = V (τ) ⊗ K. For the adjoint representation τ_{ad} of G on 𝔤 = Lie(G), the adjoint-regular and regular representations ρ_{adj} and ρ_{reg} of G on A(G) = Γ(G, O_{G}), we obtain respectively: a K-norm α_{ad}(x) on 𝔤_{K} = Lie(G_{K}) whose closed balls give the Moy-Prasad filtration of x on 𝔤_{K} [35], the K-norm α_{adj}(x) in G_{K}^{an} constructed in [38], and an embedding x ↦ α_{reg}(x) of the extended Bruhat-Tits building in the analytic Berkovich space G_{K}^{an} attached to G_{K}. Our isomorphism α also induces an explicit G(O)-equivariant identification between the “tangent space” of B^{e}(G_{K}) at ∘_{G} and the vectorial Tits building F(G_{k}) of the special fiber G_{k} of G over the residue field k of O, as expected from [10, 4.6.35-45].

In Wilson’s Tannakian formalism for Bruhat-Tits buildings [50], alcoves and their parahorics played the leading role. His Moy-Prasad filtrations are the lattice chains of closed balls of our norms. We owe to his work the essential shape of our formalism, if not the very idea that such a formalism was indeed possible: we were first naively looking for a base-point free description of the Bruhat-Tits buildings. His point of view is more adapted to the study of the simplicial structure of these buildings, but only covers split groups over discrete valuation rings. Our approach covers unramified groups over fields equipped with a Henselian absolute value. It lacks an intrinsic description of (1) the equivalence relation on F(G_{K}) defined by F ∼ F′ ⟺ ∘_{G} + F = ∘_{G} + F′, and of (2) the image B(ω_{G}^{∘}, K) = α_{G} + F(G_{K}) of α in the larger space of all K-norms on the fiber functor ω_{G}^{∘}.

Finally, we would like to mention that some of our results should extend to more general fiber functors, using the Schäppi/Lurie generalization of Deligne’s theorem as mentioned in the previous footnote.

This work grew out of a question by J-F. Dat and many discussions with D. Mauger on buildings and cocharacters. I am very grateful to G. Rousseau and A. Parreau, who always had answers to my numerous questions. Apart from the emphasis on the boundary, most of the definitions and results of chapter 5 are either taken from his survey [40] or from her preprint [36]. P. Deligne kindly provided the patch at the very end of the proof of the splitting theorem, dealing with groups of type G2 in characteristic 2, and M. Hils the proof of lemma 134.

CHAPTER 2 The group theoretical formalism

For a reductive group scheme G over an arbitrary base scheme S, we will define and study a cartesian diagram of smooth and separated schemes over S,

\xymatrix{𝔾^{Γ}(G) \ar@{->>}[r]^{Fil} \ar[d]_{F} & 𝔽^{Γ}(G) \ar@{->>}[r]^{t} \ar[d]_{F} & ℂ^{Γ}(G) \ar[d]_{F} \\ 𝕆ℙℙ(G) \ar@{->>}[r]^{p_{1}} & ℙ(G) \ar@{->>}[r]^{t} & 𝕆(G)}

Our main background reference for this chapter is SGA3 [20, 1, 21].

2.1. Γ-graduations on smooth affine groups

Theorem 1. Let H and G be group schemes over a base scheme S, with H of multiplicative type and quasi-isotrivial, G smooth and affine. Then the functor

\underline{Hom}_{S−Group}(H, G) : (Sch/S)^{∘} → Set, T ↦ Hom_{T −Group}(H_{T} , G_{T})

is representable by a smooth and separated scheme over S.

Remark 2. When H is of finite type, it is quasi-isotrivial by [1, X 4.5]. The theorem is then due to Grothendieck, see [1, XI 4.2]. The proof given there relies on the density theorem of [1, IX 4.7], definitely a special feature of finite type multiplicative groups. When H is trivial, we may still reduce the proof of the above theorem to the finite type case, as explained in remark 12 below. For the general case, we have to find another road through SGA3, passing through [1, X 5.6] which has no finite type assumption on H but requires H and G to be of multiplicative type and quasi-isotrivial:

Proposition 3. Let H and G be group schemes of multiplicative type over S, with H quasi-isotrivial and G of finite type. Then \underline{Hom}_{S−Group}(H, G) is representable by a quasi-isotrivial twisted constant group scheme X over S.

Proof. This is [1, X 5.6], since G is also quasi-isotrivial by [1, X 4.5]. □

Lemma 4. Let X be a quasi-isotrivial twisted constant scheme over S. Then X is separated and étale over S, satisfies the valuative criterion of properness, and:

(1) If S is irreducible and geometrically unibranch with generic point η, then

X = _{∐λ∈X_{η}}X(λ) with X(λ) = \overline{\{λ\}} open and closed in X,

each X(λ) is a connected finite étale cover of S and Γ(X/S) = Γ(X_{η}/η).

(2) If S is local henselian with closed point s, then

X = ∐_{x∈X_{s}}X(x) with X(x) = Spec O_{X,x} open and closed in X,

each X(x) is a connected finite étale cover of S, and Γ(X/S) = Γ(X_{s}/s).

Proof. The morphism X → S is separated by [26, 2.7.1] and étale by [28, 17.7.3]. Since valuation rings are normal integral domains, thus irreducible and geometrically unibranch, it remains to establish (1) and (2).

Suppose first that S is irreducible and geometrically unibranch with generic point η. Then by [28, 18.10.7] applied to X → S,

X = _{∐λ∈X_{η}}X(λ) with X(λ) = \overline{\{λ\}} open and closed in X,

thus X(λ) is already étale over S. Fix an étale covering {S_{i} → S} trivializing X, so that X ×_{S} S_{i} = Q_{i,S_{i}} for some set Q_{i}. Using [28, 18.10.7] again, we may assume that each S_{i} is connected, in which case we obtain decompositions

Q_{i} = _{∐λ∈X_{η}}Q_{i}(λ) with X(λ) ×_{S} S_{i} = Q_{i}(λ)_{S_{i}}.

Since the generic fiber λ → η of X(λ) → S is finite of degree n(λ) = [k(λ) : k(η)], each Q_{i}(λ) is a finite subset of Q_{i} of order n(λ), therefore X(λ) ×_{S} S_{i} is finite over S_{i} and X(λ) is finite over S by [26, 2.7.1]. Being finite and étale over the connected S, X(λ) is a finite étale cover of S. Being irreducible, it is also connected. By [28, 17.4.9], the map which sends a section g of X → S to its image g(S) identifies Γ(X/S) with the set of connected components X(λ) of X for which X(λ) → S is an isomorphism, i.e. such that n(λ) = 1. Therefore Γ(X/S) = Γ(X_{η}/η).

Suppose next that S is local henselian with closed point s. Since X → S is quasi-finite at every x ∈ X_{s} by [28, 17.6.1], it follows from [28, 18.5.11.c] that

X ⊃ X′ = _{∐x∈X_{s}}X(x) with X(x) = Spec O_{X,x} open and closed in X,

and X(x) is finite and étale over S. By assumption, there is a surjective étale morphism S_{0} → S trivializing X, so that X ×_{S} S_{0} = Q_{S_{0}} for some set Q. Using [28, 18.5.11.c] again, we may assume that S_{0} is a local scheme, finite and étale over S, say with closed point s_{0} lying above s. Since X′ ×_{S} S_{0} is open in X ×_{S} S_{0} and contains its special fiber X_{s_{0}}, we have X′ ×_{S} S_{0} = X ×_{S} S_{0}, thus actually X′ = X by [26, 2.7.1]. Finally Γ(X/S) = Γ(X_{s}/s) by [28, 18.5.12]. □

Lemma 5. Let f : H → G be a morphism of group schemes over S, with H of multiplicative type and G separated of finite presentation. Then there is a unique closed multiplicative subgroup Q of G such that f factors through a faithfully flat morphism f′ : H → Q. Moreover f′ is also uniquely determined by f.

Proof. Everything being local for the fpqc topology, we may assume that S is affine and H = 𝔻_{S}(M) for some abstract commutative group M. Then M = \underrightarrow{lim}M′ where M′ runs through the filtered set F(M) of finitely generated subgroups of M, thus also 𝔻_{S}(M) = \underleftarrow{lim}𝔻_{S}(M′). Since 𝔻_{S}(M′) is affine for all M′ and G → S is locally of finite presentation, it follows from [27, 8.13.1] that f factors through f_{1} : 𝔻_{S}(M′) → G for some M′ ∈ F(M). Applying [1, IX 6.8] to f_{1} yields a closed multiplicative subgroup Q of G such that f_{1} factors through a faithfully flat (and affine) morphism f′_{1} : 𝔻_{S}(M′) → Q, whose composite with the faithfully flat (and affine) morphism 𝔻_{S}(M) → 𝔻_{S}(M′) is the desired factorization. Since Q is then also the image of f in the category of fpqc sheaves on Sch/S, it is already unique as a subsheaf of G. Since Q → G is a monomorphism, also f′ is unique. □

Definition 6. We call Q the image of f and denote it by Q = im(f).

Lemma 7. Let f : H → G be a morphism of group schemes over S, with H of multiplicative type and G smooth and affine. Then the centralizer of f is equal to the centralizer of its image, and is representable by a closed smooth subgroup of G.

Proof. Let f = ι ∘ f′ be the factorization of the previous lemma. Since f′ is faithfully flat (and quasi-compact, being a morphism between affine S-schemes, therefore even affine), it is an epimorphism in the category of schemes. It then follows from the definitions in [20, VIB §6] that the centralizers of f, ι and im(f) are equal. By [1, XI 5.3], the centralizer of ι is a closed smooth subgroup of G. □

Lemma 8. Let f : H → Q be a morphism of group schemes of multiplicative type over S, with Q of finite type. Define U = {s ∈ S : f_{s} is faithfully flat}. Then U is open and closed in S and f_{U} : H_{U} → Q_{U} is faithfully flat.

Proof. Let I be the image of f. Then U is the set of points s ∈ S where I_{s} = Q_{s}. Now apply [1, IX 2.9] to I ↪ Q. □

We may now prove theorem 1. Define presheaves A, B, C on Sch/S by

C(S′) = {multiplicative subgroups Q of G_{S′}} ,
B(S′) = {(Q, f′) : Q ∈ C(S′) and f : H_{S′} → Q is a morphism} ,
A(S′) = {(Q, f′) ∈ B(S′) with f′ faithfully flat} .

Then C is representable, smooth and separated by [1, XI 4.1], B → C is relatively representable by étale and separated morphisms by proposition 3 and lemma 4, A → B is relatively representable by open and closed immersions by lemma 8 and finally A is isomorphic to \underline{Hom}_{S−Group}(H, G) by lemma 5, which is therefore indeed representable by a smooth and separated scheme over S.

Definition 9. For an abstract commutative group Γ = (Γ, +) and a smooth and affine group scheme G over S, we set 𝔾^{Γ}(G) = \underline{Hom}_{S−Group}(𝔻_{S}(Γ), G). Thus

𝔾^{Γ}(G) : (Sch/S)^{∘} → Set

is representable by a smooth and separated scheme over S.

Proposition 10. Let f : 𝔻_{S}(Γ) → G be a morphism of group schemes over S, with G separated and of finite presentation. Then for each s in S,

Γ(s) = {γ ∈ Γ : γ is trivial on ker(f_{s})}

belongs to the set F(Γ) of finitely generated subgroups of Γ. For each Λ ∈ F(Γ),

S(Λ) = {s ∈ S : Γ(s) = Λ}

is open and closed in S, and finally

ker(f)_{S(Λ)} = 𝔻_{S(Λ)}(Γ/Λ) and im(f)_{S(Λ)} = 𝔻_{S(Λ)}(Λ).

Proof. We may assume that S is affine and G is of multiplicative type (using lemma 5 for the latter). Since 𝔻_{S}(Γ) = \underleftarrow{lim}𝔻_{S}(Λ), it follows again from [27, 8.13.1] that there is some Λ in F(Γ) such that f factors through g : 𝔻_{S}(Λ) → G, i.e. 𝔻_{S}(Γ/Λ) ⊂ ker(f). But then Γ(s) ⊂ Λ for every s ∈ S, which proves the first claim. Applying now [1, IX 2.11 (i)] to g gives a finite partition of S into open and closed subsets S_{i}, together with a collection of distinct subgroups Λ_{i} of Λ such that ker(g)_{S_{i}} = 𝔻_{S_{i}}(Λ/Λ_{i}) and im(g)_{S_{i}} ≃ 𝔻_{S_{i}}(Λ_{i}). But then ker(f)_{S_{i}} = 𝔻_{S_{i}}(Γ/Λ_{i}), im(f)_{S_{i}} ≃ 𝔻_{S_{i}}(Λ_{i}) and S_{i} = S(Λ_{i}), which proves the remaining claims. □

Corollary 11. If Γ is torsion free, im(f) is a locally trivial subtorus of G.

Remark 12. The above proposition suggests another proof of theorem 1 when H = 𝔻_{S}(Γ). It shows indeed that the Zariski sheaf 𝔾^{Γ}(G) is the disjoint union of relatively open and closed subsheaves 𝔾^{Γ}(G)(Λ), indexed by Λ ∈ F(Γ). Moreover, 𝔾^{Γ}(G)(Λ) is isomorphic to the subsheaf 𝔾^{Λ}(G)(Λ) of 𝔾^{Λ}(G), which is representable by a smooth and separated scheme over S by [1, XI 4.2].

2.2. Γ-filtrations on reductive groups

Let S be a scheme, G a reductive group over S, 𝔤 = Lie(G) its Lie algebra. Let Γ = (Γ, +, ≤) be a non-trivial totally ordered commutative group.

2.2.1. Recall from [21, XXVI 3.5 ] that the sheaf

ℙ(G) : (Sch/S)^{∘} → Set

whose sections over an S-scheme T are given by

ℙ(G)(T) = {parabolic subgroups P of G_{T} }

is representable, smooth and projective over S, with Stein factorization

ℙ(G) ⟶^{t} 𝕆(G) → S

where 𝕆(G) is the S-scheme of open and closed subschemes of the Dynkin S-scheme 𝔻𝕐ℕ(G) of the reductive group G/S, see [21, XXIV 3.3]. Both 𝔻𝕐ℕ(G) and 𝕆(G) are twisted constant finite schemes over S, thus finite étale over S by [26, 2.7.1.xv] and [28, 17.7.3], and 𝕆(G) is actually a finite étale cover of S. The morphism t is smooth, projective, with non-empty geometrically connected fibers; it classifies the parabolic subgroups of G in the following sense: two parabolic subgroups P_{1} and P_{2} of G are conjugated locally in the fpqc topology on S if and only if t(P_{1}) = t(P_{2}).

2.2.2. For a parabolic subgroup P of G with unipotent radical U, we denote by \overline{R}(P) the radical of P/U [21, XXII 4.3.6]. For the universal parabolic subgroup P_{u} of G_{ℙ(G)}, we obtain a ℙ(G)-torus R_{ℙ(G)} = \overline{R}(P_{u}). We claim that it descends canonically to an 𝕆(G)-torus R_{𝕆(G)} over 𝕆(G). Since t is faithfully flat and quasicompact, it is a morphism of effective descent for affine group schemes by [2, VIII 2.1], thus also for tori by definition [1, IX 1.3]. Our claim now follows from:

Lemma 13. There exists a canonical descent datum on R_{ℙ(G)} with respect to t.

Proof. We have to show that for any T → S and any pair of parabolic subgroups P_{1} and P_{2} of G_{T} such that t(P_{1}) = t(P_{2}), there exists a canonical isomorphism \overline{R}(P_{1}) ≃ \overline{R}(P_{2}). Let M_{i} = P_{i}/U_{i} be the maximal reductive quotient of P_{i}, so that R_{i} = \overline{R}(P_{i}) is the radical of M_{i}. We may assume that T = S and, by a descent argument, that P_{2} = Int(g)(P_{1}) for some g ∈ G(S). Then Int(g) induces isomorphisms P_{1} → P_{2}, M_{1} → M_{2} and R_{1} → R_{2}. Since g is well-defined up to right multiplication by an element of P_{1}(S) thanks to [21, XXVI 1.2], M_{1} → M_{2} is well-defined up to an inner automorphism of M_{1} and R_{1} → R_{2} does not depend upon any choice: this is our canonical isomorphism. □

2.2.3. By [21, XXVI 4.3.4 and 4.3.5], the formula

𝕆ℙℙ(G)(T) = {(P_{1}, P_{2}) pair of opposed parabolic subgroups of G_{T} }

defines an open subscheme 𝕆ℙℙ(G) of ℙ(G)^{2} and the two projections

p_{1}, p_{2} : 𝕆ℙℙ(G) → ℙ(G)

are isomorphic U_{u}-torsors, thus affine smooth surjective morphisms with geometrically connected fibers. Here U_{u} is the unipotent radical of the universal parabolic subgroup P_{u} of G_{ℙ(G)}, it acts by conjugation on the fibers, and the isomorphism is the involution ι(P_{1}, P_{2}) = (P_{2}, P_{1}) of the S-scheme 𝕆ℙℙ(G). We denote by (P_{u}^{1}, P_{u}^{2}) = (p_{1}^{∗}P_{u}, p_{2}^{∗}P_{u}) the universal pair of opposed parabolic subgroups of G_{𝕆ℙℙ(G)}, by U_{u}^{i} = p_{i}^{∗}U_{u} the unipotent radical of P_{u}^{i}, and by R_{𝕆ℙℙ(G)} the radical of the corresponding universal Levi subgroup L_{u} = P_{u}^{1} ∩ P_{u}^{2} of G_{𝕆ℙℙ(G)}. Thus

L_{u} ≃ P_{u}^{i}/U_{u}^{i} ≃ p_{i}^{∗}(P_{u}/U_{u}) and R_{𝕆ℙℙ(G)} ≃ p_{i}^{∗}R_{ℙ(G)}.

We also denote by ι the opposition involution on 𝕆(G), see [21, XXVI 4.3.1]. Thus

t ∘ p_{2} = t ∘ p_{1} ∘ ι = ι ∘ t ∘ p_{1}.

2.2.4. The S-scheme 𝔾^{Γ}(R_{𝕆ℙℙ(G)}) represents the functor mapping T → S to the set of triples (P_{1}, P_{2}, f) where (P_{1}, P_{2}) is a pair of opposed parabolic subgroups of G_{T} with Levi subgroup L = P_{1} ∩ P_{2}, and f : 𝔻_{T} (Γ) → L is a central morphism. The next proposition uses the total ordering on Γ = (Γ, +, ≤) to define a section

𝔾^{Γ}(G) ↪ 𝔾^{Γ}(R_{𝕆ℙℙ(G)}), f ↦ (P_{f}, P_{ιf}, f)

of the obvious forgetful morphism of S-schemes

𝔾^{Γ}(R_{𝕆ℙℙ(G)}) → 𝔾^{Γ}(G), (P_{1}, P_{2}, f) ↦ f.

Proposition 14. Let f : 𝔻_{S}(Γ) → G be a morphism and write 𝔤 = ⊕_{γ}𝔤_{γ} for the corresponding weight decomposition of ad ∘ f : 𝔻_{S}(Γ) → GL_{S}(𝔤). There exists a unique parabolic subgroup P_{f} of G containing the centralizer L_{f} of f such that

Lie(P_{f}) = ⊕_{γ≥0}𝔤_{γ}.

Moreover L_{f} is a Levi subgroup of P_{f}, thus P_{f} = U_{f} ⋊L_{f} where U_{f} is the unipotent radical of P_{f}. For ιf = f^{−1}, P_{ιf} is opposed to P_{f}, L_{f} = P_{f} ∩ P_{ιf} and

\begin{matrix}Lie(P_{f}) = ⊕_{γ≥0}𝔤_{γ} \\ Lie(U_{f}) = ⊕_{γ>0}𝔤_{γ} \end{matrix} \begin{matrix}Lie(P_{ιf}) = ⊕_{γ≤0}𝔤_{γ} \\ Lie(U_{ιf}) = ⊕_{γ<0}𝔤_{γ} \end{matrix} and Lie(L_{f}) = 𝔤_{0}.

Proof. Let Q be the image of f. Then L_{f} is the centralizer of Q by lemma 7 and Q is a locally trivial subtorus of G by proposition 10 (since Γ is torsion free). We may assume that Q is trivial, i.e. Q ≃ 𝔻_{S}(Λ) for some finitely generated subgroup Λ of Γ. The proposition then follows from [21, XXVI 6.1]. □

Proposition 15. The morphism 𝔾^{Γ}(G) → 𝔾^{Γ}(R_{𝕆ℙℙ(G)}) is an open and closed immersion, and 𝔾^{Γ}(G) → 𝕆ℙℙ(G) is a quasi-isotrivial twisted constant morphism.

Proof. The second assertion follows from the first one by Grothendieck’s proposition 3. Given a section (P_{1}, P_{2}, f) of 𝔾^{Γ}(R_{𝕆ℙℙ(G)}) over some S-scheme T, we have to show that the condition (P_{1}, P_{2}) = (P_{f}, P_{ιf}) is representable by an open and closed subscheme of T. It is plainly representable by the inverse image of the diagonal of 𝕆ℙℙ(G) under the S-morphism T → 𝕆ℙℙ(G)^{2} defined by our two pairs (P_{1}, P_{2}) and (P_{f}, P_{ιf}), which is a closed subscheme of T since 𝕆ℙℙ(G) is separated over S. On the other hand, since the Levi subgroup L = P_{1} ∩ P_{2} of G is contained in L_{f} = P_{f} ∩ P_{ιf}, our condition (P_{1}, P_{2}) = (P_{f}, P_{ιf}) is equivalent to (𝔭_{1}, 𝔭_{2}) = (⊕_{γ≥0}𝔤_{γ}, ⊕_{γ≤0}𝔤_{γ}) where 𝔭_{i} = Lie(P_{i}): this last claim is local in the fpqc topology on T, we may thus assume that L contains a maximal torus of G and then apply [21, XXII 5.3.5]. Now write 𝔲_{i} = ⊕𝔲_{i,γ} for the weight decomposition of the Lie algebra of the unipotent radical of P_{i}, and set 𝔲_{i}^{±} = ⊕_{±γ≥0}𝔲_{i,γ}. Then our Lie algebra condition is equivalent to the vanishing of the locally free sheaf 𝔲_{1}^{−} ⊕ 𝔲_{2}^{+}, and it is therefore representable by the open complement of its support. □

Remark 16. This gives yet another proof of theorem 1 (using Grothendieck’s proposition 3), when G is reductive and Γ torsion free (using [33] to construct a total order ≤ on Γ).

2.2.5. The cartesian diagram (in the fibered category of tori over schemes):

\xymatrix{R_{𝕆ℙℙ(G)} \ar@{->>}[r]^{p_{1}} \ar[d] & R_{ℙ(G)} \ar@{->>}[r]^{t} \ar[d] & R_{𝕆(G)} \ar[d] \\ 𝕆ℙℙ(G) \ar@{->>}[r]^{p_{1}} & ℙ(G) \ar@{->>}[r]^{t} & 𝕆(G)}

induces an analogous cartesian diagram (in the fibered category of quasi-isotrivial twisted constant group schemes over schemes):

\xymatrix{𝔾^{Γ}(R_{𝕆ℙℙ(G)}) \ar@{->>}[r]^{Fil} \ar[d]_{F} & 𝔾^{Γ}(R_{ℙ(G)}) \ar@{->>}[r]^{t} \ar[d]_{F} & 𝔾^{Γ}(R_{𝕆(G)}) \ar[d]_{F} \\ 𝕆ℙℙ(G) \ar@{->>}[r]^{p_{1}} & ℙ(G) \ar@{->>}[r]^{t} & 𝕆(G)}

which is given on T -valued points by the following formulas:

[figure: (P_{1}, P_{2}, f); Fil; F; (P_{1}, \overline{f}); t; F; (t(P_{1}), \overline{f}); F; (P_{1}, P_{2}); p_{1}; P_{1}; t; t(P_{1})]

Here \overline{f} : 𝔻_{T} (Γ) → \overline{R}(P_{1}) is defined by the diagram

\xymatrix{𝔻_{T} (Γ) \ar[r]^{f} \ar[dr]_{\overline{f}} & R(L) \ar@{^{(}->}[r] \ar[d]^{≃} & L \ar[d]^{≃} \\  & \overline{R}(P_{1}) \ar@{^{(}->}[r] & P_{1}/U_{1}}

where L = P_{1} ∩ P_{2} and U_{1} is the unipotent radical of P_{1}.

Lemma 17. The open and closed subscheme 𝔾^{Γ}(G) of 𝔾^{Γ}(R_{𝕆ℙℙ(G)}) is saturated with respect to 𝔾^{Γ}(R_{𝕆ℙℙ(G)}) → 𝔾^{Γ}(R_{ℙ(G)}) and 𝔾^{Γ}(R_{𝕆ℙℙ(G)}) → 𝔾^{Γ}(R_{𝕆(G)}).

Proof. It is sufficient to establish that it is saturated with respect to the second map. We have to show: for an S-scheme T, a morphism f : 𝔻_{T} (Γ) → G_{T}, a pair of opposed parabolic subgroups (P_{1}, P_{2}) of G_{T} with Levi L = P_{1} ∩ P_{2}, and a central morphism h : 𝔻_{T} (Γ) → L, if (P_{f}, P_{ιf}, f) and (P_{1}, P_{2}, h) have the same image in 𝔾^{Γ}(R_{𝕆(G)})(T), then (P_{1}, P_{2}) = (P_{h}, P_{ιh}). This is local in the fpqc topology on T. Since t(P_{f}) = t(P_{1}) by assumption, we may assume that there is a g ∈ G(T) such that Int(g)(P_{f}, P_{ιf}) = (P_{1}, P_{2}) by [21, 4.3.4 iii]. But then also Int(g) ∘ f = h (by assumption), thus (P_{1}, P_{2}) = Int(g)(P_{f}, P_{ιf}) = (P_{h}, P_{ιh}). □

2.2.6. By an elementary case of fpqc descent (along p_{1} and t), we thus obtain a cartesian diagram of open and closed embeddings of smooth S-schemes,

\xymatrix{𝔾^{Γ}(G) \ar@{->>}[r]^{Fil} \ar@{^{(}->}[d] & 𝔽^{Γ}(G) \ar@{->>}[r]^{t} \ar@{^{(}->}[d] & ℂ^{Γ}(G) \ar@{^{(}->}[d] \\ 𝔾^{Γ}(R_{𝕆ℙℙ(G)}) \ar@{->>}[r]^{Fil} & 𝔾^{Γ}(R_{ℙ(G)}) \ar@{->>}[r]^{t} & 𝔾^{Γ}(R_{𝕆(G)})}

which in turns gives our fundamental cartesian diagram of smooth S-schemes

\xymatrix{𝔾^{Γ}(G) \ar@{->>}[r]^{Fil} \ar[d]_{F} & 𝔽^{Γ}(G) \ar@{->>}[r]^{t} \ar[d]_{F} & ℂ^{Γ}(G) \ar[d]_{F} \\ 𝕆ℙℙ(G) \ar@{->>}[r]^{p_{1}} & ℙ(G) \ar@{->>}[r]^{t} & 𝕆(G)}

The S-group scheme G acts on both diagrams by conjugation and their last column are the quotients of the other two columns by the action of G in the category of fpqc sheaves on S. The morphism Fil : 𝔾^{Γ}(G) → 𝔽^{Γ}(G) is a U_{𝔽^{Γ}(G)}-torsor, where U_{𝔽^{Γ}(G)} is the unipotent radical of the pull-back P_{𝔽^{Γ}(G)} of the universal parabolic subgroup P_{u} of G_{ℙ(G)}. In particular, it is affine smooth surjective with geometrically connected fibers. The morphism t : 𝔽^{Γ}(G) → ℂ^{Γ}(G) is projective smooth surjective with geometrically connected fibers. The three facet morphisms F are quasi-isotrivial twisted constant (i.e. locally constant in the étale topology on their base), in particular they are separated and étale by lemma 4. We will see in due time that they are also surjective (4.1.11). Since 𝕆(G) is a finite étale cover of S, everyone is smooth, surjective and separated over S. We denote by

0 : S → 𝔾^{Γ}(G), 0 : S → 𝔽^{Γ}(G) and 0 : S → ℂ^{Γ}(G)

the element of 𝔾^{Γ}(G)(S) corresponding to the trivial morphism 𝔻_{S}(Γ) → G or its images in 𝔽^{Γ}(G)(S) or ℂ^{Γ}(G)(S). They respectively map to (G, G) ∈ 𝕆ℙℙ(G)(S), G ∈ ℙ(G) and 𝔻𝕐ℕ(G) ∈ 𝕆(G). Being sections of separated S-schemes, these 0-sections are closed immersions and the last one is also open.

If S is irreducible and geometrically unibranch or local henselian, then so are the connected components of 𝕆(G) by [28, 18.10.1 and 18.5.10]. Over each of them, the facet map F : ℂ^{Γ}(G) → 𝕆(G) is then merely an infinite disjoint union of connected finite étale covers, see lemma 4. The same decomposition then also holds for its pull-backs over ℙ(G) or 𝕆ℙℙ(G).

2.2.7. For an S-scheme T and morphisms x, y : 𝔻_{T} (Γ) → G_{T}, we have
Fil(x) = Fil(y) in 𝔽^{Γ}(G)(T) ⟺ ∃p ∈ P_{x}(T) : Int(p)(x) = y
⟺ ∃u ∈ U_{x}(T) : Int(u)(x) = y

and then such a u is unique. This equivalence relation is known as the Parequivalence and denoted by x ∼_{Par} y. If T is an (absolutely) affine scheme, then

𝔽^{Γ}(G)(T) = 𝔾^{Γ}(G)(T)/ ∼_{Par}

by [21, XXVI 2.2]. On the other hand,

t ∘ Fil(x) = t ∘ Fil(y) in ℂ^{Γ}(G)(T)
⟺ fpqc locally on T, ∃g ∈ G(T) : Int(g)(x) = y.

If T is semi-local, then by [21, XXVI 5.2],

t ∘ Fil(x) = t ∘ Fil(y) in ℂ^{Γ}(G)(T)
⟺ ∃g ∈ G(T) : Int(g)(x) = y.

2.2.8. For an S-scheme T and F in 𝔽^{Γ}(G)(T), we denote by (P_{F}, \overline{F}) the image of F in 𝔾^{Γ}(R_{ℙ(G)})(T). Thus P_{F} = F (F) is a parabolic subgroup of G_{T}, equal to the stabilizer of F in G_{T} by [21, XXVI 1.2] and \overline{F} : 𝔻_{T} (Γ) → \overline{R}(P_{F}) is a morphism of tori over T. We write U_{F} = R^{u}P_{F} for the unipotent radical of P_{F}, so that \overline{R}(P_{F}) is the radical of P_{F}/U_{F}. If L is a Levi subgroup of P_{F} = U_{F} ⋊ L and f : 𝔻_{T} (Γ) → L is the corresponding central morphism lifting \overline{F}, then

F = Fil(f) and L_{f} = L.

The inversion f ↦ f^{−1} yields compatible involutions on 𝔾^{Γ}(G) and ℂ^{Γ}(G), which we shall both denote by ι. By proposition 14, they are also compatible with the eponymous involutions on 𝕆ℙℙ(G) and 𝕆(G):

F ∘ ι = ι ∘ F on 𝔾^{Γ}(G) or ℂ^{Γ}(G).

2.2.9. Functoriality. The formation of our fundamental diagram

\xymatrix{𝔾^{Γ}(G) \ar@{->>}[r]^{Fil} \ar[d]_{F} & 𝔽^{Γ}(G) \ar@{->>}[r]^{t} \ar[d]_{F} & ℂ^{Γ}(G) \ar[d]_{F} \\ 𝕆ℙℙ(G) \ar@{->>}[r]^{p_{1}} & ℙ(G) \ar@{->>}[r]^{t} & 𝕆(G)}

is plainly compatible with base change on S. We will see later on (corollary 64) that the first line is covariantly functorial in Γ and G. This is obvious for 𝔾^{Γ}(G) and easy for ℂ^{Γ}(G) = G\𝔾^{Γ}(G), but not so for 𝔽^{Γ}(G): the Γ-functoriality of 𝔾^{Γ}(G) is not compatible with the facet maps, and the second line of our diagram is simply not functorial in G. To showcase the first (bad) behavior, note that we will eventually have an action of the set End(Γ, +, ≤) of non-decreasing homomorphisms of Γ by morphisms of S-schemes on the first line, simply denoted by (λ, x) ↦ λ · x. Then x ↦ 0 · x is nothing but the structural morphism of the S-scheme 𝔾^{Γ}(G), 𝔽^{Γ}(G) or ℂ^{Γ}(G), followed by the corresponding 0-section. Thus F (0 · x) = 𝔻𝕐ℕ(G) in 𝕆(G) for any x ∈ ℂ^{Γ}(G). However, for a monomorphism γ : (Γ_{1}, +, ≤) ↪ (Γ_{2}, +, ≤), the induced morphisms γ in the commutative diagram

\xymatrix{𝔾^{Γ_{1}}(G) \ar@{->>}[r]^{Fil} \ar@{^{(}->}[d]_{γ} & 𝔽^{Γ_{1}}(G) \ar@{->>}[r]^{t} \ar@{^{(}->}[d]_{γ} & ℂ^{Γ_{1}}(G) \ar@{^{(}->}[d]_{γ} \\ 𝔾^{Γ_{2}}(G) \ar@{->>}[r]^{Fil} & 𝔽^{Γ_{2}}(G) \ar@{->>}[r]^{t} & ℂ^{Γ_{2}}(G)}

are open and closed immersions which commute with the facet maps: this follows from proposition 10 and 14, given the construction of our fundamental diagram.

2.2.10. ℂ^{Γ}(G) is a commutative monoid. There is natural structure of commutative monoid on the S-scheme 𝕆(G), given by the intersection morphism

∩ : 𝕆(G) ×_{S} 𝕆(G) → 𝕆(G) (a, b) ↦ a ∩ b

Let 𝕆′(G) be the open and closed subscheme of 𝕆(G) ×_{S} 𝕆(G) on which a ∩ b = a, i.e. a ⊂ b. Let p_{1} and p_{2} : 𝕆′(G) → 𝕆(G) be the two projections. We claim:

Lemma 18. There exists a canonical morphism p_{2}^{∗}R_{𝕆(G)} → p_{1}^{∗}R_{𝕆(G)}.

Proof. Let ℙ′(G) be the inverse image of 𝕆′(G) in ℙ(G) ×_{S} ℙ(G), and denote by q_{1} and q_{2} : ℙ′(G) → ℙ(G) the two projections. Then q_{i}^{∗}(R_{ℙ(G)}) = (p_{i}^{∗}R_{𝕆(G)})_{ℙ′(G)} for i ∈ {1, 2}. We have to show that there is a canonical morphism

q_{2}^{∗}R_{ℙ(G)} → q_{1}^{∗}R_{ℙ(G)}

compatible with the descent data on both sides. This boils down to: for any S′ → S and (P_{1}, P_{2}) ∈ ℙ′(G)(S′), there exists a canonical morphism \overline{R}(P_{2}) → \overline{R}(P_{1}). We may assume that S′ = S. Since t(P_{1}) ⊂ t(P_{2}), there exists by [21, XXVI 3.8] a unique parabolic subgroup P′_{2} of G, containing P_{1}, such that t(P_{2}) = t(P′_{2}). Using the canonical isomorphism \overline{R}(P′_{2}) ≃ \overline{R}(P_{2}), we may thus assume that P′_{2} = P_{2}, i.e. P_{1} ⊂ P_{2}. Let U_{i} be the unipotent radical of P_{i}, so that U_{2} ⊂ U_{1} is a normal subgroup of P_{1}. Then P_{1}/U_{2} is a parabolic subgroup of P_{2}/U_{2} with maximal reductive quotient P_{1}/U_{1}, which reduces us further to the case where G = P_{2}. Then P_{1} contains the radical \overline{R}(G) = R(G) of G, and P_{1} → P_{1}/U_{1} maps R(G) to the radical \overline{R}(P_{1}) of P_{1}/U_{1}. This yields our canonical morphism \overline{R}(P_{2}) → \overline{R}(P_{1}). □

Pulling back the above morphism through

𝕆(G) ×_{S} 𝕆(G) → 𝕆′(G)
(a, b) ↦ (a ∩ b, b)

we obtain a morphism p_{2}^{∗}R_{𝕆(G)} → (∩)^{∗}R_{𝕆(G)} of tori over 𝕆(G) ×_{S} 𝕆(G). By symmetry, there is also a morphism p_{1}^{∗}R_{𝕆(G)} → (∩)^{∗}R_{𝕆(G)}. The product of these two yields a morphism in the fibered category of tori over Sch/S,

\xymatrix{R_{𝕆(G)} ×_{S} R_{𝕆(G)} \ar[r] \ar[d] & R_{𝕆(G)} ×_{𝕆(G)} R_{𝕆(G)} \ar[d] \\ 𝕆(G) ×_{S} 𝕆(G) \ar[r]^{∩} & 𝕆(G)}

Composing it with the multiplication map on the 𝕆(G)-torus R_{𝕆(G)}, we obtain yet another such morphism, namely

\xymatrix{R_{𝕆(G)} ×_{S} R_{𝕆(G)} \ar[r] \ar[d] & R_{𝕆(G)} \ar[d] \\ 𝕆(G) ×_{S} 𝕆(G) \ar[r]^{∩} & 𝕆(G)}

Applying now the 𝔾^{Γ}(−) construction to the latter diagram yields a morphism

\xymatrix{𝔾^{Γ}(R_{𝕆(G)}) ×_{S} 𝔾^{Γ}(R_{𝕆(G)}) \ar[r] \ar[d] & 𝔾^{Γ}(R_{𝕆(G)}) \ar[d] \\ 𝕆(G) ×_{S} 𝕆(G) \ar[r]^{∩} & 𝕆(G)}

in the fibered category of commutative group schemes over Sch/S. The top map of this diagram defines a commutative monoid structure on the S-scheme 𝔾^{Γ}(R_{𝕆(G)}). By construction, the structural morphism 𝔾^{Γ}(R_{𝕆(G)}) → 𝕆(G) is compatible with the monoid structures on both sides.

Lemma 19. The S-scheme ℂ^{Γ}(G) is a submonoid of 𝔾^{Γ}(R_{𝕆(G)}).

Proof. Using additive notations, we have to show that for S′ → S and c_{1}, c_{2} in ℂ^{Γ}(G)(S′), there exists an fpqc cover S′′ → S′ and an element f ∈ 𝔾^{Γ}(G)(S′′) such that c_{1} + c_{2} = t ∘ Fil(f) in 𝔾^{Γ}(R_{𝕆(G)}). We may assume that S′ = S and c_{i} = t ∘ Fil(f_{i}) for some f_{i} : 𝔻_{S}(Γ) → G. Using [21, XXVI 1.14 and XXIV 1.5], we may also assume that there is an épinglage (G, T, Δ, ⋯) which is adapted to P_{1} = P_{f_{1}} and P_{2} = P_{f_{2}}. Then by [21, XXVI 1.6 and 1.8], we may assume that L_{1} = L_{f_{1}} and L_{2} = L_{f_{2}} both contain the maximal torus T of G, so that both f_{1} and f_{2} factor through T. Let f = f_{1} + f_{2} : 𝔻_{S}(Γ) → T ↪ G and P = P_{f}. We claim that c_{1} + c_{2} = t ∘ Fil(f). By [21, XXVI 3.2], t(P_{i}) = Δ(P_{i})_{S} where Δ(P_{i}) ⊂ Δ ⊂ Hom(T, 𝔾_{m,S}) is the set of simple roots occurring in Lie(L_{i}), i.e. Δ(P_{i}) = {α ∈ Δ : α ∘ f_{i} = 0 ∈ Γ}. By construction, α ∘ f_{i} ≥ 0 in Γ for every α ∈ Δ, thus also α ∘ f = α ∘ f_{1} + α ∘ f_{2} ≥ 0 in Γ for every α ∈ Δ, with α ∘ f = 0 if and only if α ∘ f_{1} = 0 = α ∘ f_{2}. It follows that our épinglage is also adapted to P, with Δ(P) = Δ(P_{1}) ∩ Δ(P_{2}), i.e. t(P) = t(P_{1}) ∩ t(P_{2}) in 𝕆(G)(S). The inclusion P ⊂ P_{i} induces the canonical morphism can_{i} : \overline{R}(P_{i}) → \overline{R}(P) and one checks easily that \overline{f} = can_{1} ∘ \overline{f}_{1} + can_{2} ∘ \overline{f}_{2}. Thus by definition,

c_{1} + c_{2} = t(P, can_{1} ∘ \overline{f}_{1} + can_{2} ∘ \overline{f}_{2}) = t(P, \overline{f}) = t ∘ Fil(f)

as was to be shown. □

Remark 20. The 0-section of ℂ^{Γ}(G) is the identity element of its monoid structure. The latter is compatible with functoriality in Γ, but not with functoriality in G: if H is a subtorus of G and f is a section of 𝔾^{Γ}(H) = 𝔽^{Γ}(H) = ℂ^{Γ}(H), then f + ιf is trivial in ℂ^{Γ}(H), but not necessarily in ℂ^{Γ}(G).

2.2.11. The split case. Suppose that (G, T, M, R) is a split reductive group over S [21, XXII 1.13]: G is a reductive group over S, M is a finite free ℤ-module, T ⊂ G is a maximal subtorus of G equipped with an isomorphism T ≃ 𝔻_{S}(M), R ⊂ M is a set of roots of T in G and for each α ∈ R, the corresponding quasicoherent sub-sheaf 𝔤_{α} of 𝔤 = Lie(G) is a free O_{S}-module (of rank 1). Let

ℛ = (M, R, M^{∗}, R^{∗})

be the induced (reduced) root system [21, XXII 1.14] with Weyl group W = W (ℛ) in Aut(M). Let W_{G}(T) = N_{G}(T)/Z_{G}(T) be the Weyl group of T in G, a constant group scheme over S identified with W_{S} through its action on T [21, XXII 3.4]. The composition of the isomorphism of group schemes over S

(Hom (M, Γ))_{S} ≃ Hom_{S−Group} (𝔻_{S}(Γ), 𝔻_{S}(M)) ≃ 𝔾^{Γ}(T)

from [20, VIII 1.5] with the morphism of S-schemes

𝔾^{Γ}(T) ↪ 𝔾^{Γ}(G) ⟶^{Fil} 𝔽^{Γ}(G) ⟶^{t} ℂ^{Γ}(G)

thus factors through a morphism of étale S-schemes,

(W \Hom (M, Γ))_{S} → ℂ^{Γ}(G).

We claim that the latter is an isomorphism. Since both sides are étale over S, it is sufficient to establish that for any geometric point Spec(k) → S, the induced map

W \Hom (M, Γ) → ℂ^{Γ}(G)(k) = G(k)\Hom_{k−Group} (𝔻_{k}(Γ), G_{k})

is a bijection. Any f : 𝔻_{k}(Γ) → G_{k} factors through a maximal torus T′ of G_{k} by corollary 11, and T′ = Int(g)(T_{k}) for some g ∈ G(k) by [1, XII 6.6.a]: our map is surjective. For φ, φ′ : M → Γ giving f, f′ : 𝔻_{k}(Γ) → T_{k} and g ∈ G(k) such that Int(g) ∘ f = f′, Int(g)(T_{k}) and T_{k} are maximal tori of L_{f′}, thus Int(hg)(T_{k}) = T_{k} for some h ∈ L_{f′}(k); but then n = hg ∈ N_{G}(T)(k) and Int(n) ∘ f = Int(h) ∘ f′ = f′, thus φ′ = wφ where w is the image of n in W = W_{G}(T)(k): our map is injective.

Fix a system of positive roots R_{+} ⊂ R [21, XXI 3.2.1], which corresponds to a Borel subgroup B of G by [21, XXII 5.5.1]. By lemma 28 below, the submonoid

Hom^{+}(M, Γ) = {f ∈ Hom(M, Γ) : ∀α ∈ R_{+}, f(α) ≥ 0}

is a fundamental domain for the action of W on Hom(M, Γ). The isomorphism

(Hom^{+} (M, Γ))_{S} ≃ (W \Hom (M, Γ))_{S} ≃ ℂ^{Γ}(G)

is then easily seen to be compatible with the S-monoid structures.

2.2.12. ℂ^{Γ}(G) is a partially ordered commutative monoid. A partial order ≤ on an S-scheme X is a subscheme R = R(≤) of X ×_{S} X such that for every S-scheme Y, the subset R(Y) of X(Y) × X(Y) defines a partial order (also denoted by ≤) on X(Y). We say that the partial order is open (resp. closed) if R ↪ X ×_{S} X is an open (resp. closed) immersion. A partial order on an S-monoid (X, ·) is a partial order on the underlying S-scheme such that for any S-scheme Y and f_{1}, f_{2}, g in X(Y), f_{1} ≤ f_{2} implies f_{1} · g ≤ f_{2} · g and g · f_{1} ≤ g · f_{2}.

If ℛ = (M, R, M^{∗}, R^{∗}) is a (not necessarily reduced) root system and R_{+} ⊂ R is a system of positive roots, the weak (≤) and strong (⪯) partial orders on the abstract monoid Hom^{+}(M, Γ) defined in section 2.4 below induce open and closed partial orders on the constant S-monoid (Hom^{+} (M, Γ))_{S}. If ℛ = ℛ(G, T, M, R) is the root system of a split reductive group (G, T, M, R), we thus obtain open and closed partial orders on the S-monoid ℂ^{Γ}(G). These partial orders then do not depend upon the chosen auxiliary data (T, M, R; R^{+}): this may be checked on geometric points, where all such data are indeed conjugated. Since every reductive group G over S is locally splittable in the étale topology on S [21, XXII 2.3], we finally obtain by étale descent: the S-monoid ℂ^{Γ}(G) is canonically equipped with weak (≤) and strong (⪯) partial orders, both open and closed.

The weak and strong partial orders are functorial in Γ, and coincide if Γ is divisible. We will see later on that the weak partial order is also functorial in G.

2.2.13. Behavior under isogenies. Suppose that the (torsion free) commutative group Γ is (uniquely) divisible, i.e. that it is a ℚ-vector space.

Proposition 21. The fundamental cartesian diagram

\xymatrix{𝔾^{Γ}(G) \ar[r]^{Fil} \ar[d]^{F} & 𝔽^{Γ}(G) \ar[r]^{t} \ar[d]^{F} & ℂ^{Γ}(G) \ar[d]^{F} \\ 𝕆ℙℙ(G) \ar[r]^{p_{1}} & ℙ(G) \ar[r]^{t} & 𝕆(G)}

is invariant under central isogenies.

Proof. The bottom line only depends upon the adjoint group G^{ad} = G/Z(G): this is true for 𝕆(G) because 𝔻𝕐ℕ(G) = 𝔻𝕐ℕ(G^{ad}) by definition of the Dynkin S-scheme [21, XXIV 3.3] in view of [21, XXII 4.3.7], and the maps P ↦ P/Z(G) and P^{ad} ↦ ad^{−1}(P^{ad}) (where ad : G → G^{ad} is the quotient map) induce mutually inverse bijections between parabolic subgroups of G and parabolic subgroups of G^{ad}, which are compatible with the type maps and with opposition. For the top line, let f : G_{1} → G_{2} be a central isogeny [21, XXII 4.2.9]. We first claim that composition with f yields an isomorphism 𝔾^{Γ}(G_{1}) → 𝔾^{Γ}(G_{2}): for split tori, this immediately follows from [20, VIII 1.5] and our assumption on Γ; for tori, our claim is local in the fpqc topology on S by [26, 2.7.1], which reduces us to the previous case; for arbitrary reductive groups, use lemma 5 and [1, XVII 7.1.1]. If now P_{1} is a parabolic subgroup of G_{1} with image P_{2} in G_{2}, then f induces an isogeny \overline{R}(P_{1}) → \overline{R}(P_{2}). Thus f yields an isogeny R_{f} : R_{ℙ(G_{1})} → R_{ℙ(G_{2})} of tori over ℙ(G_{1}) ≃ ℙ(G_{2}). The induced isomorphism 𝔾^{Γ}(R_{ℙ(G_{1})}) ≃ 𝔾^{Γ}(R_{ℙ(G_{2})}) is compatible with the morphisms 𝔾^{Γ}(G_{i}) → 𝔾^{Γ}(R_{ℙ(G_{i})}), therefore also 𝔽^{Γ}(G_{1}) ≃ 𝔽^{Γ}(G_{2}). Since R_{f} is also compatible with the canonical descent data of lemma 13, it descends to an isogeny R_{f} : R_{𝕆(G_{1})} → R_{𝕆(G_{2})} of tori over 𝕆(G_{1}) ≃ 𝕆(G_{2}). The induced isomorphism 𝔾^{Γ}(R_{𝕆(G_{1})}) ≃ 𝔾^{Γ}(R_{𝕆(G_{2})}) is again compatible with the morphisms 𝔽^{Γ}(G_{i}) → 𝔾^{Γ}(R_{𝕆(G_{i})}), therefore also ℂ^{Γ}(G_{1}) ≃ ℂ^{Γ}(G_{2}). □

Plainly, the above diagrams are also compatible with products. Considering the canonical diagram of central isogenies [21, XXII 4.3 & 6.2]

R(G) × G^{der} → G → G^{ab} × G^{ss} → G^{ab} × G^{ad}

where R(G) is the radical of G, G^{der} its derived group, G^{ab} = G/G^{der} its coradical, G^{ss} = G/R(G) its semi-simplification and G^{ad} = G/Z(G) its adjoint group, we obtain compatible canonical decompositions

𝔾^{Γ}(G) = 𝔾^{Γ}(G)^{r} × 𝔾^{Γ}(G)^{c}
𝔽^{Γ}(G) = 𝔽^{Γ}(G)^{r} × 𝔽^{Γ}(G)^{c}
ℂ^{Γ}(G) = ℂ^{Γ}(G)^{r} × ℂ^{Γ}(G)^{c}

with 𝔾^{Γ}(G)^{c} = 𝔽^{Γ}(G)^{c} = ℂ^{Γ}(G)^{c} = 𝔾^{Γ}(R(G)) = 𝔾^{Γ}(G^{ab}) = 𝔾^{Γ}(Z(G)) and

𝔾^{Γ}(G)^{r} = 𝔾^{Γ}(G^{der}) = 𝔾^{Γ}(G^{ss}) = 𝔾^{Γ}(G^{ad}),
𝔽^{Γ}(G)^{r} = 𝔽^{Γ}(G^{der}) = 𝔽^{Γ}(G^{ss}) = 𝔽^{Γ}(G^{ad}),
ℂ^{Γ}(G)^{r} = ℂ^{Γ}(G^{der}) = ℂ^{Γ}(G^{ss}) = ℂ^{Γ}(G^{ad}).

The decomposition of ℂ^{Γ}(G) is compatible with the partially ordered (weak=strong) monoid structures: for x = (x^{r}, x^{c}) and y = (y^{r}, y^{c}) in ℂ^{Γ}(G) = ℂ^{Γ}(G)^{r} × ℂ^{Γ}(G),

x + y = (x^{r} + y^{r}, x^{c} + y^{c}) and x ≤ y ⟺ (x^{r} ≤ y^{r} and x^{c} = y^{c}) .

This is easily checked by reduction to the split case, cf. section 2.4.10 below.

2.3. Relative positions of Γ-filtrations

Let G be a reductive group over S.

2.3.1. Standard positions. Recall that two parabolic subgroups P_{1} and P_{2} of G are said to be in standard (relative) position if and only if they satisfy the equivalent conditions of [21, XXVI 4.5.1], in particular: (i) P_{1} ∩ P_{2} is smooth over S, or (ii) P_{1} ∩ P_{2} is a subgroup of type (R) of G, or (iv) P_{1} ∩ P_{2} contains, locally on S for the Zariski topology, a maximal torus of G. Then all such maximal tori are, locally on S for the étale topology, conjugated in P_{1} ∩ P_{2} [1, XII 7.1]. In any case, P_{1} ∩ P_{2} has geometrically connected fibers [5, 4.5]. For an S-scheme T, we set

𝕊𝕋𝔻(G)(T) = {(P_{1}, P_{2}) ∈ ℙ(G)^{2}(T) : P_{1} and P_{2} are in standard position} .

By [21, XXVI 4.5.3], this defines a representable subsheaf of ℙ(G)^{2} with Stein factorization

𝕊𝕋𝔻(G) ⟶^{t_{2}} 𝕋𝕊𝕋𝔻(G) ⟶ S

fitting in a commutative (but not cartesian) diagram

\xymatrix{𝕊𝕋𝔻(G) \ar@{^{(}->}[d] \ar[r]^{t_{2}} & 𝕋𝕊𝕋𝔻(G) \ar[d]_{q} \\ ℙ(G)^{2} \ar[r]^{t^{2}} & 𝕆(G)^{2}}

where q is a finite étale surjective morphism while t_{2} is a smooth, surjective, finitely presented morphism with geometrically connected fibers which is a quotient of 𝕊𝕋𝔻(G) by the diagonal action of G in the category of fpqc sheaves on S. By [21, XXVI 4.2.5 & 4.4.3], the morphism q has two canonical sections

tr, os : 𝕆(G)^{2} → 𝕋𝕊𝔻𝕋(G)

corresponding respectively to the transverse and osculatory (standard) positions. By [21, XXVI 4.2.4], t_{2}^{−1}(im(tr)) is a relatively dense open S-subscheme 𝔾𝔼ℕ(G) of ℙ(G)^{2}. Pulling back everything through the surjective étale facet morphism F^{2} : ℂ^{Γ}(G)^{2} → 𝕆(G)^{2}, we thus obtain a commutative diagram

\xymatrix{𝕊𝕋𝔻^{Γ}(G) \ar@{^{(}->}[d] \ar[r]^{t_{2}} & 𝕋𝕊𝕋𝔻^{Γ}(G) \ar[d]_{q} \\ 𝔽^{Γ}(G)^{2} \ar[r]^{t^{2}} & ℂ^{Γ}(G)^{2} \ar[u]_{tr,os}}

where t_{2} and q still have the properties listed above, together with a relatively dense open S-subscheme 𝔾𝔼ℕ^{Γ}(G) of 𝔽^{Γ}(G)^{2}. For a scheme Z over ℙ(G)^{2}, we set

𝕊𝕋𝔻(Z) = Z ×_{ℙ(G)^{2}} 𝕊𝕋𝔻(G).

For instance, 𝕊𝕋𝔻^{Γ}(G) = 𝕊𝕋𝔻 (𝔽^{Γ}(G) ×_{S} 𝔽^{Γ}(G)).

Remark 22. The monomorphisms 𝕊𝕋𝔻(G) ↪ ℙ(G)^{2} and 𝕊𝕋𝔻^{Γ}(G) ↪ 𝔽^{Γ}(G)^{2} are surjective. More precisely, for any S-scheme T = Spec(k) with k a field,

𝕊𝕋𝔻(G)(k) = ℙ(G)^{2}(k) and 𝕊𝕋𝔻^{Γ}(G)(k) = 𝔽^{Γ}(G)^{2}(k)

by Bruhat’s theorem [21, XXVI 4.1.1].

2.3.2. The addition map on Γ-filtrations.

Proposition 23. There is an S-morphism

+ : 𝕊𝕋𝔻^{Γ}(G) → 𝔽^{Γ}(G), (F, G) ↦ F + G

such that for every S-scheme T, (F, G) ∈ 𝕊𝕋𝔻^{Γ}(G)(T) and H ∈ 𝔽^{Γ}(G)(T),

F + G = G + F and H + 0 = 0 + H = H in 𝔽^{Γ}(G)(T).

Proof. Since (P_{F}, P_{G}) ∈ 𝕊𝕋𝔻(G)(T), there is, locally on T for the Zariski topology, a maximal torus H of G_{T} inside P_{F} ∩ P_{G} [21, XXVI 4.5.1]. Let L_{F} and L_{G} be the Levi subgroups of P_{F} and P_{G} containing H [21, XXVI 1.6], let f : 𝔻_{T} (Γ) → L_{F} and g : 𝔻_{T} (Γ) → L_{G} be the corresponding central morphisms lifting \overline{F} and \overline{G}. Then f and g both factor through H, and their product f + g in the commutative group H is a group homomorphism 𝔻_{T} (Γ) → G_{T}. We claim that F + G = Fil(f + g) does not depend upon the choice of the maximal torus H – the whole construction is then indeed local in the Zariski topology on T as well as functorial in the S-scheme T, and the resulting S-morphism + : 𝕊𝕋𝔻^{Γ}(G) → 𝔽^{Γ}(G) obviously has the required properties. Let thus H′ be another maximal torus of G_{T} inside K = P_{F} ∩P_{G}, giving rise to f′, g′ and f′+g′ : 𝔻_{T} (Γ) → H′ ⊂ G. Then, locally on T for the étale topology, there is a k ∈ K(T) such that H′ = Int(k)(H) by [1, XII 7.1], in which case also f′ = Int(k)∘f, g′ = Int(k)∘g and f′+g′ = Int(k)∘(f +g). It is thus sufficient to establish that K ⊂ P_{f+g}. This second claim is again local in the étale topology on T, which reduces us further to the following case: (G, H, M, R) is a split group over S = T (i.e. H ≃ 𝔻_{S}(M) and R ⊂ M is the set of roots of H in Lie(G)) with f and g respectively induced by morphisms f^{♯} and g^{♯} : M → Γ, so that f + g is induced by (f + g)^{♯} = f^{♯} + g^{♯}. For a closed subset R′ of R, we denote by H_{R′} ⊃ H the corresponding subgroup of G of type (R), as in [21, XXII 5.4] (thus H = H_{∅} and G = H_{R}). Then P_{F} = H_{R(f)}, P_{G} = H_{R(g)} and P_{f+g} = H_{R(f+g)} where R(h) = {α ∈ R : h^{♯}(α) ≥ 0} by definition of these parabolic subgroups of G. Thus K = H_{R(f)∩R(g)} is contained in H_{R(f+g)} = P_{f+g} by [21, XXII 5.4.5]. □

Proposition 24. For any S-scheme T and (F, G) ∈ 𝕊𝕋𝔻^{Γ}(G)(T),
t(F + G) ⪯ t(F) + t(G) in ℂ^{Γ}(G)(T)

with equality if F and G are in osculatory position.

Proof. We may assume that T = s is a geometric point, with F and G lifting to morphisms f, g : 𝔻_{s}(Γ) → H for some maximal (split) subtorus H ≃ 𝔻_{s}(M) of G_{s}, corresponding to morphisms f^{♯}, g^{♯} : M → Γ as above. Let R ⊂ M be the roots of H in Lie(G_{s}). By [21, XXI 3.3.6], there is a system of positive roots R_{+} ⊂ R such that (f^{♯} + g^{♯})(R_{+}) ⊂ Γ_{+}, i.e. f^{♯} + g^{♯} ∈ Hom^{+}(M, Γ) in the notations of section 2.4. Thus if ϑ : Hom(M, Γ) ↠ Hom^{+}(M, Γ) is the retraction from lemma 28, then ϑ(f^{♯} + g^{♯}) = f^{♯} + g^{♯} and f^{♯} ⪯ ϑ(f^{♯}), g^{♯} ⪯ ϑ(g^{♯}) in Hom(M, Γ) by lemma 29, therefore ϑ(f^{♯} + g^{♯}) ⪯ ϑ(f^{♯}) + ϑ(g^{♯}) in Hom^{+}(M, Γ), i.e. t(F + G) ⪯ t(F) + t(G) in ℂ^{Γ}(G)(s) by definition. If F and G are in osculatory position, there is a Borel subgroup B of G_{s} inside P_{F} ∩ P_{G} [21, XXVI 4.4.1]. We may then take H inside B and for R_{+}, the roots of H in Lie(B), so that f^{♯} and g^{♯} already belong to Hom^{+}(M, Γ), ϑ(f^{♯} + g^{♯}) = ϑ(f^{♯}) + ϑ(g^{♯}) and indeed t(F + G) = t(F) + t(G). □

2.3.3. We record here a special case of the functoriality of 𝔽^{Γ}(−).

Proposition 25. Let L be a Levi subgroup of a parabolic subgroup P of G with unipotent radical U. Then: (1) for f : 𝔻_{S}(Γ) → L inducing g : 𝔻_{S}(Γ) → G and h : 𝔻_{S}(Γ) → P/U, the parabolic subgroups P and P_{g} of G are in standard relative position, K = P ∩ P_{g} is a smooth subgroup scheme of G, K · U is a parabolic subgroup of G with Levi L_{f}, K ∩ L = (K · U) ∩ L = P_{f} and K · U/U = P_{h} in P/U.

(2) There is a unique morphism ι : 𝔽^{Γ}(L) → 𝔽^{Γ}(G) such that the diagram

\xymatrix{𝔾^{Γ}(L) \ar@{^{(}->}[r] \ar[d]_{Fil} & 𝔾^{Γ}(G) \ar[d]^{Fil} \\ 𝔽^{Γ}(L) \ar[r]^{ι} & 𝔽^{Γ}(G)}

is commutative.

Proof. Everything in (1) is local for the fpqc topology on S. We may thus assume that L_{f} = Z_{L}(f) and G are split with respect to a maximal torus H of G contained in L_{f} with H trivial, i.e. H = 𝔻_{S}(M) for some finitely generated abelian group M [21, XXII 2.3]. Then H ⊂ L_{f} = L ∩ L_{g} ⊂ P ∩ P_{g}, thus P and P_{g} are in standard relative position, K = P ∩ P_{g} is a smooth subgroup of G and K · U is a parabolic subgroup of G by [21, XXVI 4.5.1] and its proof. More precisely, let R ⊂ M be the roots of H in Lie(G), so that R = R_{L} ∐ R_{U} ∐ −R_{U} where R_{L} and R_{U} are respectively the roots of H in Lie(L) and Lie(U). For X in {∅, L, U} let R_{X} = R_{X}^{0} ∐ R_{X}^{+} ∐ R_{X}^{−} be the decomposition of R_{X} induced by g, i.e.

R_{X}^{±} = {α ∈ R_{X} : ±α ∘ g > 0 in Γ} and R_{X}^{0} = {α ∈ R_{X} : α ∘ g = 0 in Γ}.

This yields a decomposition of R in nine pieces, as shown in the following table:

L_{g}  U_{g}  U_{ιg}
L R_{L}^{0} R_{L}^{+} R_{L}^{−} = −R_{L}^{+}
U R_{U}^{0} R_{U}^{+}  R_{U}^{−}
−R_{U}^{0} −R_{U}^{−}  −R_{U}^{+}

For a closed subset R′ of R, let H(R′) be the subgroup scheme of G of type (R) which is determined by R′, see [21, XXII 5.4.2-7]. Thus L = H(R_{L}), P = H(R_{L} ∪ R_{U}), L_{g} = H(R^{0}), P_{g} = H(R^{0} ∪ R^{+}), L_{f} = H(R_{L}^{0}) and P_{f} = H(R_{L}^{0} ∪ R_{L}^{+}) while

K = H(R_{L}^{0} ∪ R_{L}^{+} ∪ R_{U}^{0} ∪ R_{U}^{+})
and K · U = H(R_{L}^{0} ∪ R_{L}^{+} ∪ R_{U}^{0} ∪ R_{U}^{+} ∪ R_{U}^{−}).

By [21, XXVI 6.1], L_{f} = H(R_{L}^{0}) is a Levi subgroup of K · U. By [21, XXII 5.4.5], P_{f} ⊂ K, thus P_{f} ⊂ K ∩ L ⊂ (K · U) ∩ L. But (K · U) ∩ L is a parabolic subgroup of L with Levi L_{f}, thus P_{f} = K ∩ L = (K · U) ∩ L and P_{f} · U = K · U by repeated applications of [21, XXVI 1.20]. Finally, P_{f} maps to P_{h} = P_{f} · U/U = K · U/U under the isomorphism L ≃ P/U, which finishes the proof of (1). Then (2) easily follows: if (f, f′) induce (g, g′) and Fil(f) = Fil(f′), then f′ = Int(p) ∘ f for some p ∈ P_{f}(S), thus also g′ = Int(p)∘g and Fil(g′) = Fil(g) since P_{f} = L∩P_{g} ⊂ P_{g}. □

2.3.4. Let G′ = P_{u}/U_{u} where P_{u} ⊂ G_{ℙ(G)} is the universal parabolic subgroup with unipotent radical U_{u} = R^{u}(P_{u}). Thus G′ is a reductive group over ℙ(G) and

𝔽^{Γ}(G′)(T) = {(P, F) : P ∈ ℙ(G)(T), F ∈ 𝔽^{Γ}(P/U)(T), U = R^{u}(P)}

for any S-scheme T.

Proposition 26. There is a canonical morphism of schemes over ℙ(G),

𝕊𝕋𝔻 (ℙ(G) ×_{S} 𝔽^{Γ}(G)) → 𝔽^{Γ}(G′) (P, F) ↦ (P, Gr_{P} (F)).

Proof. Start with (P, F) ∈ 𝕊𝕋𝔻 (ℙ(G) ×_{S} 𝔽^{Γ}(G)) (T) and put K = P ∩ P_{F}. Then K is a smooth subgroup scheme of G_{T} which contains, locally on T for the Zariski topology, a maximal subtorus H of G_{T} [21, XXVI 4.5.1]. Let L and L_{F} be the Levi subgroups of P and P_{F} containing H [21, XXVI 1.6]. Let f : 𝔻_{T} (Γ) → L_{F} be the central morphism lifting \overline{F} : 𝔻_{T} (Γ) → \overline{R}(P_{F}), so that F = Fil(f) and L_{f} = L_{F}. Then f factors through the maximal subtorus H of L_{F}, which is also a maximal subtorus of L. Let h : 𝔻_{T} (Γ) → P/U be the induced morphism. By the previous proposition, P_{h} = K · U/U, thus K fixes Fil(h) ∈ 𝔽^{Γ}(P/U)(T). If H′ is another maximal subtorus of G contained in K, then, locally on T for the étale topology, H′ = Int(k)(H) for some k ∈ K(T) by [1, XII 7.1]. But then L′ = Int(k)(L), L′_{F} = Int(k)(L_{F}), f′ = Int(k) ∘ f and h′ = Int(k) ∘ h are the objects associated to H′ as above, thus Fil(h′) = k · Fil(h) = Fil(h) since K fixes Fil(h). It follows that Gr_{P} (F) = Fil(h) does not depend upon the choice of H, and also that the whole construction is indeed local in the Zariski topology on T. □

Remark 27. The pull-back of this morphism through p_{1} : 𝕆ℙℙ(G) → ℙ(G) has a canonical section: for an S-scheme T, the latter is given by the formula

(P_{1}, P_{2}, F) ↦ (P_{1}, P_{2}, ι(F_{L}))

where (P_{1}, P_{2}) = (U_{1} ⋊ L, U_{2} ⋊ L) is a pair of opposed parabolic subgroups of G_{T} with common Levi subgroup L = P_{1} ∩ P_{2}, F is an element of 𝔽^{Γ}(P_{1}/U_{1})(T), F_{L} is its unique lift in 𝔽^{Γ}(L)(T), and ι : 𝔽^{Γ}(L) → 𝔽^{Γ}(G_{T}) is the morphism of proposition 25 (thus indeed P_{1} and P_{ι(F_{L})} are in standard relative position).

2.4. Interlude on the dominance partial orders

Let Γ = (Γ, +, ≤) be a non-trivial totally ordered commutative group. We set

Γ_{+} = {γ ∈ Γ : γ ≥ 0}.

2.4.1. Let ℛ = (M, R, M^{∗}, R^{∗}) be a root system [21, XXI 1.1.1] with Weyl group W = W (ℛ) [21, XXI 1.1.8]. Fix a system of positive roots R_{+} ⊂ R [21, XXI 3.2.1] and let Δ ⊂ R_{+} be the corresponding simple roots [21, XXI 3.2.8]. Then

Lemma 28. The submonoid of dominant morphisms in Hom(M, Γ),

Hom^{+}(M, Γ) = {f ∈ Hom(M, Γ) : ∀α ∈ R_{+}, f(α) ≥ 0}
= {f ∈ Hom(M, Γ) : ∀α ∈ Δ, f(α) ≥ 0}

is a fundamental domain for the action of W on Hom(M, Γ).

Proof. For any morphism f : M → Γ, define

R_{f≥0} = {α ∈ R : f(α) ≥ 0} ,
R_{f>0} = {α ∈ R : f(α) > 0} ,
R_{f=0} = {α ∈ R : f(α) = 0} .

Thus R_{f=0} is closed and symmetric, R = R_{f>0} ∐ R_{f=0} ∐ −R_{f>0} and

f is dominant ⟺ R_{+} ⊂ R_{f≥0} ⟺ R_{f>0} ⊂ R_{+}.

By [21, XXI 3.3.6], there exists w ∈ W such that

R_{+} ⊂ wR_{f≥0} = R_{wf≥0},

therefore wf is dominant. If f and wf are dominant, then

R_{+} = R_{f>0} ∐ R_{+}^{1} and w^{−1}R_{+} = R_{f>0} ∐ R_{+}^{2}

where R_{+}^{1} = R_{+} ∩ R_{f=0} and R_{+}^{2} = w^{−1}R_{+} ∩ R_{f=0} are systems of positive roots in the closed symmetric subset R_{f=0} of R. Thus by [21, XXI 3.4.1 and 3.3.7], there is an w_{0} in the Weyl group W_{f} ⊂ W of R_{f=0} such that w_{0}R_{+}^{2} = R_{+}^{1}. Now W_{f} is spanned by the reflections {s_{α} : α ∈ R_{f=0}} and for any m ∈ M and α ∈ R_{f=0},

(s_{α}f)(m) = f(s_{α}m) = f (m − ⟨m, α^{∗}⟩ α) = f(m) − ⟨m, α^{∗}⟩ f(α) = f(m),

thus w_{0} fixes f, stabilizes R_{f>0} and maps w^{−1}R_{+} to R_{+}. But then w = w_{0} by [21, XXI 5.4] hence wf = w_{0}f = f, which proves the lemma. □

2.4.2. Applying the lemma to the dual root system ℛ^{∗} = (M^{∗}, R^{∗}, M, R) with Γ = ℤ, we obtain the well known fact that the cone of dominant weights

M_{d} = {m ∈ M : ∀α ∈ R_{+}, ⟨m, α^{∗}⟩ ≥ 0}
= {m ∈ M : ∀δ ∈ Δ, ⟨m, δ^{∗}⟩ ≥ 0}

is a fundamental domain for the action of W on M.

2.4.3. The coroot cone and coroot lattice defined by

Γ_{+}R_{+}^{∗} = {m ↦ ∑_{α∈R_{+}} ⟨m, α^{∗}⟩ γ_{α} : ∀α ∈ R_{+}, γ_{α} ∈ Γ_{+}}
ΓR^{∗} = {m ↦ ∑_{α∈R} ⟨m, α^{∗}⟩ γ_{α} : ∀α ∈ R, γ_{α} ∈ Γ}

are a submonoid and a subgroup of Hom(M, Γ), and so are their saturations

(Γ_{+}R_{+}^{∗})_{sat} = {f ∈ Hom(M, Γ) : ∃n ∈ ℕ^{×} such that nf ∈ Γ_{+}R_{+}^{∗}}
(ΓR^{∗})_{sat} = {f ∈ Hom(M, Γ) : ∃n ∈ ℕ^{×} such that nf ∈ ΓR^{∗}}

in Hom(M, Γ). Inside Hom(M, Γ ⊗ ℚ), any f ∈ (ΓR^{∗})_{sat} can be written as

f(−) = ∑_{δ∈Δ} ⟨−, ind(δ^{∗})⟩ γ_{δ}

for a unique (γ_{δ}) ∈ (Γ ⊗ ℚ)^{Δ}, where ind(δ^{∗}) is the simple coroot corresponding to δ ∈ Δ, namely ind(δ^{∗}) = δ^{∗} if 2δ ∉ R and ind(δ^{∗}) = \frac{1}{2}δ^{∗} = (2δ)^{∗} otherwise. Then

f ∈ (Γ_{+}R_{+}^{∗})_{sat} ⟺ ∃n ∈ ℕ^{×} such that n(γ_{δ}) ∈ Γ_{+}^{Δ},
f ∈ ΓR^{∗} ⟺ (γ_{δ}) ∈ Γ^{Δ},
f ∈ Γ_{+}R_{+}^{∗} ⟺ (γ_{δ}) ∈ Γ_{+}^{Δ}.

In particular, (Γ_{+}R_{+}^{∗})_{sat} ∩ −(Γ_{+}R_{+}^{∗})_{sat} = {0}. Moreover by duality,

(Γ_{+}R_{+}^{∗})_{sat} = {f ∈ Hom(M, Γ) : ∀m ∈ M_{d}, f(m) ≥ 0} .

2.4.4. The weak dominance partial order ≤ on Hom(M, Γ) is defined by

f_{1} ≤ f_{2} ⟺ ∀m ∈ M_{d} : f_{1}(m) ≤ f_{2}(m),
⟺ f_{2} − f_{1} ∈ (Γ_{+}R_{+}^{∗})_{sat} .

The strong dominance partial order ⪯ on Hom(M, Γ) is defined by

f_{1} ⪯ f_{2} ⟺ f_{2} − f_{1} ∈ Γ_{+}R_{+}^{∗}.

They are both compatible with the addition map: for f_{1}, f_{2}, g_{1}, g_{2} ∈ Hom(M, Γ),

(f_{1} ≤ g_{1} and f_{2} ≤ g_{2}) ⟹ f_{1} + f_{2} ≤ g_{1} + g_{2},
(f_{1} ⪯ g_{1} and f_{2} ⪯ g_{2}) ⟹ f_{1} + f_{2} ⪯ g_{1} + g_{2}.

They are related as follows: for any f_{1}, f_{2} ∈ Hom(M, Γ), we have

f_{1} ⪯ f_{2} ⟺ f_{1} ≤ f_{2} and π(f_{1}) = π(f_{2})

where π : Hom(M, Γ) ↠ Hom(M, Γ)/ΓR^{∗} is the projection. Note that

f_{1} ≤ f_{2} ⟹ π(f_{2} − f_{1}) ∈ (ΓR^{∗})_{sat}/ΓR^{∗}.

In particular since (ΓR^{∗})_{sat}/ΓR^{∗} is torsion,

f_{1} ≤ f_{2} ⟺ ∃n ∈ ℕ^{×} : nf_{1} ⪯ nf_{2}.

2.4.5. Since W M_{d} = M, both partial orders restrict to the identity on the fixed point set of W in Hom(M, Γ): for any W -invariant f_{1}, f_{2} ∈ Hom(M, Γ),

\xymatrix{f_{1} ⪯ f_{2} \ar@{=>}[r] & f_{1} ≤ f_{2} \ar@{=>}[r] & ∀m ∈ M_{d} : f_{1}(m) ≤ f_{2}(m) \ar@{=>}[d] \\ f_{1} = f_{2} \ar@{=>}[u] &  & ∀m ∈ M : f_{1}(m) ≤ f_{2}(m) \ar@{=>}[ll]}

2.4.6. These partial orders yield the following characterisation of Hom^{+}(M, Γ):

Lemma 29. The projection π is W -invariant and for every f ∈ Hom(M, Γ),

f ∈ Hom^{+}(M, Γ) ⟺ ∀w ∈ W : wf ≤ f,
⟺ ∀w ∈ W : wf ⪯ f.

In particular, Hom(M, Γ)^{W} ⊂ Hom^{+}(M, Γ).

Proof. For any f ∈ Hom(M, Γ) and α ∈ R,

f − s_{α}f = ⟨−, α^{∗}⟩ f(α) in Hom(M, Γ).

Thus π is W -invariant, wf ≤ f ⟺ wf ⪯ f for w ∈ W and

f ∈ Hom^{+}(M, Γ) ⟺ ∀α ∈ R_{+} : s_{α}f ⪯ f.

It remains to establish that

∀α ∈ R_{+} : s_{α}f ⪯ f ⟹ ∀w ∈ W : wf ⪯ f

and we argue by induction on the length ℓ(w) of w in the coxeter group (W, (s_{α})_{α∈Δ}). If ℓ(w) > 1, then w = w′s_{α} for some α ∈ Δ, w′ ∈ W with ℓ(w′) < ℓ(w). Thus

f − wf = (f − w′f) + w′ (f − s_{α}f) = (f − w′f) + ⟨−, w′α^{∗}⟩ f(α).

Now f −w′f ∈ Γ_{+}R_{+}^{∗} by induction, f(α) ≥ 0 by assumption and w′α = −wα ∈ R^{+} by [7, VI, §1, n^{∘}1.6, Corollaire 2], therefore f − wf ∈ Γ_{+}R_{+}^{∗}, i.e. wf ⪯ f. □

2.4.7. If Γ is (uniquely) divisible, the weak and strong dominance order coincide. Moreover, for any f ∈ Hom^{+}(M, Γ), lemma 29 implies that f^{♭} ≤ f where

f^{♭} = \frac{1}{♯W f} ∑_{f′∈W f}f′ ∈ Hom(M, Γ)^{W} ⊂ Hom^{+}(M, Γ).

Thus Hom(M, Γ)^{W} is then precisely the set of minimal elements in Hom^{+}(M, Γ). For Γ = ℤ and ℛ reduced, semi-simple and adjoint (i.e. ℤR = M), the strong dominance order on Hom^{+}(M, ℤ) is studied in [45]. Its minimal elements are the linear forms f : M → ℤ such that f(R_{+}) ∈ {0, 1}.

2.4.8. Applying lemma 29 to the dual root system ℛ^{∗} with Γ = ℤ, we obtain:

(1) Let M′ be the kernel of the coinvariant map M ↠ M_{W}. Then M′ ⊂ ℤR. Since also α ≡ −α in M_{W} for every α ∈ R, actually 2ℤR ⊂ M′ ⊂ ℤR, therefore M′ and ℤR have the same saturation in M. And: (2) For every m ∈ M,

m ∈ M_{d} ⟺ ∀w ∈ W : m − wm ∈ ℕR_{+} (or: (ℕR_{+})_{sat}).

Returning to the original root system ℛ and the general Γ, we thus find:

Hom(M, Γ)^{W} = Hom(M/M′, Γ) = Hom(M/ℤR, Γ) = Hom(M/(ℤR)_{sat}, Γ)

and for every f ∈ Hom^{+}(M, Γ) and m ∈ M_{d},

f(m) = max f(W m) in Γ.

2.4.9. This last property yields the following characterisation of the restriction of the weak order to the cone Hom^{+}(M, Γ). Any morphism f : M → Γ induces a ring homomorphism f : ℤ[M] → ℤ[Γ]. For x ∈ ℤ[Γ], we denote by max(x) ∈ Γ the largest element in the finite support of x if x ≠ 0, and set max(0) = 0. Then:

Lemma 30. For f_{1}, f_{2} ∈ Hom^{+}(M, Γ),

f_{1} ≤ f_{2} ⟺ ∀x ∈ ℕ[M]^{W} ,: max (f_{1}(x)) ≤ max (f_{2}(x)) .

Proof. Since M_{d} is a fundamental domain for the action of W on M,

ℕ[M]^{W} = {x = ∑_{m∈M_{d}} x_{m}e_{m} : x_{m} ∈ ℕ, {x_{m} ≠ 0} finite}

where e_{m} = ∑_{m′∈W m} m′. For any f : M → Γ and x ∈ ℕ[M]^{W} with x ≠ 0, f(x) is also nonzero with support ∪_{m∈M_{d},x_{m}≠0}f(W m). Thus if f is moreover dominant,

max (f(x)) = max {f(m) : m ∈ M_{d}, x_{m} ≠ 0} .

The lemma easily follows. □

2.4.10. Let ℛ_{ss} = (M_{ss}, R_{ss}, M_{ss}^{∗}, R_{ss}^{∗}) be the semi-simplification of ℛ, as defined in [21, XXI 6.5]. Thus M_{ss} = ℤR_{sat}, R_{ss} = R, M_{ss}^{∗} is the dual of M_{ss} and R_{ss}^{∗} is the image of R^{∗} under the transpose map M^{∗} ↠ M_{ss}^{∗}. The restriction map f ↦ f_{ss} = f|M_{ss} yields an epimorphism Hom(M, Γ) ↠ Hom(M_{ss}, Γ) with kernel Hom(M, Γ)^{W}, inducing epimorphism of monoids Hom^{+}(M, Γ) ↠ Hom^{+}(M_{ss}, Γ) and Γ_{+}R_{+}^{∗} ↠ Γ_{+}R_{ss,+}^{∗} – the former is therefore also compatible with the weak and strong partial orders. If Γ is divisible, the average map f ↦ f^{♭} of section 2.4.7 gives a retraction of Hom(M, Γ)^{W} ↪ Hom(M, Γ), and it follows that f ↦ (f_{ss}, f^{♭}) yields an isomorphism of partially ordered monoids

Hom^{+}(M, Γ) ≃ Hom^{+}(M_{ss}, Γ) × Hom(M, Γ)^{W} .

The (weak=strong) partial order on the product is then given by

(f_{ss}, f^{♭}) ≤ (g_{ss}, g^{♭}) ⟺ f_{ss} ≤ g_{ss} and f^{♭} = g^{♭}.

2.4.11. If Γ = ℝ, then Hom^{+}(M, ℝ) is a closed cone in the finite dimensional ℝ-vector space Hom(M, ℝ). The (weak=strong) dominance partial order then has the following intrinsic characterisation: for every f_{1}, f_{2} ∈ Hom^{+}(M, ℝ),

f_{1} ≤ f_{2} ⟺ f_{1} lies in the convex hull of W · f_{2}.

Indeed, suppose first that f_{1} ≤ f_{2}. If f_{1} does not belong to the convex hull of W ·f_{2}, there is a linear form F on Hom(M, ℝ) such that F (f_{1}) > F (wf_{2}) for every w ∈ W, which means that there is an x in M ⊗ℝ such that f_{1}(x) > f_{2}(wx) for every w ∈ W. Since M ⊗ ℚ is dense in ℝ, we may assume that x ∈ M ⊗ ℚ, and then rescaling that actually x ∈ M. Let y = wx be the unique element in W x ∩ M_{d} (2.4.2). Then f_{1}(y) ≥ f_{1}(x) since f_{1} ∈ Hom^{+}(M, ℝ) (2.4.8) and f_{1}(x) > f_{2}(y) by construction, thus f_{1}(y) > f_{2}(y) with y ∈ M_{d}, a contradiction. Suppose conversely that

f_{1} = _{∑w∈W} λ_{w}wf_{2} in Hom(M, ℝ) with λ_{w} ∈ [0, 1], ∑_{w∈W} λ_{w} = 1.

Since f_{2} ∈ Hom^{+}(M, ℝ), wf_{2} ≤ f_{2} for every w ∈ W by lemma 29, thus

f_{1} = _{∑w∈W} λ_{w}wf_{2} ≤ _{∑w∈W} λ_{w}f_{2} = f_{2} in Hom(M, ℝ).

2.4.12. The partial orders on W \Hom(M, Γ) which are induced by the restriction of ⪯ and ≤ to the fundamental domain Hom^{+}(M, Γ) ≃ W \Hom(M, Γ) of lemma 28 do not depend upon the chosen system of positive roots R^{+} – indeed, all such systems are conjugated under W. The weak order even does not depend upon the root system giving rise to W: any orbit [f] ∈ W \Hom(M, Γ) yields a well-defined function [f] : ℤ[M]^{W} → ℤ[Γ], and for every [f_{1}], [f_{2}] ∈ W \Hom(M, Γ),

[f_{1}] ≤ [f_{2}] ⟺ ∀x ∈ ℕ[M]^{W} : max[f_{1}](x) ≤ max[f_{2}](x)

by lemma 30. The strong order moreover depends upon ΓR^{∗}:

[f_{1}] ⪯ [f_{2}] ⟺ [f_{1}] ≤ [f_{2}] and π[f_{1}] = π[f_{2}]

where π : W \Hom(M, Γ) ↠ Hom(M, Γ)/ΓR^{∗} is the projection from lemma 29.

2.4.13. We record here a technical result comparing the partial orders attached to, respectively, the relative and absolute root systems of a reductive group G over a field k. Let S a maximal split torus in G, T a maximal torus in the centralizer Z_{G}(S) of S, k^{s} a separable closure of k, Gal_{k} = Gal(k^{s}/k). Denote by

\overline{ℛ} = ℛ (G_{k^{s}}, T_{k^{s}}) = (\overline{M}, \overline{R}, \overline{M}^{∗}, \overline{R}^{∗}) and ℛ = ℛ(G, S) = (M, R, M^{∗}, R^{∗})

the absolute and relative root systems [21, XXVI 7.12], with Weyl groups

\overline{W} = W (\overline{ℛ} ) = W (G_{k^{s}}, T_{k^{s}}) and W = W (ℛ) = W (G, S).

We will also consider the subgroups \overline{W} _{S}^{0} ⊂ \overline{W}_{S} ⊂ \overline{W} defined by

N_{G}(T) ∩ Z_{G}(S)/Z_{G}(T) ⊂ N_{G}(T) ∩ N_{G}(S)/Z_{G}(T) ⊂ N_{G}(T)/Z_{G}(T).

The embedding S ↪ T induces a pair of dual morphisms

[figure: \overline{M} × \overline{M}^{∗}; res; ⟨−,−⟩; ℤ; M × M^{∗}; _{res_{∗}}; ⟨−,−⟩; ℤ]

with R ⊂ res(\overline{R}) ⊂ R ∪ \{0\}. Set \overline{R}(0) = \{\overline{α} ∈ \overline{R} : res(\overline{α}) = 0\}, so that

\overline{ℛ} (0) = ℛ (Z_{G}(S)_{k^{s}}, T_{k^{s}}) = (\overline{M}, \overline{R}(0), \overline{M}^{∗}, \overline{R}(0)^{∗})

is the absolute root system of Z_{G}(S), with Weyl group

\overline{W} _{S}^{0} = W (\overline{ℛ} (0)) = W (Z_{G}(S)_{k^{s}}, T_{k^{s}}) .

By [5, 5.5], the natural map \overline{W}_{S} → W identifies W with \overline{W}_{S}/\overline{W} _{S}^{0}. Since the restriction res : \overline{M} ↠ M is equivariant with respect to \overline{W}_{S} ↠ W, it induces a map

W \Hom (M, Γ) ↪ \overline{W}_{S}\Hom (\overline{M}, Γ) ↠ \overline{W} \Hom (\overline{M}, Γ) .

Proposition 31. The map W \Hom (M, Γ) → \overline{W} \Hom (\overline{M}, Γ) is injective. Moreover for any f, g ∈ W \Hom (M, Γ) with image \overline{f}, \overline{g} ∈ \overline{W} \Hom (\overline{M}, Γ),

f ⪯ g ⟹ \overline{f} ⪯ \overline{g} ⟹ \overline{f} ≤ \overline{g} ⟺ f ≤ g.

Remark 32. We do not know if \overline{f} ⪯ \overline{g} implies f ⪯ g. The proof given below relates this to the following question. Recall that G is simply connected if and only if ℤ\overline{R}^{∗} = \overline{M}^{∗}. Is it true that then also ℤR^{∗} = M^{∗}? The dual question has a positive answer: G is adjoint if and only if ℤ\overline{R} = \overline{M}, in which case also ℤR = M.

2.4.14. Let G_{der} ⊂ G be the derived group of G [21, XXII 6.2], π : G_{sc} ↠ G_{der} the simply connected cover of G_{der} [13, A.4.11]. Then T_{der} = T ∩ G_{der} is a maximal torus in G_{der} [21, XXII 6.2.8] and T_{sc} = π^{−1}(T_{der}) is a maximal torus in G_{sc} [1, XVII 7.1.1]. Let S_{der} ⊂ T_{der} and S_{sc} ⊂ T_{sc} be their maximal split subtori and denote by ℛ_{der}, ℛ_{sc}, \overline{ℛ} _{der} and \overline{ℛ} _{sc} the corresponding relative and absolute root systems. By [21, XXII 6.2.7] and the definition of G_{sc}, the morphisms T_{sc} ↠ T_{der} ↪ T induce compatible bijections \overline{R} ≃ \overline{R}_{der} ≃ \overline{R}_{sc} and \overline{R}_{sc}^{∗} ≃ \overline{R}_{der}^{∗} ≃ \overline{R}^{∗}. They also induce morphisms S_{sc} ↠ S_{der} ↪ S, and S = S_{der} · R(G)_{sp} since T = T_{der} · R(G) [21, XXII 6.2.8], where R(G)_{sp} is the maximal split subtorus of the radical R(G) of G. Since R, R_{der} and R_{sc} are the nonzero restrictions of the elements of \overline{R}, \overline{R}_{der} and \overline{R}_{sc} to respectively S, S_{der} and S_{sc}, it follows that our morphisms also induce bijections R ≃ R_{der} ≃ R_{sc}. Finally, the morphisms G_{sc} ↠ G_{der} ↪ G induce embeddings W_{sc} ↪ W_{der} ↪ W between the Weyl groups of the maximal tori S_{sc} ⊂ G_{sc}, S_{der} ⊂ G_{der} and S ⊂ G. It then follows from the unicity of the relative coroots, or from their actual construction in [21, XXVI 7.4], that S_{sc} ↠ S_{der} ↪ S also induces compatible bijections R_{sc}^{∗} ≃ R_{der}^{∗} ≃ R^{∗} (and the Weyl groups maps are bijective). Since composition with S_{sc} ↪ T_{sc} maps ℤR_{sc}^{∗} into X_{∗}(T_{sc}) = ℤ\overline{R}_{sc}^{∗}, we obtain

res^{∗}(ℤR^{∗}) ⊂ ℤ\overline{R}^{∗}.

2.4.15. Fix a minimal parabolic subgroup Z_{G}(S) ⊂ P ⊂ G, a Borel subgroup T_{k^{s}} ⊂ B ⊂ P_{k^{s}} ⊂ G_{k^{s}}, let Δ ⊂ R_{+} ⊂ R and \overline{Δ} ⊂ \overline{R}_{+} ⊂ \overline{R} be the corresponding simple and positive roots, \overline{Δ}(0) ⊂ \overline{R}(0)_{+} ⊂ \overline{R}(0) the simple and positive roots attached to Z_{G}(S)_{k^{s}} ∩ B, so that \overline{Δ}(0) = \overline{Δ} ∩ \overline{R}(0), \overline{R}(0)_{+} = \overline{R}_{+} ∩ \overline{R}(0),

R_{+} ⊂ res (\overline{R}_{+}) ⊂ R_{+} ∪ \{0\} and Δ ⊂ res (\overline{Δ}) ⊂ Δ ∪ \{0\}.

In particular, the morphism res_{Γ}^{∗} : Hom (M, Γ) ↪ Hom (\overline{M}, Γ) maps Hom^{+} (M, Γ) to Hom^{+} (\overline{M}, Γ). The first assertion of Proposition 31 thus follows from Lemma 28.

For the remaining claims, we have to establish the following inclusions:

Γ_{+}R_{+}^{∗} ⊂ (res_{Γ}^{∗})^{−1} (Γ_{+}\overline{R}_{+}^{∗}) ⊂ (res_{Γ}^{∗})^{−1} ((Γ_{+}\overline{R}_{+}^{∗})_{sat}) = (Γ_{+}R_{+}^{∗})_{sat} .

We may assume that Γ = ℤ, in which case res_{Γ}^{∗} = res^{∗} : M^{∗} ↪ \overline{M}^{∗} and we want:

ℕR_{+}^{∗} ⊂ (res^{∗})^{−1} (ℕ\overline{R}_{+}^{∗}) ⊂ (res^{∗})^{−1} ((ℕ\overline{R}_{+}^{∗})_{sat}) = (ℕR_{+}^{∗})_{sat} .

The central inclusion is obvious. Since we already know that res^{∗}(ℤR^{∗}) ⊂ ℤ\overline{R}^{∗} and

ℕR_{+}^{∗} = ℤR^{∗} ∩ (ℕR_{+}^{∗})_{sat} , ℕ\overline{R}_{+}^{∗} = ℤ\overline{R}^{∗} ∩ (ℕ\overline{R}_{+}^{∗})_{sat}

it only remains to establish the following lemma.

Lemma 33. With notations as above,

(res^{∗})^{−1} ((ℕ\overline{R}_{+}^{∗})_{sat}) = (ℕR_{+}^{∗})_{sat} and (res (\overline{M}_{d}))_{sat} = M_{d}.

Note that the second formula follows from the first one by duality.

2.4.16. The dual and bidual cones of the coroot cones ℕR_{+}^{∗} and ℕ\overline{R}_{+}^{∗} are respectively equal to the cones of dominant weights M_{d} and \overline{M}_{d}, and to their own saturations (ℕR_{+}^{∗})_{sat} and (ℕ\overline{R}_{+}^{∗})_{sat}. By 2.4.8, the restriction map res : \overline{M} ↠ M sends \overline{M}_{d} into M_{d}. Indeed for m = res(\overline{m}) with \overline{m} ∈ \overline{M}_{d} and any w ∈ W lifting to \overline{w} ∈ \overline{W}_{S}, m − wm = res(\overline{m} − \overline{w}\overline{m}) belongs to ℕR_{+} since \overline{m} − \overline{w}\overline{m} belongs to ℕ\overline{R}_{+}. Passing to the bidual cones, we thus obtain the easiest inclusion:

res^{∗} (ℕR_{+}^{∗})_{sat} ⊂ (ℕ\overline{R}_{+}^{∗})_{sat} .

2.4.17. For the opposite inclusion, we will need a few more notations:

(1) Recall from [21, XXI 1.2.1] that the formulas

p(x) = _{∑\overline{α}∈\overline{R}} ⟨x, \overline{α}^{∗}⟩ \overline{α}^{∗} and ℓ(x) = ⟨x, p(x)⟩ (x ∈ \overline{M})

define a morphism p : \overline{M} → \overline{M}^{∗} and a map ℓ : \overline{M} → ℕ such that

∀\overline{α} ∈ \overline{R} : ℓ(\overline{α}) > 0 and 2p(\overline{α}) = ℓ(\overline{α})\overline{α}^{∗},

(2) The Galois group Gal_{k} acts on \overline{M}, \overline{M}^{∗}, \overline{W}, \overline{W}_{S} and \overline{W} _{S}^{0}, the morphisms res and res^{∗} are \overline{W}_{S} ⋊ Gal_{k}-equivariant, the latter identifies M^{∗} with (\overline{M}^{∗})^{Gal^{k}}, the subset \overline{R} ⊂ \overline{M} and \overline{R}^{∗} ⊂ \overline{M}^{∗} are \overline{W} ⋊ Gal_{k}-stable, and

∗ : \overline{R} → \overline{R}^{∗}, p : \overline{M} → \overline{M}^{∗}, ℓ : \overline{M} → ℕ

are also \overline{W} ⋊ Gal_{k}-equivariant (with the trivial action on ℕ).

(3) For every γ ∈ Gal_{k}, there is a unique w_{γ} ∈ \overline{W} _{S}^{0} such that

w_{γ}γ\overline{R}(0)_{+} = \overline{R}(0)_{+}

by [21, XXI 3.3.7], in which case also w_{γ}γ\overline{R}_{+} = \overline{R}_{+} since

\overline{R}_{+} \ \overline{R}(0)_{+} = \overline{R} ∩ res^{−1}(R_{+})

is already stable under \overline{W} _{S}^{0} ⋊ Gal_{k}. The twisted action of Gal_{k} on \overline{ℛ} [5, 6.2] is given by γ· = w_{γ}γ. The above maps are equivariant for the twisted action, which moreover preserves \overline{Δ} and \overline{Δ}(0). For every α ∈ Δ, the twisted action is transitive on \overline{Δ}(α) = res^{−1}(α) ∩ \overline{Δ} by [5, 6.4.2 & 6.8].

(4) A root \overline{α} ∈ \overline{R} maps to α ∈ Δ if and only if it is the sum of a (unique) simple root \overline{δ} ∈ \overline{Δ}(α) and some element of ℕ\overline{R}(0)_{+}. This yields a partition of \overline{R}(α) = \overline{R} ∩ res^{−1}(α) indexed by \overline{Δ}(α), whose parts are permuted transitively by the twisted action of Gal_{k}. It follows that

∑_{\overline{α}∈\overline{R}(α)}\overline{α} = n_{α} · ∑_{\overline{δ}∈\overline{Δ}(α)}\overline{δ} + α̃_{0} in \overline{M}

with n_{α} ∈ ℕ^{×} and α̃_{0} ∈ ℕ\overline{R}(0)_{+}. Applying the morphism 2p, we obtain

∑_{\overline{α}∈\overline{R}(α)}ℓ(\overline{α})\overline{α}^{∗} = n_{α}ℓ_{α} · ∑_{\overline{δ}∈\overline{Δ}(α)}\overline{δ}^{∗} + 2p(α̃_{0}) in \overline{M}^{∗}

with p(α̃_{0}) ∈ ℕ\overline{R}(0)_{+}^{∗} and ℓ_{α} = ℓ(\overline{δ}) ∈ ℕ^{×} for any \overline{δ} ∈ \overline{Δ}(α).

2.4.18. Fix α ∈ Δ and change (G, S, T, P, B) to (H, S, T, P ∩ H, B ∩ H_{k^{s}}) where H is the unique reductive subgroup of G containing Z_{G}(S) with

Lie(H) = Lie(G)_{0} ⊕ ⊕_{β∈ℤα∩R}Lie(G)_{β}

This changes our absolute and relative based root data to respectively

(\overline{M}, \overline{R}_{α}, \overline{M}^{∗}, \overline{R}_{α}^{∗}; \overline{Δ}(α) ∪ \overline{Δ}(0)) and (M, ℤα ∩ R, M^{∗}, (ℤα ∩ R)^{∗}; \{α\})

where \overline{R}_{α} = \{\overline{β} ∈ \overline{R} : res(\overline{β}) ∈ ℤα\}. We thus already know that

res^{∗} (ind(α^{∗})) = _{∑\overline{δ}∈\overline{Δ}(α)}λ\overline{δ}\overline{δ}^{∗} + α_{0}^{∗} in ℕ\overline{R}_{+} ⊂ \overline{M}^{∗}

with λ\overline{δ} ∈ ℕ and α_{0}^{∗} ∈ ℕ\overline{R}(0)_{+}^{∗}. Since res^{∗}(ind(α^{∗})) is fixed by the twisted action, the coefficient map \overline{δ} ↦ λ\overline{δ} is constant on the (twisted) Gal_{k}-orbit Δ(α), thus

(2.4.1)  res^{∗} (ind(α^{∗})) = λ_{α} ·_{∑\overline{δ}∈\overline{Δ}(α)}\overline{δ}^{∗} + α_{0}^{∗} in \overline{M}^{∗}

with λ_{α} ∈ ℕ, therefore also

n_{α}ℓ_{α} · res^{∗} (ind(α^{∗})) = λ_{α} · ∑_{\overline{α}∈\overline{R}(α)}ℓ(\overline{α})\overline{α}^{∗} + (n_{α}ℓ_{α} · α_{0}^{∗} − 2λ_{α} · p(α̃_{0})) .

Since res^{∗} (ind(α^{∗})) and ∑_{\overline{α}∈\overline{R}(α)}ℓ(\overline{α})\overline{α}^{∗} are fixed by the usual (untwisted) action of Gal_{k} on \overline{M}^{∗}, so is the remaining term, which thus belongs to res^{∗}(M^{∗}). But

res^{∗}(M^{∗}) ∩ ℤ\overline{R}(0)^{∗} = 0

since any element of res^{∗}(M^{∗}) pairs trivially with all of \overline{R}(0) while the restriction of the pairing \overline{M} × \overline{M}^{∗} → ℤ to ℤ\overline{R}(0) × ℤ\overline{R}(0)^{∗} is non-degenerate by [21, XXI 1.2.5]. We thus obtain the following equalities in \overline{M}^{∗}: n_{α}ℓ_{α} · α_{0}^{∗} = 2λ_{α} · p(α̃_{0}) and

n_{α}ℓ_{α} · res^{∗} (ind(α^{∗})) = λ_{α} · ∑_{\overline{α}∈\overline{R}(α)}ℓ(\overline{α})\overline{α}^{∗}.

In particular, λ_{α} ∈ ℕ^{×} since res^{∗} (ind(α^{∗})) ≠ 0.

2.4.19. Suppose now that x ∈ M^{∗} ⊗ ℚ is such that

res^{∗}(x) = _{∑\overline{δ}∈\overline{Δ}}y\overline{δ} · \overline{δ}^{∗} with y\overline{δ} ∈ ℚ_{+}.

Since the left hand side is invariant under the twisted action of Gal_{k},

res^{∗}(x) = ∑_{α∈Δ}y_{α} · ∑_{\overline{δ}∈\overline{Δ}(α)} \overline{δ}^{∗} + x_{0}^{∗} with y_{α} ∈ ℕ, x_{0}^{∗} ∈ ℚ_{+}\overline{R}(0)_{+}^{∗}.

Using (2.4.1) and ℚ\overline{R}(0)^{∗} ∩ res^{∗}(M^{∗} ⊗ ℚ) = 0, we obtain

(x_{0}^{∗} − ∑_{α∈Δ}y_{α}λ_{α}^{−1} · α_{0}^{∗}) = res^{∗} (x − ∑_{α∈Δ}y_{α}λ_{α}^{−1} · ind(α^{∗})) = 0

2.4. INTERLUDE ON THE DOMINANCE PARTIAL ORDERS 33

thus x = ∑_{α∈Δ} y_{α}λ_{α}^{−1} · ind(α^{∗}) belongs to ℚ_{+}R_{+}. It follows that

(res^{∗})^{−1} ((ℕ\overline{R}_{+}^{∗})_{sat}) ⊂ (ℕR_{+}^{∗})_{sat} in M^{∗},

which completes the proof of lemma 33 and proposition 31.

CHAPTER 3 The Tannakian formalism

Let G be an affine and flat group scheme over S and let Γ = (Γ, +, ≤) be a non-trivial, totally ordered commutative group. We will define below an equivariant diagram of fpqc sheaves (Sch/S)^{∘} → Group or (Sch/S)^{∘} → Set:

\xymatrix{G \ar[d]^{ι} & acting on & 𝔾^{Γ}(G) \ar[d]^{ι} \ar[r]^{Fil} & 𝔽^{Γ}(G) \ar[d]^{ι} \\ Aut^{⊗}(V) \ar[d] & ⋯ & 𝔾^{Γ}(V) \ar[d] \ar[r]^{Fil} & 𝔽^{Γ}(V) \ar[d] \\ Aut^{⊗}(V^{∘}) or Aut^{⊗}(ω) \ar[d] & ⋯ & 𝔾^{Γ}(V^{∘}) or 𝔾^{Γ}(ω) \ar[d] \ar[r]^{Fil} & 𝔽^{Γ}(V^{∘}) or 𝔽^{Γ}(ω) \ar[d] \\ Aut^{⊗}(ω^{∘}) & ⋯ & 𝔾^{Γ}(ω^{∘}) \ar[r]^{Fil} & 𝔽^{Γ}(ω^{∘})}

The main result of this chapter will then be the following theorem:

Theorem 34. If G is a reductive group over S, then

G  = Aut^{⊗}(V) = Aut^{⊗}(V^{∘}) = Aut^{⊗}(ω)
𝔾^{Γ}(G) = 𝔾^{Γ}(V) = 𝔾^{Γ}(V^{∘}) = 𝔾^{Γ}(ω)
𝔽^{Γ}(G) = 𝔽^{Γ}(V) = 𝔽^{Γ}(V^{∘}) ⊂ 𝔽^{Γ}(ω)

If moreover G is isotrivial and S quasi-compact, then also

G = Aut^{⊗}(ω^{∘}), 𝔾^{Γ}(G) = 𝔾^{Γ}(ω^{∘}) and 𝔽^{Γ}(G) = 𝔽^{Γ}(ω) = 𝔽^{Γ}(ω^{∘}).

More precisely, we will first show that for any affine flat group scheme G over S,

G  = Aut^{⊗}(V) = Aut^{⊗}(ω)
𝔾^{Γ}(G) = 𝔾^{Γ}(V) = 𝔾^{Γ}(ω)
𝔽^{Γ}(G) ⊂ 𝔽^{Γ}(V) ⊂ 𝔽^{Γ}(ω)

Then, under technical assumptions which are satisfied by all reductive groups (resp. all isotrivial reductive groups over quasi-compact bases), we will also establish that

Aut^{⊗}(V) = Aut^{⊗}(V^{∘}) 𝔾^{Γ}(V) = 𝔾^{Γ}(V^{∘}) 𝔽^{Γ}(V) ⊂ 𝔽^{Γ}(V^{∘}) (resp. \begin{matrix}Aut^{⊗}(ω) & = & Aut^{⊗}(ω^{∘}) \\ 𝔾^{Γ}(ω) & = & 𝔾^{Γ}(ω^{∘}) \\ 𝔽^{Γ}(ω), 𝔽^{Γ}(V^{∘}) & ⊂ & 𝔽^{Γ}(ω^{∘}) \end{matrix} ) .

We will finally show that for G reductive and isotrivial over a quasi-compact S, the morphism 𝔾^{Γ}(G) → 𝔽^{Γ}(ω^{∘}) is an epimorphism of fpqc sheaves on S. Thus

𝔽^{Γ}(G) = 𝔽^{Γ}(V) = 𝔽^{Γ}(V^{∘}) = 𝔽^{Γ}(ω) = 𝔽^{Γ}(ω^{∘})

in this case, and the remaining statement, namely

𝔽^{Γ}(G) = 𝔽^{Γ}(V) = 𝔽^{Γ}(V^{∘})

for a reductive group G over an arbitrary S easily follows.

Remark 35. As will be clear from the definitions below, the assertions about ω^{∘} and V correspond to the two extreme cases of a variety of possible statements about filtrations on fiber functors. These cases were not clearly distinguished in [41, Chapitre IV], which lead us to revisit its proofs. Our definition of isotriviality for reductive groups in section 3.6.3 is taylor-made to fit the ω^{∘}-case: it is not even local for the Zariski topology on the base. The corresponding Zariski-local notion was defined in [21, XXIV 4.1.2]. For a locally isotrivial reductive group, the above theorem works with a suitably (Zariski) localized version of the fiber functor ω^{∘}.

3.1. Γ-graduations and Γ-filtrations on quasi-coherent sheaves

3.1.1. Let M be a quasi-coherent sheaf on a scheme X.

Definition 36. A Γ-graduation on M is a collection G = (G_{γ})_{γ∈Γ} of quasicoherent subsheaves of M such that M = ⊕_{γ∈Γ}G_{γ}. A Γ-filtration on M is a collection F = (F^{γ})_{γ∈Γ} of quasi-coherent subsheaves of M such that, locally on X for the fpqc topology, there exists a Γ-graduation G = (G_{γ})_{γ∈Γ} on M for which F^{γ} = ⊕_{η≥γ}G_{η}. We call any such G a splitting of F and write F = Fil(G). We set

F_{+}^{γ} = ∪_{η>γ}F^{η} and Gr_{F}^{γ}M = F^{γ}/F_{+}^{γ}.

Lemma 37. Let F be a Γ-filtration on M. Then γ ↦ F^{γ} is non-increasing, exhaustive (∪F^{γ} = M), separated (∩F^{γ} = 0), and for every γ ∈ Γ,

0 → F^{γ} → M → M/F^{γ} → 0 and 0 → F_{+}^{γ} → F^{γ} → Gr_{F}^{γ}(M) → 0

are pure exact sequences of quasi-coherent sheaves (see 3.13).

Proof. This is local in the fpqc topology on X, trivial if F has a splitting. □

3.1.2. These definitions give rise to a diagram of fpqc stacks over Sch

\xymatrix{Gr^{Γ}QCoh \ar[r]^{Fil} & Fil^{Γ}QCoh \ar[l]^{Gr} \ar[r]^{forg} & QCoh}

whose fiber over a scheme X is the diagram of exact ⊗-functors

\xymatrix{Gr^{Γ}QCoh(X) \ar[r]^{Fil} & Fil^{Γ}QCoh(X) \ar[l]^{Gr} \ar[r]^{forg} & QCoh(X)}

where QCoh(X) is the abelian ⊗-category of quasi-coherent sheaves M on X, Gr^{Γ}QCoh(X) is the abelian ⊗-category of Γ-graded quasi-coherent sheaves (M, G) on X, and Fil^{Γ}QCoh(X) is the exact (in Quillen’s sense) ⊗-category of Γ-filtered quasi-coherent sheaves (M, F) on X. The morphisms in these last two categories are the morphisms of the underlying quasi-coherent sheaves which preserve the given collections of subsheaves, and the ⊗-products are given by the usual formulas

(M_{1}, G_{1}) ⊗ (M_{2}, G_{2}) = (M_{1} ⊗ M_{2}, G) with G_{γ} = ⊕_{γ_{1}+γ_{2}=γ}G_{1,γ_{1}} ⊗ G_{2,γ_{2}},
(M_{1}, F_{1}) ⊗ (M_{2}, F_{2}) = (M_{1} ⊗ M_{2}, F) with F^{γ} = ∑_{γ_{1}+γ_{2}=γ} F_{1}^{γ_{1}} ⊗ F_{2}^{γ_{2}}.

The second formula makes sense by the purity mentioned above, and indeed defines a Γ-filtration on M_{1} ⊗ M_{2}: if G_{i} splits F_{i} for i ∈ {1, 2}, then G splits F. We have

F_{+}^{γ} = _{∑γ_{1}+γ_{2}>γ}F_{1}^{γ_{1}} ⊗ F_{2}^{γ_{2}}
and Gr_{F}^{γ}(M_{1} ⊗ M_{2}) ≃ ⊕_{γ_{1}+γ_{2}=γ}Gr_{F_{1}}^{γ_{1}}(M_{1}) ⊗ Gr_{F_{2}}^{γ_{2}}(M_{2}).

The first formula is trivial and gives the morphism (from right to left) in the second formula, which is easily seen to be an isomorphism by localization to an fpqc cover of X over which F_{1} and F_{2} both acquire a splitting. The neutral objects for ⊗ are

1_{X} = (O_{X}, G of F) with G_{γ} = \begin{cases}O_{X} & for γ = 0, \\ 0 & otherwise \end{cases} and F^{γ} = \begin{cases}O_{X} & for γ ≤ 0, \\ 0 & otherwise. \end{cases}

A morphism (M_{1}, F_{1}) → (M_{2}, F_{2}) is strict if Im(F_{1}^{γ}) = F_{2}^{γ} ∩ Im(M_{1}) in M_{2} for every γ ∈ Γ. The short exact sequences of Fil^{Γ}QCoh(X) are those made of strict arrows whose underlying sequence of sheaves is short exact. The formulas

Fil(M, G) = (M, Fil(G)), Gr(M, F) = ⊕_{γ}Gr_{F}^{γ}M and forg(M, −) = M

define the exact ⊗-functors between our three categories. Finally the “base change functors” defining the fibered category structures on Gr^{Γ}QCoh and Fil^{Γ}QCoh are induced by the base change functors on QCoh (thanks to the purity of the subsheaves). It is well-known that QCoh is an fpqc stack over Sch (see for instance [48, Theorem 4.23]) and it follows rather formally from their definitions that the other two fibered categories are also fpqc stacks over Sch. We denote by

\xymatrix{Gr^{Γ}QCoh/S \ar[r]^{Fil} & Fil^{Γ}QCoh/S \ar[l]^{Gr} \ar[r]^{forg} & QCoh} /S

the corresponding stacks over Sch/S where S is any base scheme.

3.2. Γ-graduations and Γ-filtrations on fiber functors

3.2.1. Let s : G → S be an affine and flat group scheme. We denote by Rep(G) the fpqc stack over Sch/S whose fiber over T → S is the abelian ⊗-category Rep(G)(T) of quasi-coherent G_{T} -O_{T} -modules as defined in [20, I 4.7.1]. Then

A(G) = s_{∗}O_{G}

is a quasi-coherent Hopf algebra over S and Rep(G)(T) is ⊗-equivalent to the category of quasi-coherent A(G_{T})-comodules where A(G_{T}) = A(G)_{T}. Let

V : Rep(G) → QCoh/S

be the forgetful functor. For any S-scheme q : T → S, we denote by

V_{T} : Rep(G_{T}) → QCoh/T and ω_{T} : Rep(G)(S) → QCoh(T)

the induced morphism of fpqc stack over Sch/T and fiber functor. Note that ω_{T} is a right exact ⊗-functor. It also commutes with arbitrary colimits and preserves pure monomorphisms and pure short exact sequences, where purity in Rep(G)(S) refers to purity of the underlying objects in QCoh(S).

3.2.2. A Γ-graduation G on V_{T} : Rep(G_{T}) → QCoh/T is a factorization

\xymatrix{Rep(G_{T}) \ar[r]^{G} & Gr^{Γ}QCoh/T \ar[r]^{forg} & QCoh/T}

of V_{T} such that if G_{γ} : Rep(G_{T}) → QCoh/T is the γ-component of G, (G0) For every T -morphism f : X → Y, ρ ∈ Rep(G)(Y) and γ ∈ Γ,

f^{∗}(G_{γ}(ρ)) = G_{γ}(f^{∗}ρ).

(G1) For every T -scheme X → T, ρ_{1}, ρ_{2} ∈ Rep(G)(X) and γ ∈ Γ,

G_{γ}(ρ_{1} ⊗ ρ_{2}) = ⊕_{γ_{1}+γ_{2}=γ}G_{γ_{1}}(ρ_{1}) ⊗ G_{γ_{2}}(ρ_{2}).

Thus (G0) says that each G_{γ} is a morphism of fibered categories over Sch/T. Then (G1) implies that G_{0}(ρ) = M and G_{γ}(ρ) = 0 for γ ≠ 0 when ρ is the trivial representation of G_{X} on M ∈ QCoh(X) (one proves it first for M = O_{X}).

3.2.3. A Γ-graduation G on ω_{T} : Rep(G)(S) → QCoh(T) is a factorization

\xymatrix{Rep(G)(S) \ar[r]^{G} & Gr^{Γ}QCoh(T) \ar[r]^{forg} & QCoh(T)}

of ω_{T} such that if G_{γ} : Rep(G)(S) → QCoh(T) is the γ-component of G, (G1) For every ρ_{1}, ρ_{2} ∈ Rep(G)(S) and γ ∈ Γ,

G_{γ}(ρ_{1} ⊗ ρ_{2}) = ⊕_{γ_{1}+γ_{2}=γ}G_{γ_{1}}(ρ_{1}) ⊗ G_{γ_{2}}(ρ_{2}).

(G2) For the trivial representation ρ of G on M ∈ QCoh(S),

G_{0}(ρ) = M and G_{γ}(ρ) = 0 if γ ≠ 0.

Note that each G_{γ} is right exact, commutes with arbitrary colimits and preserves pure monomorphisms and pure short exact sequences.

3.2.4. A Γ-filtration F on V_{T} : Rep(G_{T}) → QCoh/T is a factorization

\xymatrix{Rep(G_{T}) \ar[r]^{F} & Fil^{Γ}QCoh/T \ar[r]^{forg} & QCoh/T}

of V_{T} such that if F^{γ} : Rep(G_{T}) → QCoh/T is the γ-component of F, (F0) For every T -morphism f : X → Y, ρ ∈ Rep(G)(Y) and γ ∈ Γ,

f^{∗}(F^{γ}(ρ)) = F^{γ}(f^{∗}ρ).

(F1) For every X → T, ρ_{1}, ρ_{2} ∈ Rep(G)(X) and γ ∈ Γ,

F^{γ}(ρ_{1} ⊗ ρ_{2}) = _{∑γ_{1}+γ_{2}=γ}F^{γ_{1}}(ρ_{1}) ⊗ F^{γ_{2}}(ρ_{2}).

(F3) For every X → T and γ ∈ Γ, F^{γ} : Rep(G)(X) → QCoh(X) is exact. Thus (F0) says that each F^{γ} is a morphism of fibered categories over Sch/T. Then again (F1) and (F3) imply that F^{γ}(ρ) = M for γ ≤ 0 and F^{γ}(ρ) = 0 for γ > 0 when ρ is the trivial representation of G on M ∈ QCoh(X).

3.2.5. A Γ-filtration F on ω_{T} : Rep(G)(S) → QCoh(T) is a factorization

\xymatrix{Rep(G)(S) \ar[r]^{F} & Fil^{Γ}QCoh(T) \ar[r]^{forg} & QCoh(T)}

of ω_{T} such that if F^{γ} : Rep(G)(S) → QCoh(T) is the γ-component of F, (F1) For every ρ_{1}, ρ_{2} ∈ Rep(G)(S) and γ ∈ Γ,

F^{γ}(ρ_{1} ⊗ ρ_{2}) = _{∑γ_{1}+γ_{2}=γ}F^{γ_{1}}(ρ_{1}) ⊗ F^{γ_{2}}(ρ_{2}).

(F2) For the trivial representation ρ of G on M ∈ QCoh(S),

F^{γ}(ρ) = M if γ ≤ 0 and F^{γ}(ρ) = 0 if γ > 0.

(F3) For every γ ∈ Γ, F^{γ} : Rep(G)(S) → QCoh(T) is right exact. Since F^{γ} preserves arbitrary direct sums (as a subfunctor of ω_{T} which does), this last axiom implies that F^{γ} commutes with arbitrary colimits. It also preserves pure monomorphisms and pure short exact sequences.

3.2.6. We may now introduce a diagram of fpqc sheaves (Sch/S)^{∘} → Set,

\xymatrix{𝔾^{Γ}(V) \ar[r]^{res} \ar[d]_{Fil} & 𝔾^{Γ}(ω) \ar[d]_{Fil} \\ 𝔽^{Γ}(V) \ar[r]^{res} & 𝔽^{Γ}(ω)}

The four presheaves map an S-scheme T to the corresponding set of Γ-graduations or Γ-filtrations on V_{T} or ω_{T}, the Fil-morphisms are given by post-composition with the eponymous functors, and the res morphisms map G or F on V_{T} to

Rep(G)(S) → Rep(G)(T) ⟶^{G_{T}} Gr^{Γ}QCoh(T) or ⋯ ⟶^{F_{T}} Fil^{Γ}QCoh(T).

The fact that all four presheaves are actually fpqc sheaves on S is essentially a formal consequence of the fact that the corresponding fibered categories of Γ-graded and Γ-filtered quasi-coherent sheaves are fpqc stacks over Sch/S.

3.2.7. The above diagram is equivariant with respect to a morphism

\xymatrix{Aut^{⊗}(V) \ar[r]^{res} & Aut^{⊗}(ω)}

of fpqc sheaves of groups on Sch/S, with Aut^{⊗}(⋆) acting on 𝔾^{Γ}(⋆) and 𝔽^{Γ}(⋆) and mapping an S-scheme T to a group Aut^{⊗}(⋆_{T}) defined as follows: Aut^{⊗}(V_{T}) is the group of all automorphisms η : V_{T} → V_{T} such that: (A0) For every T -morphism f : X → Y and ρ ∈ Rep(G)(Y),

η_{f^{∗}(ρ)} = f^{∗}(η_{ρ}).

(A1) For every T -scheme X → T and ρ_{1}, ρ_{2} ∈ Rep(G)(X),

η_{ρ_{1}⊗ρ_{2}} = η_{ρ_{1}} ⊗ η_{ρ_{2}}.

These conditions imply as above that η_{ρ} = Id_{M} when ρ is the trivial representation of G_{X} on a quasi-coherent O_{X}-module M. Similarly, Aut^{⊗}(ω_{T}) is the group of all automorphisms η : ω_{T} → ω_{T} such that: (A1) For every ρ_{1}, ρ_{2} ∈ Rep(G)(S),

η_{ρ_{1}⊗ρ_{2}} = η_{ρ_{1}} ⊗ η_{ρ_{2}}.

(A2) For the trivial representation ρ of G on M ∈ QCoh(S),

η_{ρ} = Id_{M}.

The fact that these two presheaves are actually fpqc sheaves on S is essentially a formal consequence of the fact that QCoh/S is a stack over Sch/S. The morphism between them sends η ∈ Aut^{⊗}(V_{T}) to the automorphism of ω_{T} which maps ρ in Rep(G)(S) to the automorphism η_{ρ_{T}} of V (ρ_{T}) = ω_{T} (ρ), the actions mentioned above are the obvious ones, and the claimed equivariance is equally straightforward.

3.2.8. For ⋆ ∈ {V, ω} and X ∈ 𝔾^{Γ}(⋆)(T) or 𝔽^{Γ}(⋆)(T), we denote by
Aut^{⊗}(X) : (Sch/T)^{∘} → Group

the stabilizer of X in the restriction Aut^{⊗}(⋆)|_{T} of Aut^{⊗}(⋆) to Sch/T. It is an fpqc subsheaf of Aut^{⊗}(⋆)|_{T}. For X = F in 𝔽^{Γ}(⋆)(T), there is also a morphism

Gr^{•} : Aut^{⊗}(F) → Aut^{⊗}(Gr_{F}^{•}).

Here Aut^{⊗}(Gr_{F}^{•}) is an fpqc sheaf of groups on Sch/T which maps X → T to a group of automorphisms of Gr_{F_{X}}^{•} = Gr^{•} ∘ F_{X} subject to conditions whose precise formulation will be left to the reader. The kernel of this morphism is an fpqc sheaf

Aut^{⊗!}(F) : (Sch/T)^{∘} → Group.

If G is a splitting of F, then Gr_{F}^{•} ≃ G, thus Aut^{⊗}(Gr_{F}^{•}) ≃ Aut^{⊗}(G) and

Aut^{⊗}(F) ≃ Aut^{⊗!}(F) ⋊ Aut^{⊗}(G).

3.2.9. There is finally another equivariant diagram of fpqc sheaves on S,

\xymatrix{G \ar[dd]^{ι} &  & 𝔾^{Γ}(G) \ar[r]^{Fil} \ar[dd]^{ι} & 𝔽^{Γ}(G) \ar[dd]^{ι} \\  & acting on \\ Aut^{⊗}(V) &  & 𝔾^{Γ}(V) \ar[r]^{Fil} & 𝔽^{Γ}(V)}

The morphism ι : G → Aut^{⊗}(V) sends g ∈ G(T) to the automorphism ι(g) of V_{T} which maps ρ ∈ Rep(G)(X) to the automorphism ρ(g_{X}) of V (ρ) – for an S-scheme T and a T -scheme X. The morphism ι : 𝔽^{Γ}(G) ↪ 𝔽^{Γ}(V) is the image of

\xymatrix{𝔾^{Γ}(G) \ar[r]^{ι} & 𝔾^{Γ}(V) \ar[r]^{Fil} & 𝔽^{Γ}(V)}

where ι : 𝔾^{Γ}(G) → 𝔾^{Γ}(V) is defined as follows. Recall from [20, I 4.7.3] that the fpqc stacks Gr^{Γ}QCoh and Rep 𝔻(Γ) over Sch are ⊗-equivalent: A Γ-graded quasicoherent sheaf M = ⊕_{γ∈Γ}G_{γ} on a scheme X is mapped to the unique representation ρ of 𝔻_{X}(Γ) on M such that for every f : Y → X and α : Γ → Γ(Y, O_{Y}^{∗}) in 𝔻_{X}(Γ)(Y), ρ(α)(x) equals α(γ) · x for every γ ∈ Γ and x ∈ Γ(Y, f^{∗}G_{γ}). Conversely, a representation ρ of 𝔻_{X}(Γ) on a quasi-coherent O_{X}-module M is sent to the Γ-grading on M defined by the eigenspace decomposition of ρ. Then ι maps a morphism χ : 𝔻_{T} (Γ) → G_{T} in 𝔾^{Γ}(G)(T) to the Γ-graduation on V_{T} defined by

\xymatrix{Rep(G_{T}) \ar[r]^{−∘χ} & Rep(𝔻_{T} (Γ)) ≃ Gr^{Γ}QCoh/T \ar[r]^{forg} & QCoh/T.}

Remark 38. We will show in corollary 55 that for a reductive group G, the definition of the fpqc sheaf 𝔽^{Γ}(G) on Sch/S given here (image of 𝔾^{Γ}(G) → 𝔽^{Γ}(V)) coincides with the definition of section 2.2 (image of 𝔾^{Γ}(G) → 𝔾^{Γ}(R_{ℙ(G)})).

3.3. The subcategories of rigid objects

We briefly discuss the −^{∘} variants of the above definitions, mostly mentioning the new features.

3.3.1. Finite locally free sheaves. Let LF → Sch be the fibered category whose fiber over X is the full subcategory LF(X) of QCoh(X) whose objects are the finite locally free sheaves on X. Then LF is a substack of QCoh by [26, 2.5.2]. Pulling back through LF ↪ QCoh, we obtain a diagram of fpqc stacks over Sch,

\xymatrix{Gr^{Γ}LF \ar[r]^{Fil} & Fil^{Γ}LF \ar[l]^{Gr} \ar[r]^{forg} & LF}

whose fiber over a scheme X is a diagram of exact (in Quillen’s sense) ⊗-functors

\xymatrix{Gr^{Γ}LF(X) \ar[r]^{Fil} & Fil^{Γ}LF(X) \ar[l]^{Gr} \ar[r]^{forg} & LF(X).}

An alternative and useful description of the objects of Fil^{Γ}LF(X) is provided by proposition 39 below, which also implies that the Gr functor is indeed well-defined. Over a base scheme S, there is the corresponding diagram of fpqc stacks:

\xymatrix{Gr^{Γ}LF/S \ar[r]^{Fil} & Fil^{Γ}LF/S \ar[l]^{Gr} \ar[r]^{forg} & LF/S}

3.3.2. These categories have compatible inner Hom’s and duals given by

\underline{Hom}(x, y) = x^{∨} ⊗ y with (M, G)^{∨} = (M^{∨}, G^{∨}) and (M, F)^{∨} = (M^{∨}, F^{∨})

where M^{∨} is the dual of M, (G^{∨})_{γ} = (G_{−γ})^{∨} and (F^{∨})^{γ} = (F_{+}^{−γ})^{⊥} = (M/F_{+}^{−γ})^{∨}. Thus if G is a splitting of F, then G^{∨} is a splitting of F^{∨}. Moreover, we have

(F^{∨})_{+}^{γ} = (F^{−γ})^{⊥} ≃ (M/F^{−γ})^{∨} and Gr_{F^{∨}}^{γ}(M^{∨}) ≃ Gr_{F}^{−γ}(M)^{∨}.

For the inner Homs, we obtain the following formula:

Gr_{F}^{γ} (\underline{Hom}(M_{1}, M_{2})) ≃ ⊕_{γ_{2}−γ_{1}=γ} \underline{Hom} (Gr_{F_{1}}^{γ_{1}}(M_{1}), Gr_{F_{2}}^{γ_{2}}(M_{2})) .

3.3.3. Γ-filtrations on finite locally free sheaves.

Proposition 39. Let M be a finite locally free sheaf on X. Let (F^{γ})_{γ∈Γ} be a non-increasing collection of quasi-coherent subsheaves of M. Then the following conditions are equivalent:

(1) For every affine open subset U of X, there is a Γ-graduation

M_{U} = ⊕_{γ∈Γ}G_{γ}

such that F_{U}^{γ} = ⊕_{η≥γ}G_{η} for every γ ∈ Γ.

(2) Locally on X for the Zariski topology, there is a Γ-graduation

M = ⊕_{γ∈Γ}G_{γ}

such that F^{γ} = ⊕_{η≥γ}G_{η} for every γ ∈ Γ.

(3) Locally on X for the fpqc topology, there exists a Γ-graduation

M = ⊕_{γ∈Γ}G_{γ}

such that F^{γ} = ⊕_{η≥γ}G_{η} for every γ ∈ Γ, i.e. F is a Γ-filtration on M.

(4) For every γ ∈ Γ, Gr_{F}^{γ}(M) is finite locally free and for every x ∈ X,

dim_{k(x)} M(x) = _{∑γ} dim_{k(x)} Gr_{F(x)}^{γ}(M(x)).

In (4), F(x) is the image of F in M(x) = M ⊗ k(x) and Gr_{F}^{γ}(M), Gr_{F(x)}^{γ}(M(x)) are defined as usual. Under the above equivalent conditions, for all γ ∈ Γ: F^{γ}, F_{+}^{γ} and Gr_{F}^{γ}(M) are finite locally free sheaves on X and for every x ∈ X,

F^{γ}(x) ≃ F^{γ} ⊗ k(x), F_{+}^{γ}(x) ≃ F_{+}^{γ} ⊗ k(x), Gr_{F(x)}^{γ}(M(x)) ≃ Gr_{F}^{γ}(M) ⊗ k(x).

Proof. Plainly (1) ⇒ (2) ⇒ (3). Moreover (3) ⇒ (4) is easy (using [26, 2.5.2.iii]) and the last assertions follow from (1). To prove that (4) ⇒ (1), we may assume that X = U is affine. Since Gr_{F}^{γ}(M) is finite locally free by assumption, it is then projective in QCoh(X) by [34, Corollary of 7.12]. Therefore, there exists a quasi-coherent subsheaf G_{γ} of F_{γ} such that F^{γ} = G_{γ} ⊕ F_{+}^{γ}. We will show that

M = ⊕_{γ∈Γ}G_{γ} and ∀γ : F^{γ} = ⊕_{η≥γ}G_{η}.

This being now a local question in the Zariski topology of X, we may assume that the rank of M is constant on X, and also nonzero. Fix x ∈ X and define

Γ(x) = {γ : Gr_{F(x)}^{γ}(M(x)) ≠ 0} = {γ_{1} < ⋯ < γ_{r}}.

Define U_{0} = Supp(M/F^{γ_{1}})^{c}, U_{i} = Supp(F_{+}^{γ_{i}}/F^{γ_{i+1}})^{c} ∩ U_{i−1} for 0 < i < r and U_{r} = Supp(F_{+}^{γ_{r}})^{c} ∩U_{r−1}. Since M is finite locally free, M/F^{γ_{1}} is finitely generated and U_{0} is open in X. Since M = F^{γ_{1}} over U_{0} and F^{γ_{1}} = F_{+}^{γ_{1}} ⊕ G_{γ_{1}} over X, M = F_{+}^{γ_{1}} ⊕ G_{γ_{1}} over U_{0}. Therefore F_{+}^{γ_{1}} is finite locally free over U_{0}. Repeating this argument successively with (M, X) replaced by (F_{+}^{γ_{1}}, U_{0}), (F_{+}^{γ_{2}}, U_{1}) etc… we obtain: U_{r} is open in X, M = ⊕_{i}G_{γ_{i}} and F^{γ} = ⊕_{i:γ_{i}≥γ}G_{γ_{i}} over U_{r} for every γ ∈ Γ, with everyone finite locally free over U_{r}. All we have to do now is to show that the formula of (4) implies that x belongs to U_{r}. The formula is equivalent to:

F^{γ}(x) = \begin{cases}M(x) & if γ ≤ γ_{1}, \\ F^{γ_{i+1}}(x) & if γ ∈]γ_{i}, γ_{i+1}], \\ 0 & if γ > γ_{r}. \end{cases}

Since M is finitely generated over X, F^{γ_{1}}(x) = M(x) implies F_{x}^{γ_{1}} = M_{x} by Nakayama’s lemma, thus x belongs to U_{0}. Since M = F^{γ_{1}} = F_{+}^{γ_{1}} ⊕ G_{γ_{1}} over U_{0}, F_{+}^{γ_{1}}(x) = F^{γ_{2}}(x) in M(x) implies F_{+,x}^{γ_{1}} = F_{x}^{γ_{2}} by Nakayama’s lemma, therefore x belongs to U_{1}. Repeating the argument, we find that indeed x belongs to U_{r}. □

Remark 40. The whole proof becomes much simpler over a Noetherian base.

Lemma 41. Let M_{α} be a finite collection of locally free sheaves of finite rank on X and for each α, let (F_{α}^{γ})_{γ∈Γ} be a non-increasing collection of quasi-coherent subsheaves of M_{α}. Set M = ⊕M_{α} and F^{γ} = ⊕F_{α}^{γ}. Then (M, (F^{γ})) satisfies the above equivalent conditions if and only if each (M_{α}, (F_{α}^{γ})) does.

Proof. For every γ ∈ Γ and x ∈ X, Gr_{F}^{γ}(M) = ⊕_{α}Gr_{F^{α}}^{γ}(M_{α}) and
M(x) = ⊕_{α}M_{α}(x), Gr_{F(x)}^{γ}(M(x)) = ⊕_{α}Gr_{F_{α}(x)}^{γ}(M_{α}(x)).

Moreover for every α and x ∈ X,

dim_{k(x)} M_{α}(x) ≥ _{∑γ} dim_{k(x)} Gr_{F_{α}(x)}^{γ}(M_{α}(x)).

The lemma easily follows. □

3.3.4. Let Rep^{∘}(G) → Sch/S be the substack of Rep(G) → Sch/S whose fiber over T → S is the exact, rigid, full sub-⊗-category Rep^{∘}(G)(T) of Rep(G)(T) whose objects are the representations of G_{T} on finite locally free sheaves on T. We write

V^{∘} : Rep^{∘}(G) → LF/S

for the forgetful functor. For an S-scheme T → S, we denote by

V_{T}^{∘} : Rep^{∘}(G_{T}) → LF/T and ω_{T}^{∘} : Rep^{∘}(G)(S) → LF(T)

the induced morphism of fpqc stack over Sch/T and fiber functor. Note that ω_{T}^{∘} is now an exact ⊗-functor, since all short exact sequences in Rep^{∘}(G)(S) are pure.

3.3.5. We obtain yet another equivariant diagram of fpqc sheaves on S,

\xymatrix{Aut^{⊗}(V^{∘}) \ar[dd]^{res} &  & 𝔾^{Γ}(V^{∘}) \ar[r]^{Fil} \ar[dd]^{res} & 𝔽^{Γ}(V^{∘}) \ar[dd]^{res} \\  & acting on \\ Aut^{⊗}(ω^{∘}) &  & 𝔾^{Γ}(ω^{∘}) \ar[r]^{Fil} & 𝔽^{Γ}(ω^{∘})}

where everything is defined as before, using V^{∘} and ω^{∘} instead of V and ω. The only differences worth mentioning are as follows: for any S-scheme T, the Γ-graduations or Γ-filtrations on ω_{T}^{∘} are automatically compatible with inner Homs and duals, and there γ-components are exact functors. We also have equivariant diagrams

\xymatrix{Aut^{⊗}(V) \ar[dd]^{res} &  & 𝔾^{Γ}(V) \ar[r]^{Fil} \ar[dd]^{res} & 𝔽^{Γ}(V) \ar[dd]^{res} \\  & acting on \\ Aut^{⊗}(V^{∘}) &  & 𝔾^{Γ}(V^{∘}) \ar[r]^{Fil} & 𝔽^{Γ}(V^{∘})}

and similarly for ω and ω^{∘}, where all the vertical maps are induced by pre-composition with the full embedding Rep^{∘}(G) ↪ Rep(G).

3.3.6. Finally, the definitions of Aut^{⊗}(G), Aut^{⊗}(F), Aut^{⊗!}(F) and Aut^{⊗}(Gr_{F}^{•}) given in section 3.2.8 carry over to the situation considered here.

3.4. Skalar extensions

The whole diagram at the beginning of this section has now been defined. It is covariantly functorial in G but not entirely compatible with base change on S: if S̃ → S is any morphism, G̃ = G ×_{S} S̃ and Ṽ, ω̃ … are the relevant functors for G̃, then 𝔾^{Γ}(G̃) = 𝔾^{Γ}(G)|S̃, 𝔽^{Γ}(G̃) = 𝔽^{Γ}(G)|S̃ and

Aut^{⊗}(X̃) = Aut^{⊗}(X)|S̃, 𝔾^{Γ}(X̃) = 𝔾^{Γ}(X)|S̃ and 𝔽^{Γ}(X̃) = 𝔽^{Γ}(X)|S̃

for X ∈ {V, V^{∘}}, but the natural morphisms of fpqc sheaves on S̃,

Aut^{⊗}(Ỹ) → Aut^{⊗}(Y)|S̃, 𝔾^{Γ}(Ỹ) → 𝔾^{Γ}(Y)|S̃ and 𝔽^{Γ}(Ỹ) → 𝔽^{Γ}(Y)|S̃

may not be isomorphisms for Y ∈ {ω, ω^{∘}}. We investigate this issue.

3.4.1. When C is a category and B is a ring object in C, we can form the category C(B) of (left) B-modules in C. Here C will be an additive ⊗-category and the ring object will be given by its multiplication morphism μ : B ⊗ B → B and unit 1 → B, where 1 is the neutral object for the tensor product, the abelian group structure on B being provided by the additive structure of C. Then C(B) is the category of pairs (M, ν) where M is an object of C and ν : B ⊗ M → M is a morphism in C subject to certain natural conditions. There is an adjunction

f^{∗} : C ↔ C(B) : f_{∗} given by f_{∗}(M, ν) = M and f^{∗}(N) = (B ⊗ N , μ ⊗ Id).

In many cases, it is also possible to equip C(B) with a ⊗-product inherited from the ⊗-product on C, with (B, μ) as neutral object. Instead of trying to develop this formal theory more rigorously, let us list some of the relevant examples:

C = QCoh(S) and B = f_{∗}O_{T} where f : T → S is an affine morphism. There is an equivalence of ⊗-categories C(B) ≃ QCoh(T) which is compatible with the usual adjunctions f^{∗} : QCoh(S) ↔ QCoh(T) : f_{∗}, see [24, 1.4].

C = Gr^{Γ}QCoh(S) and B as above with the trivial Γ-graduation. The first example induces an equivalence of ⊗-categories C(B) ≃ Gr^{Γ}QCoh(T) which is again compatible with the natural adjunctions.

C = Fil^{Γ}QCoh(S) and B as above with the trivial Γ-filtration. The first example now only induces a fully faithful exact ⊗-functor C(B) ↪ Fil^{Γ}QCoh(T). The essential image is made of those Γ-filtered quasi-coherent sheaves (M, F) on T such that, locally on S (as opposed to T) for the fpqc topology, F has a splitting.

C = Rep(G)(S) and B as above with the trivial action of G. The first example again induces an equivalence of ⊗-categories C(B) ≃ Rep(G)(T) which is compatible with the adjunctions given on the comodules by the following formulas:

f^{∗} (V (ρ) ⟶^{c_{ρ}} V (ρ) ⊗_{O_{S}} A(G)) = (V (f^{∗}ρ) ⟶^{c_{f^{∗}ρ}} V (f^{∗}ρ) ⊗_{O_{T}} A(G_{T})) ,
f_{∗} (V (ρ) ⟶^{c_{ρ}} V (ρ) ⊗_{O_{T}} A(G_{T})) = (V (f_{∗}ρ) ⟶^{c_{f∗ρ}} V (f_{∗}ρ) ⊗_{O_{S}} A(G)) .

C = LF(S) and B = f_{∗}O_{T} where f : T → S is a finite étale morphism. The first example induces an equivalence of ⊗-categories C(B) ≃ LF(T). We have to show that for a quasi-coherent sheaf M on T, M is a finite locally free O_{T} -module if and only if f_{∗}M is a finite locally free O_{S}-module (the direct implication is easy, and only requires f to be finite and locally free). By [26, 2.5.2], our claim is local in the fpqc topology on S. But, locally on S for the étale topology, our finite étale morphism f is simply a finite disjoint union of open and closed embeddings (this follows from [28, 17.9.3]), for which the claim is now obvious.

Combining this last example with the previous three, we obtain:

C = Gr^{Γ}LF(S) and B as above with the trivial Γ-graduation. Then

C(B) ≃ Gr^{Γ}LF(T).

C = Fil^{Γ}LF(S) and B as above with the trivial Γ-filtration. Then

C(B) ≃ Fil^{Γ}LF(T).

C = Rep^{∘}(G)(S) and B as above with the trivial action. Then

C(B) ≃ Rep^{∘}(G)(T).

3.4.2. The point of this abstract nonsense is that, if α : C → D is a ⊗-functor and B is a ring object in C, then α(B) is a ring object in D and α extends to a ⊗- functor α(B) : C(B) → D(α(B)) which we call the skalar extension of α. Similarly, if η is a ⊗-automorphism of α such that η_{B} is the identity of α(B), then η extends to a ⊗-automorphism η(B) of α(B) which we call the skalar extension of η.

Proposition 42. (1) Let f : S̃ → S be a finite étale morphism and denote by ω̃ the fiber functors for G̃ = GS̃. Then we have isomorphisms of fpqc sheaves on S̃:

Aut^{⊗}(ω^{∘})|S̃ = Aut^{⊗}(ω̃^{∘}), 𝔾^{Γ}(ω^{∘})|S̃ = 𝔾^{Γ}(ω̃^{∘}) and 𝔽^{Γ}(ω^{∘})|S̃ = 𝔽^{Γ}(ω̃^{∘}).

(2) If f is merely affine, then 𝔽^{Γ}(ω)|S̃ = 𝔽^{Γ}(ω̃).

Proof. (1) Let T be an S̃-scheme. We have to define mutually inverse maps

α : Aut^{⊗}(ω̃^{∘})(T) ⟷ Aut^{⊗}(ω^{∘})(T) 𝔾^{Γ}(ω̃^{∘})(T) ⟷ 𝔾^{Γ}(ω^{∘})(T) 𝔽^{Γ}(ω̃^{∘})(T) ⟷ 𝔽^{Γ}(ω^{∘})(T) : β

functorial in T. The α maps are induced by precomposition with the base change map Rep^{∘}(G)(S) → Rep^{∘}(G)(S̃). The β maps are defined by composing the skalar extension maps with the base change maps for the S̃-section ι : T → T̃ of the projection f_{T} : T̃ = T ×_{S} S̃ → T given by the structural morphism T → S̃:

β : Aut^{⊗}(ω^{∘})(T) ⟶ Aut^{⊗}(ω̃^{∘})(T̃) ⟶ Aut^{⊗}(ω̃^{∘})(T) 𝔾^{Γ}(ω^{∘})(T) ⟶ 𝔾^{Γ}(ω̃^{∘})(T̃) ⟶ 𝔾^{Γ}(ω̃^{∘})(T) 𝔽^{Γ}(ω^{∘})(T) ⟶ 𝔽^{Γ}(ω̃^{∘})(T̃) ⟶ 𝔽^{Γ}(ω̃^{∘})(T)

Explicitly, for η, G and F in the source sets and ρ̃ ∈ Rep^{∘}(G̃)(S̃), we first view f_{∗}ρ̃ as a B-module in Rep^{∘}(G)(S) where B = f_{∗}OS̃ with trivial G-action. Then:

• η_{f_{∗}ρ̃} is a B_{T} -linear isomorphism of ω_{T}^{∘} (f_{∗}ρ̃) = (f_{T})_{∗}ω̃∘ ; T̃(ρ̃). It thus corresponds to an isomorphism of ω̃∘ ; T̃(ρ̃) whose pull-back to ι^{∗}ω̃∘ ; T̃(ρ̃) = ω̃_{T}^{∘} (ρ̃) is an isomorphism β(η)ρ̃. By construction, there is a commutative diagram

\xymatrix{ω_{T}^{∘} (f_{∗}ρ̃) = ω̃_{T}^{∘} (f^{∗}f_{∗}ρ̃) \ar@{->>}[r] \ar[d]_{η_{f∗ρ̃}} & ω̃_{T}^{∘} (ρ̃) \ar[d]^{β(η)ρ̃} \\ ω_{T}^{∘} (f_{∗}ρ̃) = ω̃_{T}^{∘} (f^{∗}f_{∗}ρ̃) \ar@{->>}[r] & ω̃_{T}^{∘} (ρ̃)}

where the horizontal map comes from the adjunction morphism

f^{∗}f_{∗}ρ̃ → ρ̃.

• G(f_{∗}ρ̃) is a B_{T} -stable Γ-graduation on (f_{T})_{∗}ω̃∘ ; T̃(ρ̃), giving a Γ-graduation on ω̃∘ ; T̃(ρ̃) whose pull-back is a Γ-graduation β(G)(ρ̃) on ω̃_{T}^{∘} (ρ̃). Thus β(G)_{γ}(ρ̃) is the image of G_{γ}(f_{∗}ρ̃) under the adjunction ω_{T}^{∘} (f_{∗}ρ̃) ↠ ω̃_{T}^{∘} (ρ̃).

• F(f_{∗}ρ̃) is a B_{T} -stable Γ-filtration on (f_{T})_{∗}ω̃∘ ; T̃(ρ̃), giving a Γ-filtration on ω̃∘ ; T̃(ρ̃) whose pull-back is a Γ-filtration β(F)(ρ̃) on ω̃_{T}^{∘} (ρ̃). Thus B(F)^{γ}(ρ̃) is the image of F^{γ}(f_{∗}(ρ̃)) under the adjunction ω_{T}^{∘} (f_{∗}ρ̃) ↠ ω̃_{T}^{∘} (ρ̃).

One checks easily that α ∘ β = Id and β ∘ α = Id. The proof of (2) is similar. □

Remark 43. We have not mentioned Aut^{⊗}(ω) and 𝔾^{Γ}(ω) in part (2) of the proposition, because we will establish a stronger result for them in the next section.

3.5. The regular representation

The single most important representation of G is the regular representation ρ_{reg}. We shall use it to establish the classical:

Theorem 44. The above morphisms of fpqc sheaves induce isomorphisms

G ≃ Aut^{⊗}(V) ≃ Aut^{⊗}(ω) and 𝔾^{Γ}(G) ≃ 𝔾^{Γ}(V) ≃ 𝔾^{Γ}(ω).

3.5.1. The regular representation ρ_{reg} of G on V (ρ_{reg}) = A(G) is defined by

(g · a)(h) = a(hg)

for T → S, a ∈ Γ(T, A(G)_{T}) = Γ(G_{T} , O_{G_{T}}) and g, h ∈ G(T). The corresponding A(G)-comodule structure morphism is the comultiplication map:

(V (ρ_{reg}) ⟶^{c_{reg}} V (ρ_{reg}) ⊗_{O_{S}} A(G)) = (A(G) ⟶^{μ^{♮}} A(G) ⊗_{O_{S}} A(G))

The O_{S}-algebra structure morphisms on A(G), namely the unit O_{S} → A(G) and the multiplication A(G) ⊗ A(G) → A(G) correspond to G-equivariant morphisms

1_{S} → ρ_{reg} and ρ_{reg} ⊗ ρ_{reg} → ρ_{reg}.

For any ρ ∈ Rep(G)(S), we denote by ρ_{0} ∈ Rep(G)(S) the trivial representation of G on V (ρ_{0}) = V (ρ). We may then view the A(G)-comodule structure morphism c_{ρ} : V (ρ) → V (ρ) ⊗_{O_{S}} A(G) of ρ as a G-equivariant morphism in Rep(G)(S)

c_{ρ} : ρ → ρ_{0} ⊗ ρ_{reg}

The underlying morphism of quasi-coherent sheaves on S is a split monomorphism since (Id ⊗ 1_{G}^{♮}) ∘ c_{ρ} = Id on V (ρ) where 1_{G}^{♮} : A(G) → O_{S} is the counit of A(G).

3.5.2. It follows that any η ∈ Aut^{⊗}(ω_{T}), G ∈ 𝔾^{Γ}(ω_{T}) or F ∈ 𝔽^{Γ}(ω_{T}) is uniquely determined by its value η_{reg}, G_{reg} or F_{reg} on ρ_{reg}. Indeed for any ρ ∈ Rep(G)(S), η_{ρ}, G(ρ) and F(ρ) will then be the automorphism, Γ-graduation and Γ-filtration on

\xymatrix{ω_{T} (ρ) \ar@{^{(}->}[r]^{ω_{T} (c_{ρ})} & ω_{T} (ρ_{0}) ⊗ ω_{T} (ρ_{reg})}

which are respectively induced by the corresponding objects for ρ_{0} ⊗ ρ_{reg}, namely

η_{ρ_{0}⊗ρ_{reg}} = Id ⊗ η_{reg},
G(ρ_{0} ⊗ ρ_{reg}) = ω_{T} (ρ_{0}) ⊗ G_{reg},
F(ρ_{0} ⊗ ρ_{reg}) = ω_{T} (ρ_{0}) ⊗ F_{reg}.

We have here used the defining axioms (A1) and (A2) for η, (G1) and (G2) for G and (F1) and (F2) for F, as well as the fact that for every γ ∈ Γ, the functors G_{γ} and F^{γ} : Rep(G)(S) → QCoh(T) both preserve pure short exact sequences.

3.5.3. By the same token, we find that the morphisms of fpqc sheaves

Aut^{⊗}(V) → Aut^{⊗}(ω), 𝔾^{Γ}(V) → 𝔾^{Γ}(ω) and 𝔽^{Γ}(V) → 𝔽^{Γ}(ω)

are monomorphisms. For instance if η ∈ Aut^{⊗}(V_{T}) induces the identity of ω_{T}, then for any f : X → T and ρ ∈ Rep(G)(X), η_{ρ} is the identity of V (ρ) because

η_{ρ_{0}⊗ρ_{reg,X}} = η_{ρ_{0}} ⊗ η_{ρ_{reg,X}} = Id_{V (ρ_{0})} ⊗ f^{∗}(η_{ρ_{reg,T}})

and η_{ρ_{reg,T}} is the trivial automorphism of V (ρ_{reg,T}) = ω_{T} (ρ_{reg}).

3.5.4. We show that G = Aut^{⊗}(ω). Fix an S-scheme T and η ∈ Aut^{⊗}(ω_{T}). Recall that η_{reg} is the O_{T} -linear automorphism of ω_{T} (ρ_{reg}) = A(G_{T}) induced by η. Since η_{1_{S}} = Id_{O_{T}} on ω_{T} (1_{S}) = O_{T} by (A2) and η_{ρ_{reg}⊗ρ_{reg}} = η_{reg} ⊗ η_{reg} on

ω_{T} (ρ_{reg} ⊗ ρ_{reg}) = A(G_{T}) ⊗ A(G_{T})

by (A1), the functoriality of η applied to 1_{S} → ρ_{reg} and ρ_{reg} ⊗ ρ_{reg} → ρ_{reg} implies that η_{reg} is an automorphism of the quasi-coherent O_{T} -algebra A(G_{T}). Similarly for any ρ ∈ Rep(G)(S), the G-equivariant morphism c_{ρ} : ρ → ρ_{0} ⊗ ρ_{reg} induces a commutative diagram of quasi-coherent O_{T} -modules

\xymatrix{ω_{T} (ρ) \ar[r]^{(c_{ρ})_{T}} \ar[d]_{η_{ρ}} & ω_{T} (ρ_{0}) ⊗_{O_{T}} A(G_{T}) \ar[d]^{Id⊗η_{reg}} \\ ω_{T} (ρ) \ar[r]^{(c_{ρ})_{T}} & ω_{T} (ρ_{0}) ⊗_{O_{T}} A(G_{T})}

Composing η_{reg} with the counit 1_{G,T}^{♮} : A(G)_{T} → O_{T}, we obtain a morphism of O_{T} -algebras s(η)^{♮} : A(G)_{T} → O_{T}, i.e. a T -valued point s(η) ∈ G(T). Now for any g ∈ G(T) corresponding to g^{♮} : A(G)_{T} → O_{T} and mapping to ι(g) ∈ Aut^{⊗}(ω_{T}), the automorphism ι(g)_{ρ} = ρ_{T} (g) of ω_{T} (ρ) is obtained by composing (c_{ρ})_{T} with

Id ⊗ g^{♮} : ω_{T} (ρ) ⊗_{O_{T}} A(G)_{T} → ω_{T} (ρ).

We thus find that s ∘ ι(g) = g since

s ∘ ι(g)^{♮} = 1_{G,T}^{♮} ∘ ι(g)_{reg} = 1_{G_{T}}^{♮} ∘ (Id ⊗ g^{♮}) ∘ μ_{T}^{♮} = (1_{G_{T}}^{♮} ⊗ g^{♮}) ∘ μ_{T}^{♮} = (1_{G_{T}} · g)^{♮} = g^{♮}.

On the other hand, ι ∘ s(η) = η since for any ρ ∈ Rep(G)(S),

(ι ∘ s)(η)_{ρ} = (Id ⊗ 1_{G,T}^{♮} ) ∘ (Id ⊗ η_{reg}) ∘ (c_{ρ})_{T}
= (Id ⊗ 1_{G,T}^{♮} ) ∘ (c_{ρ})_{T} ∘ η_{ρ}
= ρ(1_{G})_{T}^{♮} ∘ η_{ρ} = η_{ρ}.

Thus G = Aut^{⊗}(ω) and by 3.5.3, also G = Aut^{⊗}(V).

3.5.5. We show that 𝔾^{Γ}(G) = 𝔾^{Γ}(ω). Let T be an S-scheme, G ∈ 𝔾^{Γ}(ω_{T}). Then for any T -scheme X, the Γ-graduation G on ω_{T} and the ⊗-equivalence

Gr^{Γ}QCoh(T) ≃ Rep(𝔻_{T} (Γ))(T)

together induce a factorization

ω_{X}^{1} : Rep(G)(S) ⟶^{G′} Rep(𝔻_{T} (Γ))(T) ⟶^{ω_{X}^{2}} QCoh(X)

of the fiber functor ω_{X}^{1} for the group scheme G over S through the fiber functor ω_{X}^{2} for the group scheme 𝔻_{T} (Γ) over T. Moreover G′ is a ⊗-functor preserving trivial representations by (G1) and (G2). It thus induces a group homomorphism

𝔻_{T} (Γ)(X) {3.5.4 \atop ≃} Aut^{⊗}(ω_{X}^{2}) → Aut^{⊗}(ω_{X}^{1}) {3.5.4 \atop ≃} G(X).

The latter being functorial in X gives a morphism s(G) : 𝔻_{T} (Γ) → G_{T} of group schemes over T, i.e. an element s(G) of 𝔾^{Γ}(G)(T). Since G ↦ s(G) is itself functorial in T, it gives a morphism of fpqc sheaves s : 𝔾^{Γ}(ω) → 𝔾^{Γ}(G) which is the inverse of ι : 𝔾^{Γ}(G) → 𝔾^{Γ}(ω). Thus 𝔾^{Γ}(G) = 𝔾^{Γ}(ω) and by 3.5.3, also 𝔾^{Γ}(G) = 𝔾^{Γ}(V).

3.6. RELATING Rep(G)(S) AND Rep^{∘}(G)(S) 47

3.6. Relating Rep(G)(S) and Rep^{∘}(G)(S)

While Rep(G)(S) already contains the interesting regular representation, it could be that Rep^{∘}(G)(S) contains no representations beyond the trivial ones, in which case Aut^{⊗}(ω^{∘}), 𝔾^{Γ}(ω^{∘}) and 𝔽^{Γ}(ω^{∘}) are the trivial sheaves represented by S. For instance, let S be one of the two curves considered in [1, X 6.4], whose enlarged fundamental group equals ℤ. Let n ≥ 2 and A ∈ GL_{n}(ℤ) be any matrix with no roots of unity as eigenvalue. Then by [1, X 7.1], this determines an n-dimensional torus G over S, and all representations ρ ∈ Rep^{∘}(G)(S) are trivial because ℤ^{n} contains no finite A-orbit except {0}.

When S is quasi-compact, we also consider the intermediate full subcategory

Rep^{∘}(G)(S) ⊂ Rep′(G)(S) ⊂ Rep(G)(S)

whose objects are the representations ρ for which ρ = \underrightarrow{lim} τ where τ runs through the partially ordered set X(ρ) of all subrepresentations of ρ which belong to Rep^{∘}(G)(S). For such ρ’s, V (ρ) = \underrightarrow{lim} V (τ) is a flat O_{S}-module and the quasi-compactness of S implies that X(ρ) is a filtered set. This subcategory is stable under tensor product and the ρ ↦ ρ_{0} construction, it contains Rep^{∘}(G)(S) as a full subcategory, and for any ρ_{1}, ρ_{2} ∈ Rep′(G)(S),

(3.6.1)  Hom_{Rep(G)}(ρ_{1}, ρ_{2}) = \underleftarrow{lim}_{τ_{1}∈X(ρ_{1})}\underrightarrow{lim}_{τ_{2}∈X(ρ_{2})}Hom_{Rep(G)}(τ_{1}, τ_{2}).

We denote by ω′_{T} : Rep′(G)(S) → QCoh(T) the restriction of ω_{T} to Rep′(G)(S) and define the fpqc sheaf Aut^{⊗}(ω′) : (Sch/S)^{∘} → Group as before, with automorphisms of ω′_{T} satisfying the axioms (A1) and (A2), thus obtaining a factorization

Aut^{⊗}(ω) → Aut^{⊗}(ω′) → Aut^{⊗}(ω^{∘}).

On the other hand, it is obvious that Aut^{⊗}(ω′) = Aut^{⊗}(ω^{∘}).

3.6.1. The following assumption implies that Rep^{∘}(G)(S) is sufficiently big: HYP(ω^{∘}) There exists a covering {S_{i} → S} by finite étale morphisms such that

for every i, G_{S_{i}}/S_{i} satisfies HYP′(ω^{∘}) where:

HYP′(ω^{∘}) S is quasi-compact and ρ_{reg} belongs to Rep′(G)(S).

Proposition 45. If G/S satisfies HYP(ω^{∘}), then

G = Aut^{⊗}(ω^{∘}), 𝔾^{Γ}(G) = 𝔾^{Γ}(ω^{∘}) and 𝔽^{Γ}(ω) ⊂ 𝔽^{Γ}(ω^{∘}).

Proof. These being fpqc sheaves on S, it is sufficient to establish the proposition for their restriction to the S_{i}’s, which by proposition 42 reduces us to the case where S is quasi-compact and ρ_{reg} belongs to Rep′(G)(S). The proof of theorem 44 then shows that G = Aut^{⊗}(ω′_{T}). Thus G = Aut^{⊗}(ω^{∘}). To prove that 𝔾^{Γ}(G) = 𝔾^{Γ}(ω^{∘}), we may test this on quasi-compact schemes, and then the proof of section 3.5.5 carries over to this case. Finally: a Γ-filtration F on ω_{T} is uniquely determined by its value on ρ_{reg} by 3.5.3, thus 𝔽^{Γ}(ω) ⊂ 𝔽^{Γ}(ω^{∘}) since ρ_{reg} ∈ Rep′(G)(S). □

3.6.2. For the V^{∘} variants of these, one needs a weaker assumption: HYP(V^{∘}) Locally on S for the fpqc topology, ρ_{reg} belongs to Rep′(G)(S).

Proposition 46. If G/S satisfies HYP(V^{∘}), then

G = Aut^{⊗}(V^{∘}), 𝔾^{Γ}(G) = 𝔾^{Γ}(V^{∘}) and 𝔽^{Γ}(V) ⊂ 𝔽^{Γ}(V^{∘}).

Proof. This being local in the fpqc topology on S, we may assume that S is quasi-compact and ρ_{reg} is in Rep′(G)(S), then G_{T} /T satisfies HYP′(ω^{∘}) for every quasi-compact T over S and the proposition easily follows from the previous one.

□

3.6.3. It remains to give some cases where our assumptions are met.

Definition 47. A reductive group G over S is called isotrivial if and only if there exists a covering {S_{i} → S} by finite étale morphisms such that each G_{S_{i}} is splittable.

For tori, this definition is slightly more general than that given in [1, IX 1.1], which requires a single finite étale cover S′ → S. If S is quasi-compact or connected, both notions coincide. For arbitrary reductive groups, [21, XXIV 4.1] only defines local and semi-local isotriviality. If S is local, these two notions coincide with ours.

Proposition 48. If S is local and either geometrically unibranch or henselian, then every reductive group G over S is isotrivial.

Proof. We may assume that G is a torus by [21, XXIV 4.1.5]. We then have to show that the connected components of R = \underline{Hom}_{S}(G, 𝔾_{m,S}) are open and finite over S by [20, X 5.11], and this follows from proposition 3 and lemma 4. The henselian case also follows directly from [1, X 4.6] or [21, XXIV 1.21]. □

Proposition 49. (1) If S = Spec(A) for a Prüfer domain A and ρ ∈ Rep(G)(S),

ρ ∈ Rep′(G)(S) ⟺ V (ρ) is a flat O_{S}-module.

(2) A split reductive group over a quasi-compact S satisfies HYP′(ω^{∘}).

(3) An isotrivial reductive group over a quasi-compact S satisfies HYP(ω^{∘}).

(4) A reductive group over any S satisfies HYP(V^{∘}).

Proof. (1) is exactly [49, Corollary 5.10]. For (2), we may assume that G is of constant type [21, XXII 2.8], thus isomorphic [21, XXIII 5.2] to the base change of a reductive group G_{0} over Spec(ℤ) [21, XXV 1.2] to which (1) now applies. Obviously (2) ⇒ (3), and (2) ⇒ (4) by [21, XXII 2.3]. □

3.6.4. Together with theorem 44, proposition 45 and 46 give many cases where automorphisms or Γ-graduations automatically extend from ω^{∘} or V^{∘} to ω or V. Assuming that S is quasi-compact, we will now do something similar for Γ-filtrations.

3.6.5. Let F be a Γ-filtration on ω_{T}^{∘}. For each γ ∈ Γ, we may extend

F^{γ} : Rep^{∘}(G)(S) → LF(T) to F^{γ} : Rep′(G)(S) → QCoh(T)

by the formula F^{γ}(ρ) = \underrightarrow{lim} F^{γ}(τ), where τ runs through X(ρ). It defines a functor by (3.6.1), and gives back F^{γ}(ρ) = F^{γ}(τ) when ρ = τ belongs to Rep^{∘}(G)(S). In general, F^{γ}(ρ) is a pure quasi-coherent subsheaf of V (ρ)_{T} = \underrightarrow{lim} V (τ)_{T} since filtered colimits are exact and commute with base change. While γ → F^{γ}(ρ) is non-increasing, it may not be a Γ-filtration on V (ρ)_{T} in our sense. However:

Lemma 50. We have the following properties: (F1) For every ρ_{1}, ρ_{2} ∈ Rep′(G)(S) and γ ∈ Γ,

F^{γ}(ρ_{1} ⊗ ρ_{2}) = _{∑γ_{1}+γ_{2}=γ}F^{γ_{1}}(ρ_{1}) ⊗ F^{γ_{2}}(ρ_{2}).

(F2) For a trivial representation ρ ∈ Rep′(G)(S) on M ∈ QCoh(S),

F^{γ}(ρ) = M if γ ≤ 0 and F^{γ}(ρ) = 0 if γ > 0.

(F3r) If ρ ↠ τ is an epimorphism with ρ ∈ Rep′(G)(S) and τ ∈ Rep^{∘}(G)(S), then

F^{γ}(ρ) ↠ F^{γ}(τ) is an epimorphism in QCoh(T) for every γ ∈ Γ. (F3l) If ρ_{reg} belongs to Rep′(G)(S) and ρ_{1} ↪ ρ_{2} is a pure monomorphism in

Rep′(G)(S), then F^{γ}(ρ_{1}) = F^{γ}(ρ_{2}) ∩ V_{T} (ρ_{1}) in V_{T} (ρ_{2}) for every γ ∈ Γ.

Proof. (F2) is obvious and (F1), (F3r) follow from the eponymous properties of F on ω_{T}^{∘} because, since S is quasi-compact, {τ_{1} ⊗ τ_{2} : (τ_{1}, τ_{2}) ∈ X(ρ_{1}) × X(ρ_{2})} and {τ′ ∈ X(ρ) : τ′ ↠ τ} are respectively cofinal in X(ρ_{1} ⊗ ρ_{2}) and X(ρ). For (F3l), we first treat the special case of the pure monomorphism c_{ρ} : ρ ↪ ρ_{0} ⊗ ρ_{reg} for an arbitrary ρ ∈ Rep′(G)(S). Given (F1) and (F2), we have to show that

F^{γ}(ρ) = ker [ω_{T} (ρ) ⟶^{ω_{T} (c_{ρ})} ω_{T} (ρ_{0}) ⊗ (ω_{T} (ρ_{reg})/F^{γ}(ρ_{reg}))] .

Since both sides are filtered limits over τ ∈ X(ρ), we may assume that ρ belongs to Rep^{∘}(G)(S). The right hand side is then the filtered limit of

ker [ω_{T} (ρ) ⟶^{ω_{T} (c_{ρ,τ})} ω_{T} (ρ_{0}) ⊗ (ω_{T} (τ)/F^{γ}(τ))] = F^{γ}(ρ, τ)

where τ ranges through the cofinal set X′ of X(ρ_{reg}) defined by

X′ = {τ : c_{ρ} factors as ρ ⟶^{c_{ρ,τ}} ρ_{0} ⊗ τ ↪ ρ_{0} ⊗ ρ_{reg}} .

Note that ρ_{0} ⊗ τ ↪ ρ_{0} ⊗ ρ_{reg} since V (ρ_{0}) is a flat O_{S}-module. For each τ in X′, the cokernel σ_{ρ,τ} of c_{ρ,τ} : ρ ↪ ρ_{0} ⊗ τ is an object of Rep^{∘}(G)(S): the counit 1_{G}^{♮} : A(G) → O_{S} gives a retraction of V (c_{ρ,τ}), whose kernel is a direct factor of V (ρ_{0} ⊗ τ) isomorphic to V (σ_{ρ,τ}). Since F^{γ} is exact on Rep^{∘}(G)(S), it follows that

F^{γ}(ρ) = ker [ω_{T} (ρ) ⟶^{ω_{T} (c_{ρ,τ})} ω_{T} (ρ_{0} ⊗ τ)/F^{γ}(ρ_{0} ⊗ τ)] = F^{γ}(ρ, τ)

for every τ ∈ X′, which proves our claim. For any morphism ρ_{1} → ρ_{2} in Rep′(G)(S) and any γ ∈ Γ, we now have a commutative diagram with exact rows

0 → F^{γ}(ρ_{1}) → ω_{T} (ρ_{1}) → ω_{T} (ρ_{1,0}) ⊗ ω_{T} (ρ_{reg})/F^{γ}(ρ_{reg})
↓ ↓ ↓
0 → F^{γ}(ρ_{2}) → ω_{T} (ρ_{2}) → ω_{T} (ρ_{2,0}) ⊗ ω_{T} (ρ_{reg})/F^{γ}(ρ_{reg})

If V (ρ_{1}) → V (ρ_{2}) is a pure monomorphism, the vertical maps are monomorphisms, therefore F^{γ}(ρ_{1}) = F^{γ}(ρ_{2}) ∩ ω_{T} (ρ_{1}) in ω_{T} (ρ_{2}): this proves (F3l). □

3.6.6. As before, for every ρ ∈ Rep′(G)(S) and γ ∈ Γ, we define

F_{+}^{γ}(ρ) = ∪_{η>γ}F^{η}(ρ) and Gr_{F}^{γ}(ρ) = F^{γ}(ρ)/F_{+}^{γ}(ρ).

Since again filtered limits are exact, we find that

F_{+}^{γ}(ρ) = \underrightarrow{lim} F_{+}^{γ}(τ) and Gr_{F}^{γ}(ρ) = \underrightarrow{lim} Gr_{F}^{γ}(τ)

where τ ranges through X(ρ). In particular, the formula

Gr_{F}^{γ}(ρ_{1} ⊗ ρ_{2}) ≃ ⊕_{γ_{1}+γ_{2}=γ}Gr_{F}^{γ_{1}}(ρ_{1}) ⊗ Gr_{F}^{γ_{2}}(ρ_{2})

also holds for ρ_{1} and ρ_{2} in Rep′(G)(S). All of the above constructions commute with arbitrary base change on T. Finally if the original Γ-filtration F on ω_{T}^{∘} already was the restriction of some Γ-filtration F′ on ω_{T}, the restriction of the latter is equal to the extension of the former on ω′_{T} since F^{′γ} commutes with arbitrary colimits.

3.6.7. We first use the above device to show that:

Proposition 51. If G/S satisfies HYP(ω^{∘}), then 𝔽^{Γ}(V^{∘}) ↪ 𝔽^{Γ}(ω^{∘}).

Proof. By proposition 42, we may assume: S is quasi-compact and ρ_{reg} is in Rep′(G)(S). We have to show that for an S-scheme T and F ∈ 𝔽^{Γ}(V_{T}^{∘}) with image F̃ ∈ 𝔽^{Γ}(ω_{T}^{∘}), for any U → T, the Γ-filtration F_{U} on Rep^{∘}(G_{U})(U) → LF(U) induced by F is determined by F̃. We may assume that T and U are quasi-compact. Then: F_{U} is determined by its extension to Rep′(G_{U})(U) → QCoh(U), which itself is determined by its value on ρ_{reg,U} ∈ Rep′(G_{U})(U) thanks to (F1-2) and (F3l) applied to the pure monomorphisms c_{ρ} : ρ → ρ_{0} ⊗ ρ_{reg,U} for all ρ’s in Rep′(G_{U})(U). Since U is quasi-compact, X(ρ_{reg})_{U} = {τ_{U} : τ ∈ X(ρ_{reg})} is cofinal in X(ρ_{reg,U}), thus F_{U}(ρ_{reg,U}) is determined by the restriction of F_{U} to X(ρ_{reg})_{U}. By the axiom (F0) for F, the latter is determined by the values of F_{T} on X(ρ_{reg})_{T}, which are the values of F̃ on X(ρ_{reg}). Thus F̃ determines F_{U} and F uniquely. □

3.6.8. Here is another useful assumption: we say that G/S is linear if there exists τ ∈ Rep^{∘}(G)(S) inducing a closed immersion τ : G ↪ GL(V (τ)). Note that upon replacing τ with τ ⊕ (det τ)^{−1}, we may then also assume that det τ = 1.

Lemma 52. The affine and flat group G over S is linear in the following cases:

(1) G is of finite type over a noetherian regular S with dim S ≤ 2.

(2) HYP(ω^{∘}) holds and S is quasi-compact and quasi-separated.

(3) G is an isotrivial reductive group over a quasi-compact S.

(4) G is a reductive group of adjoint type over any S.

Proof. (1) is [20, VIB 13.2]. For (2), let f : S′ → S be a finite étale cover such that HYP′(ω^{∘}) holds for G′ = G_{S′}. Then S′ is also quasi-compact and quasiseparated, thus by [25, 1.7.9], the finitely generated quasi-coherent O_{S′}-algebra A(G′) is generated by a finitely generated quasi-coherent O_{S′}-submodule E. By assumption HYP′(ω^{∘}) for G′, we may replace E by a larger V (τ′) for some τ′ in X(f^{∗}ρ_{reg}). The proof of [20, VIB 13.2] then shows that τ′ : G′ → GL(V (τ′)) is a closed immersion. Put τ = f_{∗}τ′, so that τ belongs to Rep^{∘}(G)(S). Then τ : G → GL(V (τ)) is a closed immersion. Indeed, it is sufficient to show that f^{∗}τ : G′ → GL(V (f^{∗}τ)) is a closed immersion by [26, 2.7.1]. But f^{∗}τ = ρ ⊗ τ′ in Rep^{∘}(G′)(S′), where ρ = f^{∗}f_{∗}1_{G′} is the trivial representation on V (ρ) = f^{∗}f_{∗}O_{S′}, i.e. f^{∗}τ is the composition

\xymatrix{G′ \ar[r]^{ρ′} & GL(V (τ′)) \ar[r]^{Id⊗−} & GL(V (ρ) ⊗ V (τ′))}

of two closed immersions, therefore itself a closed immersion. For (3): it is wellknown that the Chevalley groups over Spec ℤ are linear (a complete overkill: use (1)), so are therefore also the split reductive groups over any base by [21, XXII 2.8, XXIII 5.2 and XXV 1.2], to which one reduces as in (2). For (4), simply take τ to be the adjoint representation ρ_{ad} of G on its Lie algebra Lie(G) = 𝔤 = V (ρ_{ad}). □

3.7. The stabilizer of a Γ-filtration, I

3.7.1. Let now G be a reductive group over S and let ρ_{ad} ∈ Rep^{∘}(G)(S) be the adjoint representation of G on V (ρ_{ad}) = 𝔤 = Lie(G). Let T be an S-scheme.

Theorem 53. Let F be a Γ-filtration on V_{T}. Then Aut^{⊗}(F) is a parabolic subgroup P_{F} of G_{T} with unipotent radical U_{F} ⊂ Aut^{⊗!}(F). Moreover,

Lie(U_{F}) = F_{+}^{0}(ρ_{ad}) and Lie(P_{F}) = F^{0}(ρ_{ad}) in V_{T} (ρ_{ad}) = 𝔤_{T} .

Remark 54. Let χ : 𝔻_{T} (Γ) → G_{T} be a morphism, G the corresponding Γgraduation and F the induced Γ-filtration. Let P_{χ} = U_{χ} ⋊ L_{χ} be the subgroups of G_{T} defined in proposition 14. Since Aut^{⊗}(F) = Aut^{⊗!}(F)⋊Aut^{⊗}(G) with Aut^{⊗}(G) equal to L_{χ} and isomorphic to Aut^{⊗}(Gr_{F}^{•}) (because G ≃ Gr_{F}^{•}), the theorem implies

P_{χ} = Aut^{⊗}(F), U_{χ} = Aut^{⊗!}(F) and P_{χ}/U_{χ} ≃ Aut^{⊗}(Gr_{F}^{•}).

Corollary 55. The quotients Fil : 𝔾^{Γ}(G) ↠ 𝔽^{Γ}(G) of 𝔾^{Γ}(G) defined in sections 2.2 and 3.2.9 are canonically isomorphic, and for any F ∈ 𝔽^{Γ}(G)(T),

P_{F} = Aut^{⊗}(ιF), U_{F} = Aut^{⊗!}(ιF) and P_{F}/U_{F} ≃ Aut^{⊗}(Gr_{ιF}^{•})

where ιF is the image of F in 𝔽^{Γ}(V_{T}).

Proof. For the first assertion, we have to show that for χ_{1}, χ_{2} : 𝔻_{T} (Γ) → G_{T},

χ_{1} ∼_{Par} χ_{2} ⟺ Fil ∘ ι(χ_{1}) = Fil ∘ ι(χ_{2}) in 𝔽^{Γ}(V_{T}).

Put G_{i} = ι(χ_{i}), F_{i} = Fil(G_{i}) and P_{i} = Aut^{⊗}(F_{i}) = P_{χ_{i}}. If χ_{1} ∼_{Par} χ_{2}, then χ_{2} = Int(p) ∘ χ_{1} for some p ∈ P_{1}(T), thus F_{2} = pF_{1} = F_{1}. If F_{1} = F_{2} = F, then P_{1} = P_{2} = P and the canonical isomorphism G_{1} ≃ Gr_{F}^{•} ≃ G_{2} gives an element of Aut^{⊗}(V_{T}) preserving F and mapping G_{1} to G_{2}, i.e. an element p ∈ P (T) such that χ_{2} = Int(p) ∘ χ_{1}, thus χ_{1} ∼_{Par} χ_{2}. The remaining assertions are local in the fpqc topology on T and thus follow from the above remark. □

3.7.2. For Γ-filtrations on ω_{T}, we need a technical assumption on G/S: TA There exists an fpqc cover {f_{i} : S_{i} → S} such that (a) each f_{i} is an affine

morphism, and (b) each G_{i} = G_{S_{i}} is linear (3.6.8). This is true for any reductive group G over a separated S: starting from a Zariski covering of S by affine U_{i}’s, we pick fpqc covers {U_{i,j} → U_{i}} splitting G_{U_{i}}, and again cover the U_{i,j}’s by affine U_{i,j,k}’s. The resulting fpqc cover {U_{i,j,k} → S} satisfies our assumption: U_{i,j,k} → U_{i} is affine as a morphism between affine schemes, U_{i} ↪ S is affine because S is separated, and G_{U_{i,j,k}} is linear by lemma 52 since it is split.

Theorem 56. Assume T A. Let F be a Γ-filtration on ω_{T}. Then Aut^{⊗}(F) is a parabolic subgroup P_{F} of G_{T} with unipotent radical U_{F} ⊂ Aut^{⊗!}(F). Moreover,

Lie(U_{F}) = F_{+}^{0}(ρ_{ad}) and Lie(P_{F}) = F^{0}(ρ_{ad}) in V_{T} (ρ_{ad}) = 𝔤_{T} .

3.7.3. If F′ is a Γ-filtration on V_{T} and F is the induced Γ-filtration on ω_{T}, then Aut^{⊗}(F′) = Aut^{⊗}(F) as subsheaves of G_{T} by 3.5.3 and theorem 44. Therefore:

(a) theorem 56 holds without the technical assumption for such filtrations on ω_{T}, and (b) theorem 53, which is local on S, follows from theorem 56 applied to any affine cover of S. We thus only have to consider the case of a Γ-filtration F on ω_{T}. The technical assumption will be used only once below, in section 3.7.9.

3.7.4. The adjoint-regular representation ρ_{adj} of G on V (ρ_{adj}) = A(G) is given by

(g · a)(h) = a(g^{−1}hg)

for T → S, a ∈ Γ(T, A(G_{T})) and g, h ∈ G(T). The unit, counit 1_{G}^{♮}, multiplication, comultiplication μ^{♮} and inversion inv^{♮} of A(G) define morphisms in Rep(G)(S):

1_{S} → ρ_{adj}, ρ_{adj} → 1_{S}, ρ_{adj} ⊗ ρ_{adj} → ρ_{adj}, ρ_{adj} → ρ_{adj} ⊗ ρ_{adj}, ρ_{adj} → ρ_{adj}.

For any ρ in Rep(G)(S), we may also view c_{ρ} as a split monomorphism

c_{ρ} : ρ → ρ ⊗ ρ_{adj} in Rep(G)(S).

If τ belongs to Rep^{∘}(G)(S), c_{τ} gives a morphism τ^{∨} ⊗ τ → ρ_{adj} which induces a G-equivariant morphism of quasi-coherent G − O_{S}-algebras

Sym^{•}(τ^{∨} ⊗ τ) → ρ_{adj}

whose underlying morphism of quasi-coherent O_{S}-algebras is given by

Sym_{O_{S}}^{•}(V (τ)^{∨} ⊗ V (τ)) ↪ Sym_{O_{S}}^{•} (\underline{End}_{O_{S}}(τ)) [\frac{1}{det}] = A (GL(V (τ))) ⟶^{τ^{♮}} A(G)

where τ^{♮} is the morphism attached to τ : G → GL(V (ρ)). In particular, if the latter is a closed embedding and det(τ) = 1, then Sym^{•}(τ^{∨}⊗τ) ↠ ρ_{adj} is an epimorphism.

3.7.5. Let ρ_{adj}^{∘} be the kernel of 1_{G}^{♮} : ρ_{adj} → 1_{S}. Thus ρ_{adj} = ρ_{adj}^{∘} ⊕ 1_{S} and V (ρ_{adj}^{∘}) is the augmentation ideal I(G) of A(G). For any n ≥ 1, the multiplication map I(G)^{⊗n+1} → I(G) defines a morphism (ρ_{adj}^{∘})^{⊗n+1} → ρ_{adj}^{∘} in Rep(G)(S). We denote by ρ^{n} ∈ Rep^{∘}(G)(S) its cokernel, a representation of G on V (ρ^{n}) = I(G)/I(G)^{n+1}, and by ρ_{n} = (ρ^{n})^{∨} ∈ Rep^{∘}(G)(S) the dual of ρ^{n}. Thus ρ_{1} = ρ_{ad}, the adjoint representation of G on V (ρ_{ad}) = 𝔤.

3.7.6. Let now I(F) and J (F) be the quasi-coherent ideals of A(G_{T}) which are respectively generated by the quasi-coherent subsheaves F_{+}^{0}(ρ_{adj}^{∘}) and F^{0}(ρ_{adj}^{∘}) of the augmentation ideal I(G_{T}) = ω_{T} (ρ_{adj}^{∘}) of A(G_{T}). Then

U_{F} =^{def} Spec (A(G_{T})/J (F)) ↪ P_{F} =^{def} Spec (A(G_{T})/I(F))

are closed subgroup schemes of G_{T}, because J (F) and I(F) are compatible with the comultiplication μ_{T}^{♮} and inversion inv_{T}^{♮} of A(G_{T}), since μ^{♮} : ρ_{adj} → ρ_{adj} ⊗ ρ_{adj} and inv^{♮} : ρ_{adj} → ρ_{adj} are morphisms in Rep(G)(S). It follows from their definition that the formation of U_{F} and P_{F} commutes with arbitrary base change on T.

3.7.7. Let N(U_{F}) and N(P_{F}) be the normalizers of U_{F} and P_{F} in G_{T}. Then

P_{F} ⊂ Aut^{⊗}(F) ⊂ N(U_{F}), N(P_{F}) and U_{F} ⊂ Aut^{⊗!}(F)

as fpqc subsheaves of G_{T}. We have to check this on sections over an arbitrary T -scheme X, but we may assume that X = T. Since G = Aut^{⊗}(ω) by theorem 44,

Aut^{⊗}(F)(T) = {g ∈ G(T)| ∀ρ, γ : ρ(g) (F^{γ}(ρ)) = F^{γ}(ρ)} .

On the other hand, for any ρ in Rep(G)(S), the morphism c_{ρ} : ρ → ρ ⊗ ρ_{adj} gives a morphism ω_{T} (c_{ρ}) : ω_{T} (ρ) → ω_{T} (ρ) ⊗ ω_{T} (ρ_{adj}) in QCoh(T) mapping F^{γ}(ρ) into

F^{γ}(ρ ⊗ ρ_{adj}) = _{∑α+β=γ}F^{α}(ρ) ⊗ F^{β}(ρ_{adj}).

(a) For g in Aut^{⊗}(F)(T), ρ_{adj}^{∘}(g) fixes F_{+}^{0}(ρ_{adj}^{∘}) = ∪_{γ>0}F^{γ}(ρ_{adj}^{∘}) as well as the A(G_{T})-ideal I(F) which it spans. It follows that the inner automorphism of G_{T} defined by g fixes P_{F}. Thus g belongs to N(P_{F})(T). Similarly, g ∈ N(U_{F})(T).

(b) For g in P_{F}(T), g^{♮} : A(G_{T}) → O_{T} is trivial on F^{β}(ρ_{adj}) for every β > 0 and thus ρ(g) = (Id ⊗ g^{♮}) ∘ ω_{T} (c_{ρ}) maps F^{γ}(ρ) into ∑_{α≥γ} F^{α}(ρ) = F^{γ}(ρ). Since g^{−1} also belongs to P_{F}(T), ρ(g) fixes F^{γ}(ρ). Thus g belongs to Aut^{⊗}(F)(T).

(c) For g in U_{F}(T), g^{♮}−1^{♮} : A(G_{T}) → O_{T} is trivial on F^{0}(ρ_{adj}) = O_{T} ⊕F^{0}(ρ_{adj}^{∘}), thus ρ(g)−ρ(1) = (Id ⊗ (g^{♮} − 1^{♮}))∘ω_{T} (c_{ρ}) maps F^{γ}(ρ) into ∑_{α>γ} F^{α}(ρ) = F_{+}^{γ}(ρ). Therefore g belongs to Aut^{⊗!}(F)(T).

3.7.8. We will establish below that the neutral components [20, VIB 3.1] U_{F}^{∘} and P_{F}^{∘} of U_{F} and P_{F} are smooth over S, using the following criterion:

Proposition 57. Let G be affine smooth over S, A = A(G) and I = I(G). Let H ⊂ G be a closed subgroup defined by a quasi-coherent ideal J of A such that

(1) J is finitely generated,

(2) J ∩ I^{2} = I · J in A, and

(3) I/J + I^{2} is finite locally free on S. Then H^{∘} is representable by a smooth open subgroup scheme of H.

Proof. By [20, VIB 3.10], we have to show that H is smooth at all points of its unit section. Let thus x ∈ H be the image of s ∈ S under 1_{H} : S → H. By [25, 1.4.3 and 1.4.5], we already know from (1) that H is locally of finite presentation over S. Thus by [28, 17.5.1] and the Jacobian criterion [25, 0_{IV} 22.6.4], we have to show that J_{x}/J_{x}^{2} ⊗_{O_{H,x}} k → Ω_{O_{G,x}/O_{S,s}}^{1} ⊗_{O_{G,x}} k is injective, where k is the common residue field of s and x, and the morphism is induced by the universal derivation d : O_{G,x} → Ω_{O_{G,x}/O_{S,s}}^{1}. This map factors through the corresponding map for I_{x}, namely I_{x}/I_{x}^{2} ⊗_{O_{S,s}} k → Ω_{O_{G,x}/O_{S,s}}^{1} ⊗_{O_{G,x}} k, which is injective (because O_{G,x}/I_{x} = O_{S,s} is formally smooth over itself!). We thus have to show that

J_{x}/J_{x}^{2} ⊗_{O_{H,x}} k = J_{x}/m_{x}J_{x} → I_{x}/m_{x}I_{x} = I_{x}/I_{x}^{2} ⊗_{O_{S,s}} k

is injective, where m_{x} is the maximal ideal of O_{G,x}. The latter map is base-changed from the morphism J_{x}/J_{x}I_{x} → I_{x}/I_{x}^{2}, which itself is the localization at x of the morphism J /J I → I/I^{2}, which is a pure monomorphism by assumption. □

3.7.9. We show that I(F) and J (F) are finitely generated, focusing on I(F) to simplify the exposition. Let {S_{i} → S} be an fpqc cover as in assumption (TA), {f_{i} : T_{i} → T } the corresponding fpqc cover of T, ω_{i} the fiber functor for G_{i} = G_{S_{i}} and F_{i} the extension of F_{T_{i}} to a Γ-filtration on ω_{i,T_{i}} – which exists by proposition 42 since f_{i} is affine. By [26, 2.5.2], it is sufficient to show that f_{i}^{∗}I(F) is finitely generated. Since f_{i} is flat, f_{i}^{∗}I(F) = I(F_{T_{i}}) and obviously I(F_{T_{i}}) = I(F_{i}). We may thus assume that G is linear over S: there exists τ ∈ Rep^{∘}(G)(S) inducing a closed embedding τ : G ↪ GL(V (τ)) with det τ ≡ 1, thus also an epimorphism S^{•}(τ) = Sym^{•}(τ^{∨} ⊗ τ) ↠ ρ_{adj} of quasi-coherent G-O_{S}-algebras. By the axiom (F3) for F, I(F) is the image of the ideal I(τ) spanned by F_{+}^{0}(S^{•}(τ)) in V (S^{•}(τ))_{T}. Using proposition 39, we may assume that there is a splitting V (τ^{∨} ⊗ τ)_{T} = ⊕_{γ}G_{γ} of F on τ^{∨} ⊗ τ. By the axioms (F1) and (F3), it induces a splitting of F on S^{•}(τ),

V (S^{n}(τ))_{T} = ⊕_{γ} ⊕_{γ_{1}+···+γ_{n}=γ} G_{γ_{1}} ⋯ G_{γ_{n}}.

It follows easily that I(τ) is spanned by the finite locally free subsheaf ⊕_{γ>0}G_{γ} of V (S^{1}(τ))_{T} = V (τ^{∨} ⊗ τ)_{T}, therefore I(τ) and I(F) are indeed finitely generated.

3.7.10. We show that I(F) ∩ I(G_{T})^{2} = I(F) · I(G_{T}) – the proof for J (F) is similar. Plainly, I(F) · I(G_{T}) ⊂ I(F) ∩ I(G_{T})^{2}. For the other inclusion, we may assume that T is affine and work with global sections. Let thus s be a (global) section of I(F), so that s = a + b with a a section of F_{+}^{0}(ρ_{adj}^{∘}) and b a section of

I(G_{T}) · F_{+}^{0}(ρ_{adj}^{∘}) ⊂ I(G_{T}) · I(F) ⊂ I(G_{T})^{2}.

Then s belongs to I(G_{T})^{2} if and only a does, i.e. a is a section of F_{+}^{0}(ρ_{adj}^{∘})∩I(G_{T})^{2}. The pure short exact sequence and epimorphism of quasi-coherent sheaves on S

0 → I(G)^{2} → I(G) → I(G)/I(G)^{2} → 0 and I(G)^{⊗2} ↠ I(G)^{2}

correspond to a pure short exact sequence and epimorphism in Rep(G)(S),

0 → ρ_{adj}^{∘(2)} → ρ_{adj}^{∘} → ρ^{1} → 0 and (ρ_{adj}^{∘})^{⊗2} ↠ ρ_{adj}^{∘(2)}

which together give, using the axioms (F1) and (F3) for F,

F_{+}^{0}(ρ_{adj}^{∘}) ∩ I(G_{T})^{2} = F_{+}^{0}(ρ_{adj}^{∘(2)}) = _{∑γ_{1}+γ_{2}>0}F^{γ_{1}}(ρ_{adj}^{∘}) · F^{γ_{2}}(ρ_{adj}^{∘})

which is contained in I(F) · I(G_{T}), thus I(F) ∩ I(G_{T})^{2} ⊂ I(F) · I(G_{T}).

3.7.11. We show that I(G_{T})/I(F) + I(G_{T})^{2} is finite locally free – the proof for J (F) is similar. By the axiom (F3), I(F) + I(G_{T})^{2}/I(G_{T})^{2} is the A(G_{T})-submodule of I(G_{T})/I(G_{T})^{2} = ω_{T} (ρ^{1}) generated by F_{+}^{0}(ρ^{1}), i.e. this O_{T} - submodule itself since A(G_{T}) acts on I(G_{T})/I(G_{T})^{2} through O_{T}. We are thus claiming that ω_{T} (ρ^{1})/F_{+}^{0}(ρ^{1}) is finite locally free, which follows from proposition 39.

3.7.12. We have just established that U_{F}^{∘} and P_{F}^{∘} are representable by smooth open subschemes of U_{F} and P_{F}. They are also finitely presented over T: they are separated over T as compositions of affine morphisms and open immersions, and they are quasi-compact over T by [20, VIB 3.9], since U_{F} and P_{F} are finitely presented over S, being locally of finite presentation by 3.7.9 and [25, 1.4.5], and affine by definition. From 3.7.7, we obtain the following chain of inclusions

\begin{aligned}U_{F}^{∘} & ⊂ & U_{F} & ⊂ & Aut^{⊗!}(F) \\ ∩ & & ∩ & & ∩ \\ P_{F}^{∘} & ⊂ & P_{F} & ⊂ & Aut^{⊗}(F) \end{aligned} and \begin{aligned}Aut^{⊗}(F) & ⊂ & N(P_{F}) & ⊂ & N(P_{F}^{∘}) \\ ∥ \\ Aut^{⊗}(F) & ⊂ & N(U_{F}) & ⊂ & N(U_{F}^{∘}) \end{aligned}

The Lie algebras of U_{F}^{∘} ⊂ U_{F} and P_{F}^{∘} ⊂ P_{F} are respectively given by

Lie(U_{F}^{∘}) = Lie(U_{F}) = (I(G_{T})/J (F) + I(G_{T})^{2})^{∨}
and Lie(P_{F}^{∘}) = Lie(P_{F}) = (I(G_{T})/I(F) + I(G_{T})^{2})^{∨}

As quasi-coherent O_{T} -submodules of

Lie(G_{T}) = 𝔤_{T} = (I(G_{T})/I(G_{T})^{2})^{∨}

they correspond to the O_{T} -linear forms on ω_{T} (ρ^{1}) = I(G_{T})/I(G_{T})^{2} vanishing on

F^{0}(ρ^{1}) = J (F) + I(G_{T})^{2}/I(G_{T})^{2} and F_{+}^{0}(ρ^{1}) = I(F) + I(G_{T})^{2}/I(G_{T})^{2}

respectively. We thus find that, as O_{T} -submodules of 𝔤_{T} = ω_{T} (ρ_{ad}) = ω_{T} (ρ_{1}),

Lie(U_{F}^{∘}) = Lie(U_{F}) = F_{+}^{0}(ρ_{ad}) and Lie(P_{F}^{∘}) = Lie(P_{F}) = F^{0}(ρ_{ad}).

3.7.13. We show that P_{F}^{∘} is a parabolic subgroup of G_{T} with unipotent radical U_{F}^{∘}. Since both groups are finitely presented and smooth over T with P_{F}^{∘} ⊂ N(U_{F}^{∘}), we may assume that T = Spec(k) for some algebraically closed field k by [21, XXVI 1.1 and 1.6]. Since then T → S is affine, we may also assume that S = Spec(k) by part (2) of proposition 42, in which case G is linear by lemma 52. Using the criterion of [41, IV 2.4.3.1], we now have to verify that

(a) dim U_{F}^{∘} = dim G/P_{F}^{∘} and (b) U_{F}^{∘} is unipotent.

The equality of dimensions follows from proposition 58 below since

dim U_{F}^{∘} = dim_{k} Lie(U_{F}^{∘}) = dim_{k} F_{+}^{0}(ρ_{ad}) = _{∑γ>0} dim_{k} Gr_{k}^{γ}(ρ_{ad})

and

dim G/P_{F}^{∘} = dim_{k} 𝔤/F^{0}(ρ_{ad}) = dim_{k} F_{+}^{0}(ρ_{ad}^{∨}) = _{∑γ>0} dim_{k} Gr_{k}^{γ}(ρ_{ad}).

For (b), pick a finite dimensional faithful representation τ of G. Then

U_{F}^{∘} ⊂ U_{F} ⊂ Aut^{⊗!}(F) ⊂ U(F(τ))

where U(F(τ)) is the unipotent subgroup of GL(V (τ)) defined by the Γ-filtration F(τ) on V (τ). Therefore U_{F}^{∘} is unipotent by [1, XVII 2.2.ii].

3.7.14. By [21, XXII 5.8.5], P_{F}^{∘} = N(P_{F}^{∘}), therefore also
P_{F}^{∘} = P_{F} = Aut^{⊗}(F) = N(P_{F}) = N(P_{F}^{∘}).

On the other hand, the above proof of (b) shows that U_{F} has unipotent geometric fibers, and then so does its quotient U_{F}/U_{F}^{∘} by [1, XVII 2.2.iii]. Since U_{F}/U_{F}^{∘} is also normal in the reductive group P_{F}^{∘}/U_{F}^{∘}, it must be trivial, thus U_{F} = U_{F}^{∘} and this finishes the proof of our theorem. Note that we can not say much more about Aut^{⊗!}(F) at this point – we do not even know that it is actually representable.

3.8. Grothendieck groups

Let again G be affine and flat over S. Let T be an S-scheme and let F be a Γ-filtration on ω_{T}^{∘}. Since F and Gr are exact ⊗-functors,

Gr_{F}^{•} : Rep^{∘}(G)(S) ⟶^{F} Fil^{Γ}LF(T) ⟶^{Gr} Gr^{Γ}LF(T)

is also an exact ⊗-functor. It thus induces a morphism between the Grothendieck ring K_{0}(G) of Rep^{∘}(G)(S) and the Grothendieck ring of Gr^{Γ}LF(T). The rank function on finite locally free sheaves over T defines a morphism from the latter ring to the ring C(T, ℤ[Γ]) of locally constant functions on T with values in the group ring ℤ[Γ] of Γ. The Γ-filtration F on ω_{T}^{∘} thus defines a ring homomorphism

κ(F) : K_{0}(G) → C(T, ℤ[Γ])

which maps the class of ρ ∈ Rep^{∘}(G)(S) in K_{0}(G) to the function

t ↦ _{∑γ∈Γ} dim_{k(t)} (Gr_{F}^{γ}(ρ) ⊗ k(t)) · e^{γ}

where e^{γ} is the basis element of ℤ[Γ] corresponding to γ. We have:

∀z ∈ K_{0}(G) : κ(F)(z^{∨}) = κ(F)(z)^{∨}

where the involutions z ↦ z^{∨} are induced by the duality ρ ↦ ρ^{∨} on Rep^{∘}(G)(S) and by ∑ x_{λ}e^{λ} ↦ ∑ x_{λ}e^{−λ} on ℤ[Γ]. When G is smooth over S, we define

κ(G) = [ρ_{ad}] − [ρ_{ad}^{∨}] ∈ K_{0}(G)
and κ(G, F) = image of κ(G) in C(T, ℤ[Γ])

The formation of κ(G, F) is compatible with arbitrary base change on T.

Proposition 58. If (1) G is an isotrivial reductive group over a quasi-compact S, or (2) G is a reductive group over S and F comes from a filtration on ω_{T}, then

κ(G, F) = 0 in C(T, ℤ[Γ]).

Proof. (1) Let {S_{i} → S} be a covering of S by finite étale morphisms such that each G_{i} = G_{S_{i}} splits. Let {T_{i} → T } be the corresponding covering of T. By part (1) of proposition 42, F_{T_{i}} extends to a Γ-filtration F_{i} on ω_{i}^{∘} : Rep^{∘}(G_{i})(S_{i}) → LF(T_{i}), and obviously κ(G_{i}, F_{i}) = κ(G, F) ∘ (T_{i} → T). We may thus assume that G splits over S, in which case the proposition follows from lemma 59 below. The proof of (2) is similar: let {t → T } be a covering of T by geometric points, thus G_{t} splits. By part (2) of proposition 42, F_{t} extends to a Γ-filtration on ω_{t}^{∘} : Rep(G_{t})(t) → LF(t) which we also denote by F_{t}, and obviously κ(G_{t}, F_{t}) = κ(G, F) ∘ (t → T). □

Lemma 59. If G is a split reductive group over a quasi-compact S, then

κ(G) = 0 in K_{0}(G).

Proof. By [21, XXII 2.8], there is a decomposition S = ∐_{i∈I} S_{i} into open and closed subschemes S_{i} ≠ ∅ of S such that for each i ∈ I, G_{S_{i}} is of constant type, thus isomorphic [21, XXIII 5.2] to the base change of a split reductive group G_{0,i} over Spec(ℤ) [21, XXV 1.2]. Since S is quasi-compact, the indexing set I is finite and K_{0}(G) ≃ ⊗_{i∈I}K_{0}(G_{S_{i}}) with κ(G) = ∑_{i∈I} κ(G)_{i} where κ(G)_{i} is the image of κ(G_{0,i}) under K_{0}(G_{0,i}) → K_{0}(G_{S_{i}}) → K_{0}(G). We may thus assume that S = Spec(A) where A a principal ideal domain. By [44, Théorème 5], we may even assume that A = K is a field. Let H be a split maximal torus in G, with character group M and Weyl group W. The restriction Rep^{∘}(G) → Rep^{∘}(H) induces a ring homomorphism K_{0}(G) → K_{0}(H) ≃ ℤ[M] which yields an isomorphism from K_{0}(G) to ℤ[M]^{W} by [44, Théorème 4]. Let R ⊂ M be the set of roots of H in the Lie algebra 𝔤 = V (ρ_{ad}). The weight decomposition of ρ_{ad}|H is then given by 𝔤 = 𝔤_{0} ⊕ ⊕_{α∈R}𝔤_{α} with dim_{K} 𝔤_{α} = 1 for α ∈ R and 𝔤_{0} = 𝔥 is the Lie algebra of H. Since R = −R, we find that ρ_{ad}|H ≃ ρ_{ad}^{∨}|H. Thus indeed κ(G) = 0 in K_{0}(G). □

3.9. The stabilizer of a Γ-filtration, II

We have the following variant of theorem 53 and 56. Let G be an isotrivial reductive group over a quasi-compact S.

Theorem 60. For an S-scheme T and a Γ-filtration F on V_{T}^{∘} or ω_{T}^{∘}, Aut^{⊗}(F) is a parabolic subgroup P_{F} of G_{T} with unipotent radical U_{F} ⊂ Aut^{⊗!}(F). Moreover,

Lie(U_{F}) = F_{+}^{0}(ρ_{ad}) and Lie(P_{F}) = F^{0}(ρ_{ad}) in V_{T} (ρ_{ad}) = 𝔤_{T} .

Corollary 61. For any S-scheme T and F ∈ 𝔽^{Γ}(G)(T),

P_{F} = Aut^{⊗}(ιF), U_{F} = Aut^{⊗!}(ιF) and P_{F}/U_{F} = Aut^{⊗}(Gr_{ιF}^{•})

where ιF stands for the image of F in either 𝔽^{Γ}(V_{T}^{∘}) or 𝔽^{Γ}(ω_{T}^{∘}).

The proof of the corollary is identical to that of its earlier counterpart.

3.9.1. By propositions 49, 45, 46 and 51, it is sufficient to establish the theorem for a Γ-filtration F on ω_{T}^{∘}. For any T -scheme X, we have

Aut^{⊗}(F)(X) = {g ∈ G(X)|∀τ, γ ∈ Rep^{∘}(G)(S) × Γ : ρ_{X}(g) (F^{γ}(τ)_{X}) = F^{γ}(τ)_{X}} ,
= {g ∈ G(X)|∀ρ, γ ∈ Rep′(G)(S) × Γ : ρ_{X}(g) (F^{γ}(ρ)_{X}) = F^{γ}(ρ)_{X}} .

We have to show that the fpqc subsheaf Aut^{⊗}(F) : (Sch/T)^{0} → Set of G_{T} is representable by a parabolic subgroup with the specified Lie algebra: this is a local question in the fpqc topology on T. Let {S_{i} → S} be a covering of S by finite étale morphisms such that G_{i} = G_{S_{i}} is split, let {T_{i} → T } be the induced covering of T, let ω_{i} denote the fiber functors for G_{i} and let F_{i} be the unique extension of F_{T_{i}} to a Γ-graduation on ω_{i,T_{i}}^{∘}. Going back to its actual definition in the proof of proposition 42, one checks easily that Aut^{⊗}(F)|_{T_{i}} = Aut^{⊗}(F_{T_{i}}) is equal to Aut^{⊗}(F_{i}) as a subsheaf of G|_{T_{i}} = Aut^{⊗}(ω^{∘})|_{T_{i}} = Aut^{⊗}(ω_{i}^{∘})|_{T_{i}}. We may (and do) therefore assume that G is a split reductive group over a quasi-compact S. By [21, XXII 2.8, XXIII 5.2 and XXV 1.2], we then have a finite partition of S = ∐ S_{i} into open and closed subschemes such that each G_{i} = G_{S_{i}} arises from a split group over Spec(ℤ), and repeating the above argument with that covering, we may thus also assume that G is the base change of a split reductive group G_{0} over Spec(ℤ).

3.9.2. In particular, the proof of part (2) of proposition 49 now shows that with ρ_{reg}, also ρ_{adj} and ρ_{adj}^{∘} belong to Rep′(G)(S), to which we have extended F in section 3.6.4. We may thus define subschemes U_{F} and P_{F} of G_{T} as in section 3.7.6, and try to follow from there on the subsequent steps of the proof of theorem 53. Of course, we have to check that we are only using our filtration where it is defined, namely on Rep′(G)(S), and that whenever the axiom (F3) was used, we could have replaced it with the weaker left and right properties (F3l) or (F3r).

3.9.3. In 3.7.9 and 3.7.10, we used the right exactness of F for (respectively)

A : S^{•}(τ) = Sym^{•}(τ^{∨} ⊗ τ) ↠ ρ_{adj} and B : (ρ_{adj}^{∘})^{⊗2} ↠ ρ_{adj}^{∘(2)}.

To deal with the first one, it would be sufficient to know that there is a cofinal set Σ ∈ X(ρ_{adj}) such that for all σ ∈ Σ, A^{−1}(σ) is still in Rep′(G)(S): then

F^{γ}(ρ_{adj}) = \underrightarrow{lim} F^{γ}(σ) =^{F3r} \underrightarrow{lim} A(F^{γ}(A^{−1}(σ)))
= A(F^{γ}(\underrightarrow{lim} A^{−1}(σ))) = A(F^{γ}(S^{•}(τ))).

Over a Dedekind domain, we have Wedhorn’s criterion: a ρ is in Rep′(G)(S) if and only V (ρ) is flat, i.e. torsion free: thus over such a domain, A^{−1}(σ) still belongs to Rep′(G)(S) for any σ ∈ X(ρ_{adj}). Applying this to G_{0} and choosing τ in 3.7.9 to also be defined over Spec(ℤ) settles the case of A, and that of B is similar.

3.9.4. Everything then goes through up to 3.7.13: U_{F}^{∘} and P_{F}^{∘} are smooth subgroups of G_{T} with the good Lie algebras, etc… In 3.7.13, we may still reduce to the case where T = Spec(k) for some algebraically closed field k and use the criterion of [41, IV 2.4.3.1], but we can not change S to Spec(k). However, since we have already reduced to the split case, proposition 58 (or lemma 59) deals perfectly well with condition (a), and lemma 52 with condition (b).

3.10. Splitting filtrations

We now come to the main statement of theorem 34. Let thus G be a reductive isotrivial group over a quasi-compact S, let T be an S-scheme and let F be a Γfiltration on ω_{T}^{∘}. We will then show that: locally on T for the étale topology, F has a splitting χ : 𝔻_{T} (Γ) → G_{T}.

3.10.1. Let f : S̃ → S be a finite étale cover splitting G and denote by F̃ the unique extension of FT̃ to a Γ-filtration on ω̃∘ ; T̃ (see proposition 42), where T̃ = TS̃ and ω̃ is the fiber functor for G̃ = GS̃. If χ : 𝔻T̃(Γ) → GT̃ is a splitting of F̃, it is a fortiori a splitting of FT̃: we may thus assume that G splits over S.

3.10.2. For a positive integer k, there is a cartesian diagram of fpqc sheaves on S,

[figure: 𝔾^{Γ}(G); Prop. 45; k_{1}; 𝔾^{Γ}(ω^{∘}); Fil; k_{2}; 𝔽^{Γ}(ω^{∘}); k_{3}; 𝔾^{Γ}(G); 45^{Prop.}; 𝔾^{Γ}(ω^{∘}); Fil; 𝔽^{Γ}(ω^{∘})]

where the k_{i}’s map χ, G and F to respectively k_{1}(χ) = χ ∘ 𝔻_{T} (k) = χ^{k},

k_{2}(G)_{γ}(ρ) = \begin{cases}0 & if γ ∉ kΓ, \\ G_{η}(ρ) & if γ = kη, \end{cases} and k_{3}(F)^{γ}(ρ) = ∪_{kη≥γ}F^{η}(ρ).

They are all obviously well-defined monomorphisms, and the image of k_{2} is the subsheaf of 𝔾^{Γ}(ω^{∘}) made of those Γ-graduation G′ for which G′_{γ} ≡ 0 for γ ∉ kΓ. The diagram is cartesian because if G′ splits k_{3}(F), then G′_{γ} ≃ Gr_{k_{3}(F)}^{γ} ≡ 0 for γ ∉ kΓ, thus G′ = k_{2}(G) for a unique G, which has to also split F since

k_{3}(F) = Fil(G′) = Fil (k_{2}(G)) = k_{3}(Fil(G)).

3.10.3. For a central isogeny f : G → G′, there is a commutative diagram

[figure: 𝔾^{Γ}(G); Prop. 45; f_{1}; 𝔾^{Γ}(ω^{∘}); Fil; f_{2}; 𝔽^{Γ}(ω^{∘}); f_{3}; 𝔾^{Γ}(G′); 45^{Prop.}; 𝔾^{Γ}(ω^{′∘}); Fil; 𝔽^{Γ}(ω^{′∘})]

where ω′ = ω ∘ f^{∗} denotes the fiber functor for G′ and the f_{i}’s map χ, G and F to respectively f_{1}(χ) = f ∘ χ, f_{2}(G) = G ∘ f^{∗} and f_{3}(F) = F ∘ f^{∗}, with

f^{∗} : Rep(G′)(S) → Rep(G)(S) f^{∗}(ρ) = ρ ∘ f.

We claim that (1) all f_{i}’s are monomorphisms, and (2) the diagram is cartesian. This is local in the finite étale topology on S by proposition 42, and we may thus assume that the kernel C of f is isomorphic to 𝔻_{S}(X) for some finite commutative group X. We fix an S-scheme T and consider sections of the above sheaves over T. If f ∘ χ_{1} = f ∘ χ_{2}, then χ_{1}^{−1}χ_{2} is a morphism 𝔻_{T} (Γ) → C_{T}, which has to be trivial since X is finite and Γ torsion free: f_{1} is injective. Any ρ ∈ Rep^{∘}(G)(S) has a finite sum decomposition ρ = ⊕ρ(x) according to the characters x ∈ X of C, and C acts trivially on ρ(x)^{⊗k(x)} where k(x) ≥ 1 is the order of x in X. If two Γ-filtrations F_{1} and F_{2} on ω_{T}^{∘} induce the same Γ-filtration on ω_{T}^{′∘}, then F_{1}(ρ) = F_{2}(ρ) for every ρ on which C acts trivially, thus F_{1}(ρ(x)) = F_{2}(ρ(x)) for every ρ and x by lemma 62 below, therefore F_{1}(ρ) = F_{2}(ρ) since ρ = ⊕ρ(x): f_{3} is injective. Similarly: f_{2} is injective. Finally, suppose that G′ splits f_{3}(F). Let χ′ : 𝔻_{T} (Γ) → G′_{T} be the corresponding morphism. Fix k ≥ 1 such that k_{1}(χ′) lifts to χ_{k} : 𝔻_{T} (Γ) → G_{T}, giving a Γ-graduation G_{k} and a Γ-filtration F_{k} on ω_{T}^{∘}. They respectively map to

f_{2}(G_{k}) = f_{2} ∘ ι(χ_{k}) = ι ∘ f_{1}(χ_{k}) = ι ∘ k_{1}(χ′) = k_{2} ∘ ι(χ′) = k_{2}(G′)

where ι is the isomorphism 𝔾^{Γ}(G) ≃ 𝔾^{Γ}(ω^{∘}), and

f_{3}(F_{k}) = f_{3} ∘ Fil(G_{k}) = Fil ∘ f_{2}(G_{k}) = Fil ∘ k_{2}(G′) = k_{3} ∘ Fil(G′) = k_{3} ∘ f_{3}(F).

Thus f_{3}(F_{k}) = f_{3} ∘ k_{3}(F) and F_{k} = k_{3}(F) since f_{3} is a monomorphism. Since G_{k} splits k_{3}(F), there is a unique G such that F = Fil(G) and k_{2}(G) = G_{k} by the cartesian diagram of the previous subsection. Moreover f_{2}(G) = G′ since

k_{2} ∘ f_{2}(G) = f_{2} ∘ k_{2}(G) = f_{2}(G_{k}) = k_{2}(G′)

and k_{2} is a monomorphism: our diagram is indeed cartesian.

Lemma 62. Let M be a finite locally free sheaf on a scheme S, k ≥ 1.

(1) Let F_{1} and F_{2} be local direct factors of M. Then:

F_{1}^{⊗k} = F_{2}^{⊗k} in M^{⊗k} ⟹ F_{1} = F_{2} in M.

(2) Let F_{1} and F_{2} be Γ-filtrations on M. Then:

F_{1}^{⊗k} = F_{2}^{⊗k} on M^{⊗k} ⟹ F_{1} = F_{2} on M.

Proof. (1) Fix s ∈ S with residue field k(s). We have to show that F_{1} = F_{2} in a neighborhood of s. Shrinking S if necessary, we may assume that F_{1} and F_{2} are free of constant rank n_{1} and n_{2}. By assumption, n_{1}^{k} = n_{2}^{k}, therefore n_{1} = n_{2} = n. If n = 0, F_{1} = 0 = F_{2} and we are done. Suppose n > 0, and choose a linear form f : M(s) → k(s) which is non-zero on F_{1}(s) and F_{2}(s). Shrinking S further, we may lift f to an O_{S}-linear map f : M → O_{S} such that f(F_{1}) = O_{S} = f(F_{2}). Then for the O_{S}-linear map F = Id ⊗ f^{k−1} : M^{⊗k} → M, we have

F_{1} = F (F_{1}^{⊗k}) = F (F_{2}^{⊗k}) = F_{2}.

(2) The question is local for the Zariski topology on S. By proposition 39, we may thus assume that both filtrations split, say

F_{1}^{γ} = ⊕_{η≥γ}G_{1}^{η} and F_{2}^{γ} = ⊕_{η≥γ}G_{2}^{η}

with G_{i}^{γ} locally free of constant rank n_{i}^{γ} for every i ∈ {1, 2} and γ ∈ Γ. We then argue by induction on the constant rank n = ∑ n_{1}^{γ} = ∑ n_{2}^{γ} of M. For n = 0, there is nothing to prove. Suppose n > 0. By assumption, for every γ ∈ Γ,

∑_{a_{1}+···+a_{k}=γ} F_{1}^{a_{1}} ⊗ ⋯ ⊗ F_{1}^{a_{k}} = ∑_{a_{1}+···+a_{k}=γ} F_{2}^{a_{1}} ⊗ ⋯ ⊗ F_{2}^{a_{k}}

which means that

⨁_{a_{1}+···+a_{k}≥γ} G_{1}^{a_{1}} ⊗ ⋯ ⊗ G_{1}^{a_{k}} = ⨁_{a_{1}+···+a_{k}≥γ} G_{2}^{a_{1}} ⊗ ⋯ ⊗ G_{2}^{a_{k}}

Let γ_{i} be the largest element of the (non-empty!) finite set {a : G_{i}^{a} ≠ 0}. Then

⊕_{a_{1}+···+a_{k}≥γ}G_{i}^{a_{1}} ⊗ ⋯ ⊗ G_{i}^{a_{k}} = \begin{cases}0 & if γ > kγ_{i}, \\ G_{i}^{γ_{i}} ⊗ ⋯ ⊗ G_{i}^{γ_{i}} & for γ = kγ_{i}, \\ ≠ 0 & if γ ≤ kγ_{i}. \end{cases}

Thus kγ_{1} = kγ_{2}, γ_{1} = γ_{2} = γ_{0} and G_{1}^{γ_{0}} ⊗ ⋯ ⊗ G_{1}^{γ_{0}} = G_{2}^{γ_{0}} ⊗ ⋯ ⊗ G_{2}^{γ_{0}} in M^{⊗k}, therefore F_{1}^{γ_{0}} = G_{1}^{γ_{0}} = G_{2}^{γ_{0}} = F_{2}^{γ_{0}} = N in M by the previous lemma. We conclude by our induction hypothesis applied to the images of F_{1} and F_{2} in M/N. □

3.10.4. Suppose that G = G_{1} ×_{S} G_{2}. Let F be a Γ-filtration on ω_{T}^{∘}. Then F induces a Γ-filtration F_{i} on the fiber functor ω_{i,T}^{∘} for G_{i} by the formulas:

F_{1}^{γ}(ρ_{1}) = F^{γ}(ρ_{1} ⊠ 1_{G_{2}}) and F_{2}^{γ}(ρ_{2}) = F^{γ}(1_{G_{1}} ⊠ ρ_{2})

We claim that if χ_{i} splits F_{i}, then χ = (χ_{1}, χ_{2}) splits F. Indeed, we may as above assume that G_{1} and G_{2} are split, and we extend F to Rep′(G)(S). We then have to show that the Γ-filtration F′ associated to χ equals F on ρ_{reg}. Since

ρ_{reg} = ρ_{1,reg} ⊠ ρ_{2,reg} = \underrightarrow{lim} τ_{1} ⊠ τ_{2}

where ρ_{i,reg} is the regular representation of G_{i} and the colimit is over τ_{i} ∈ X(ρ_{i,reg}), it is also sufficient to establish that F′ equals F on ρ = τ_{1} ⊠ τ_{2}, τ_{i} ∈ Rep^{∘}(G)(S). Note that ρ = ρ_{1} ⊗ ρ_{2} where ρ_{1} = τ_{1} ⊠ 1_{G_{2}} and ρ_{2} = 1_{G_{1}} ⊠ τ_{2}. We thus find

F^{γ}(ρ) = _{∑γ_{1}+γ_{2}=γ}F^{γ_{1}}(ρ_{1}) ⊗ F^{γ_{2}}(ρ_{2})
= _{∑γ_{1}+γ_{2}=γ}F_{1}^{γ_{1}}(τ_{1}) ⊗ F_{2}^{γ_{2}}(τ_{2})
= ⊕_{γ_{1}+γ_{2}≥γ}G_{1}^{γ_{1}}(τ_{1}) ⊗ G_{2}^{γ_{2}}(τ_{2})
= ⊕_{η≥γ}G^{η}(τ_{1} ⊠ τ_{2})
= F^{′γ}(ρ)

where G and the G_{i}’s are the Γ-graduations induced by χ and the χ_{i}’s.

3.10.5. Applying 3.10.1, 3.10.3 with the central isogeny from G to the product of its adjoint group and its coradical, and finally 3.10.4, we may assume that G is either a split torus or a split reductive group of adjoint type.

3.10.6. Let thus first G = 𝔻_{S}(M) for some M ≃ ℤ^{d} and let F be a Γfiltration on ω_{T}^{∘} for an S-scheme T, which we may assume to be (absolutely) affine. Let ρ_{m} be the representation of G on V (ρ_{m}) = O_{S} on which G acts by the character m ∈ M. By proposition 39, there exists a Γ-graduation O_{T} = ⊕_{γ}I_{γ}(m) such that

∀γ ∈ Γ : F^{γ}(ρ_{m}) = ⊕_{η≥γ}I_{η}(m).

Let T_{γ}(m) be the support of I_{γ}(m), so that T = ∐_{γ} T_{γ}(m) and T_{γ}(m) is open and closed in T. For t ∈ T and m ∈ M, we denote by f(t)(m) the unique element γ in Γ such that t belongs to T_{γ}(m). Thus F_{t}^{γ}(ρ_{m}) = k(t) if γ ≤ f(t)(m) and 0 otherwise, where k(t) is the residue field at t. Since ρ_{0} = 1_{G}, f(t)(0) = 0 by the axiom (F2) for F. Since ρ_{m_{1}} ⊗ ρ_{m_{2}} = ρ_{m_{1}+m_{2}}, f(t)(m_{1} + m_{2}) = f(t)(m_{1}) + f(t)(m_{2}) by the axiom (F1) for F. Therefore f(t) : M → Γ is a group homomorphism. Since M is finitely generated, f : T → Hom_{Group}(M, Γ) is locally constant, and thus corresponds to a global section χ : 𝔻_{T} (Γ) → G_{T} of the locally constant sheaf (see [1, VIII 1.5])

Hom(M, Γ)_{T} = \underline{Hom}(M_{T} , Γ_{T}) = \underline{Hom}(𝔻_{T} (Γ), 𝔻_{T} (M)) = \underline{Hom}(𝔻_{T} (Γ), G_{T}).

Let F′ be the corresponding Γ-filtration on ω_{T}. For any morphism ϕ : M → Γ, let T (ϕ) be the open and closed subset of T where f ≡ ϕ, so that T = ∐ T (ϕ) and

T (ϕ) = ∩_{m∈M}T_{ϕ(m)}(m) = ∩_{i=1}^{r}T_{ϕ(m_{i})}(m_{i})

if {m_{1}, …, m_{r}} ⊂ M spans M. On T (ϕ), we find that

F_{T (ϕ)}^{′γ}(ρ_{m}) = \{ \begin{matrix}O_{T (ϕ)} & if γ ≤ ϕ(m) \\ 0 & if γ > ϕ(m) \end{matrix} \} = F_{T (ϕ)}^{γ}(ρ_{m}).

Thus F′(ρ_{m}) = F(ρ_{m}) for every m. Extending F as in 3.6.4, also F′(ρ_{reg}) = F(ρ_{reg}) since ρ_{reg} = ⊕_{m∈M}ρ_{m} . Finally F′(ρ) = F(ρ) for any ρ by (F3l) applied to c_{ρ}. Therefore χ is a splitting of F – it is in fact the unique such splitting.

3.10.7. Suppose finally that G is a split reductive group of adjoint type over S, let T be an S-scheme, and let F be a Γ-filtration on ω_{T}^{∘}. We have just recalled that F is uniquely determined by the value of its extension to Rep′(G)(S) on ρ_{reg}, but we now also have this: there is at most one Γ-filtration F′ on ω_{T} which equals F on the adjoint representation ρ_{ad} of G on V (ρ_{ad}) = 𝔤 = Lie(G). In particular, any morphism χ : 𝔻_{T} (Γ) → G_{T} inducing F on ρ_{ad} is a splitting of F. To establish our claim, we consider the G-equivariant epimorphism of quasi-coherent G-O_{S}-algebras

f : Sym_{O_{S}}^{•}(ρ_{ad,0}^{∨} ⊗ ρ_{ad}) ↠ ρ_{reg}

which is defined as in section 3.7.4, starting from c_{ad} : ρ_{ad} → ρ_{ad,0} ⊗ ρ_{reg} for the closed embedding ρ_{ad} : G → GL(𝔤). If F′ equals F on ρ_{ad}, they are also equal on Sym^{•} (ρ_{ad,0}^{∨} ⊗ ρ_{ad}) by the axioms (F1-3) for Γ-filtrations on ω_{T}^{∘}, thus also

F^{′γ}(ρ_{reg}) ⊂ F^{γ}(ρ_{reg})

for every γ ∈ Γ by the axiom (F3) for the Γ-filtration F′ on ω_{T} — it is not yet known to be satisfied by the extension of F to Rep′(G)(S), unless we appeal to the arguments of section 3.9.3, which is not necessary: then F^{′γ}(ρ) ⊂ F^{γ}(ρ) for every ρ in Rep^{∘}(G)(S) by (F3l) with c_{ρ}, therefore also F_{+}^{′γ}(ρ) ⊂ F_{+}^{γ}(ρ); applying the latter inclusion to ρ^{∨} and dualizing gives F^{γ}(ρ) ⊂ F^{′γ}(ρ). Thus F = F′ on ω_{T}^{∘}.

3.10.8. By theorem 60, P_{F} = Aut^{⊗}(F) is a parabolic subgroup of G_{T}. Since our problem is local for the étale topology on T, we may assume that T is affine and the pair (G_{T} , P_{F}) has an épinglage E = (H, M, R, ⋯) [21, XXVI 1.14]. Thus H = 𝔻_{T} (M) is a trivialized split maximal torus of G_{T} contained in P_{F}, R ⊂ M is the set of roots of H in 𝔤_{T} and if 𝔤_{T} = 𝔤_{0} ⊕ ⊕_{α∈R}𝔤_{α} is the corresponding weight decomposition (so that 𝔤_{0} = Lie(H)), then Lie(P_{F}) = 𝔤_{0} ⊕ ⊕_{α∈R′}𝔤_{α} for some subset R′ of R as in [21, XXVI 1.4]. The maximal torus H ⊂ P_{F} gives rise to a Levi decomposition P_{F} = U_{F} ⋊ L_{F} with H ⊂ L_{F}, Lie(L_{F}) = 𝔤_{0} ⊕ ⊕_{α∈R′_{1}}𝔤_{α} and Lie(U_{F}) = ⊕_{α∈R′_{2}}𝔤_{α} where R′_{1} = {α ∈ R′ : −α ∈ R′} and R′_{2} = {α ∈ R′ : −α ∉ R′} [21, XXII 5.11.3]. We will then show that F has a splitting χ : 𝔻_{T} (Γ) → G_{T}.

3.10.9. Since H ⊂ P_{F} = Aut^{⊗}(F), the Γ-filtration F is stable under H and

∀γ ∈ Γ, ρ ∈ Rep^{∘}(G)(S) : F^{γ}(ρ) = ⊕_{m∈M}F_{m}^{γ}(ρ)

where F_{m}^{γ}(ρ) is the m-th eigenspace of F^{γ}(ρ), viewed as a representation of H. Since Lie(U_{F}) = F_{+}^{0}(ρ_{ad}) and Lie(P_{F}) = F^{0}(ρ_{ad}) by theorem 60, F_{α}^{γ}(ρ_{ad}) = 0 for (γ > 0 and α ∉ R′_{2}) or (γ = 0 and α ∉ R′ ∪ {0}) while F_{α}^{γ}(ρ_{ad}) = 𝔤_{α} when γ ≤ 0 and α ∈ R′ ∪ {0}. This determines F_{α}^{γ}(ρ_{ad}) for α ∈ R′_{1} ∪ {0}:

∀α ∈ R′_{1} ∪ \{0\} : F_{α}^{γ}(ρ_{ad}) = \begin{cases}𝔤_{α} & if γ ≤ 0, \\ 0 & if γ > 0. \end{cases}

For the remaining α’s (those in ±R′_{2}), 𝔤_{α} is free of rank 1. Using lemma 41, we obtain a partition T = ∐ T (f) into non-empty open and closed subschemes T (f) of T indexed by certain functions f : ±R′_{2} → Γ such that, over T (f),

∀α ∈ ±R′_{2} : F_{α}^{γ}(ρ_{ad}) = \begin{cases}𝔤_{α} & if γ ≤ f(α), \\ 0 & if γ > f(α). \end{cases}

We extend these functions to R ∪ {0} by setting f(R′_{1} ∪ {0}) = 0. Thus over T (f),

F^{γ}(ρ_{ad}) = ⊕_{α∈R∪{0}:f(α)≥γ}𝔤_{α}

Moreover f(α) > 0 (resp. < 0) if and only if α ∈ R′_{2} (resp. −R′_{2}).

3.10.10. We will establish below that each of these f’s extends to a group homomorphism f : M → Γ. The locally constant function T → Hom(M, Γ) mapping t ∈ T (f) to f thus defines a morphism χ : 𝔻_{T} (Γ) → 𝔻_{T} (M) = H ↪ G_{T}. By construction, χ splits F on ρ_{ad}, therefore χ splits F everywhere by 3.10.7.

3.10.11. To show that f extends to a group homomorphism f : M → Γ, we may assume that T = T (f) = Spec(k) where k is a field. By the definition of adjoint groups in [21, XXII 4.3.3] and using [21, XXI 3.5.5], we have to show that

(1) f(−α) = −f(α) for every α ∈ R and

(2) f(α + β) = f(α) + f(β) for every α, β ∈ R such that also α + β ∈ R.

3.10.12. Since H ⊂ P_{F} = Aut^{⊗}(F) fixes F, there is a factorization of Gr_{F}^{•}:

\xymatrix{Rep^{∘}(G)(S) \ar[r] & Gr^{Γ}Rep^{∘}(H)(k) \ar[r] & Gr^{Γ}LF(k)}

where Gr^{Γ}Rep^{∘}(H)(k) is the abelian ⊗-category of Γ-graded objects in Rep^{∘}(H)(k). Both functors are exact ⊗-functors, and we thus obtain a factorization of κ(F):

\xymatrix{K_{0}(G) \ar[r]^{κ} & K_{0}(H)[Γ] = ℤ[M][Γ] \ar[r] & ℤ[Γ]}

The morphism κ maps the class of ρ ∈ Rep^{∘}(G)(S) to

κ[ρ] = _{∑m,γ}x_{m}^{γ}[ρ] · ϵ^{m}e^{γ}

where ϵ^{m} ∈ ℤ[M] and e^{γ} ∈ ℤ[Γ] are the basis elements corresponding to m ∈ M and γ ∈ Γ and x_{m}^{γ}[ρ] is the dimension of the m-th eigenspace of Gr_{F}^{γ}(ρ). Thus

κ[ρ_{ad}] = (dim_{k}(𝔤_{0}) · ϵ^{0} + ∑_{α∈R′_{1}} ϵ^{α}) · e^{0} + ∑_{α∈±R′_{2}} ϵ^{α}e^{f(α)}.

Since the above functors are compatible with dualities,

κ[ρ_{ad}^{∨}] = (dim_{k}(𝔤_{0}) · ϵ^{0} + ∑_{α∈R′_{1}} ϵ^{−α}) · e^{0} + ∑_{α∈±R′_{2}} ϵ^{−α}e^{−f(α)}

= (dim_{k}(𝔤_{0}) · ϵ^{0} + ∑_{α∈R′_{1}} ϵ^{α}) · e^{0} + ∑_{α∈±R′_{2}} ϵ^{α}e^{−f(−α)}.

Since [ρ_{ad}] = [ρ_{ad}^{∨}] in K_{0}(G) by lemma 59, f(−α) = −f(α) for every α ∈ R.

3.10.13. We have already defined the dual ρ_{n} of

ρ^{n} = Coker((ρ_{adj}^{∘})^{⊗n+1} → ρ_{adj}^{∘})

in section 3.7.5. These representations act compatibly (as n varies), functorialy (as ρ varies) and G-equivariantly on any representation ρ ∈ Rep(G)(S) by

\xymatrix{ρ_{n} ⊗ ρ \ar[r]^{Id⊗c_{ρ}} & ρ_{n} ⊗ ρ ⊗ ρ_{adj} \ar@{->>}[r]^{Id⊗proj} & ρ_{n} ⊗ ρ ⊗ ρ_{adj}^{∘} \ar@{->>}[r]^{Id⊗proj} & ρ_{n} ⊗ ρ ⊗ ρ^{n} \ar@{->>}[r]^{^{eval^{n}}} & ρ}

For n = 1, we retrieve the usual adjoint G-equivariant action

ad(ρ) : ρ_{ad} ⊗ ρ → ρ

of 𝔤 on V (ρ), which for ρ = ρ_{ad} is nothing but the usual Lie bracket

[−, −] : ρ_{ad} ⊗ ρ_{ad} → ρ_{ad}.

We also denote by [−, −] : ρ_{n} ⊗ ρ_{ad} → ρ_{ad} the above actions on ρ_{ad}. Thus

∀γ ∈ Γ, ∀α, β ∈ M : [F_{α}^{γ}(ρ_{n}), 𝔤_{β}] ⊂ F_{α+β}^{γ+f(β)}(ρ_{ad}).

In particular, [F_{α}^{γ}(ρ_{n}), 𝔤_{β}] ≠ 0 implies α + β, β ∈ R ∪ {0} and

f(α + β) ≥ γ + f(β).

3.10.14. Suppose that α, β and α+β all belong to R, with ℓ(α) ≤ ℓ(β) where ℓ is the length. Let q and p be the positive integers (with 2 ≤ p + q ≤ 4) such that

{β + nα ∈ R : n ∈ ℤ} = {β − (p − 1)α, ⋯, β, β + α, ⋯, β + qα}

see [21, XXI 2.3.5 and 1]. By Chevalley’s rule [21, XXIII 6.5],

[𝔤_{α}, 𝔤_{β}] = p𝔤_{α+β} and [𝔤_{−α}, 𝔤_{−β}] = p𝔤_{−α−β}.

Thus if p ≠ 0 in k, [𝔤_{α}, 𝔤_{β}] ≠ 0 and [𝔤_{−α}, 𝔤_{−β}] ≠ 0, therefore

f(α + β) ≥ f(α) + f(β) and f(−α − β) ≥ f(−α) + f(−β)

which implies (2) by (1), i.e.

f(α + β) = f(α) + f(β).

If q = 1, Chevalley’s rule gives [𝔤_{α}, 𝔤_{−α−β}] ≠ 0 and [𝔤_{−α}, 𝔤_{α+β}] ≠ 0, thus again f(α + β) = f(α) + f(β). This leaves a single case: p = q = 2 = char(k), where the same method already gives f(β) = f(β − α) + f(α). We will see below that also

[F_{2α}^{2f(α)}(ρ_{2}), 𝔤_{β−α}] = 𝔤_{α+β} and [F_{−2α}^{−2f(α)}(ρ_{2}), 𝔤_{α+β}] = 𝔤_{β−α}.

Therefore f(α + β) = 2f(α) + f(β − α), thus again f(α + β) = f(α) + f(β).

3.10.15. The pure short exact sequences of finite locally free sheaves on S

0 → Sym_{O_{S}}^{2} (\frac{I(G)}{I(G)^{2}} ) →  \frac{I(G)}{I(G)^{3}}  →  \frac{I(G)}{I(G)^{2}}  → 0
0 →  ker  → (\frac{I(G)}{I(G)^{2}} )^{⊗2} → Sym_{O_{S}}^{2} (\frac{I(G)}{I(G)^{2}} ) → 0

give rise to pure short exact sequences in Rep^{∘}(G)(S) which dualize to

0 → ρ_{ad} → ρ_{2} → Γ^{2}(ρ_{ad}) → 0
0 → Γ^{2}(ρ_{ad}) → ρ_{ad}^{⊗2} → Λ^{2}(ρ_{ad}) → 0

where Γ^{2}(ρ) = Sym^{2}(ρ^{∨})^{∨} = ker (ρ^{⊗2} → Λ^{2}(ρ)). Therefore

[ρ_{2}] = [ρ_{ad}] + [ρ_{ad}]^{2} − [Λ^{2}(ρ_{ad})] in K_{0}(G).

Since 𝔤_{2α} = 0 = Λ^{2}(𝔤)_{2α}, the coefficients of ϵ^{2α} in κ[ρ_{2}] and κ[ρ_{ad}^{⊗2}] = κ[ρ_{ad}]^{2} are both equal to e^{2f(α)}. Thus if 𝔡 = ⊕𝔡_{m} is the weight decomposition of 𝔡 = ω_{k}^{∘}(ρ_{2}), then 𝔡_{2α} is 1-dimensional and contained in F^{γ}(ρ_{2}) if and only if γ ≤ 2f(α). In particular, F_{2α}^{2f(α)}(ρ_{2}) = 𝔡_{2α}, and similarly for −α. We thus want:

[𝔡_{2α}, 𝔤_{β−α}] = 𝔤_{β+α} and [𝔡_{−2α}, 𝔤_{β+α}] = 𝔤_{β−α}.

3.10.16. This now only involves the split group G_{k} and its épinglage, all of which descends to Spec(ℤ) by [21, XXIII 5.1 and XXV 1.2]. We may thus assume that G and E = (H, M, R, ⋯) are defined over S = Spec(ℤ). The épinglage comes along with simple roots Δ ⊂ R and, for each α ∈ R, a basis X_{α} of 𝔤_{α}, which extends to a Chevalley system {X_{α} : α ∈ R} by [21, XXIII 6.2], giving rise to isomorphisms u_{α}(t) = exp(tX_{α}) from 𝔾_{a} = Spec(ℤ[t]) to the root subgroup U_{α} of α ∈ R. As a linear form on I(G)/I(G)^{2}, X_{α} is the composition of u_{α}^{♮} : I(G) → I(𝔾_{a}) with the linear form on I(𝔾_{a}) = tℤ[t] given by the coefficient of t. If instead we take the coefficient of t^{2}, we obtain a linear form on I(G)/I(G)^{3} which is a basis X_{2α} of 𝔡_{2α}. The action of X_{α} on the regular representation is given by

A(G) → A(G × 𝔾_{a}) = A(G)[t] → A(G)

where the first map takes f in A(G) to the function (g, t) ↦ f (u_{α}(t)gu_{α}^{−1}(t)), and the second takes the coefficient of t (or evaluates \frac{d}{dt} at t = 0). The action of X_{2α} is obtained by replacing the second map with the coefficient of t^{2}, thus 2X_{2α} = X_{α}^{2} on ρ_{reg}, therefore 2X_{2α} = X_{α}^{2} on all ρ’s. Let us now return to our chain of roots

{β − α, β, β + α, β + 2α} ⊂ R.

By Chevalley’s rule [21, XXIII 6.5]

[X_{α}, X_{β−α}] = ±X_{β} and [X_{α}, X_{β}] = ±2X_{β+α}.

Therefore [X_{2α}, X_{β−α}] = ±X_{β+α} since (we are now over ℤ!)

2[X_{2α}, X_{β−α}] = [2X_{2α}, X_{β−α}] = [X_{α}, [X_{α}, X_{β−α}]] = ±2X_{β+α}.

Similarly, [X_{−2α}, X_{β+α}] = ±X_{β−α}, and this completes our proof.

3.11. Consequences

Let G be a reductive group over S.

3.11.1. Proof of theorem 34. The assertions concerning automorphisms and Γ-graduations follow from theorem 44 and propositions 45, 46 and 49. If G is an isotrivial reductive group over a quasi-compact S, we have monomorphisms

[figure: 𝔽^{Γ}(V^{∘}); 51^{Prop.}; 𝔽^{Γ}(G); Cor. 55; 𝔽^{Γ}(V); 46^{Prop.}; 3.5.3; 𝔽^{Γ}(ω^{∘}); 𝔽^{Γ}(ω); 45^{Prop.}]

and we have just seen that 𝔾^{Γ}(G) → 𝔽^{Γ}(G) → 𝔽^{Γ}(ω^{∘}) is an epimorphism, therefore

𝔽^{Γ}(G) = 𝔽^{Γ}(V) = 𝔽^{Γ}(V^{∘}) = 𝔽^{Γ}(ω) = 𝔽^{Γ}(ω^{∘})

in this case, from which easily follows that also

𝔽^{Γ}(G) = 𝔽^{Γ}(V) = 𝔽^{Γ}(V^{∘})

for any reductive group over any S – and this is contained in 𝔽^{Γ}(ω) by 3.5.3.

3.11.2. Since the S-scheme 𝔾^{Γ}(G) and 𝔽^{Γ}(G) of chapter 2 represent the functors indicated in theorem 34, there is a universal Γ-graduation G_{univ} on V_{𝔾^{Γ}(G)} (inducing universal Γ-graduations on V_{𝔾^{Γ}(G)}^{∘}, ω_{𝔾^{Γ}(G)} and ω_{𝔾^{Γ}(G)}^{∘}) and a universal Γ-filtration F_{univ} on V_{𝔽^{Γ}(G)} (inducing universal Γ-filtrations on V_{𝔽^{Γ}(G)}^{∘}, ω_{𝔽^{Γ}(G)} and ω_{𝔽^{Γ}(G)}^{∘}) from which all other Γ-graduations or Γ-filtrations over some base T can be retrieved by pull-back through unique morphisms T → 𝔾^{Γ}(G) or T → 𝔽^{Γ}(G) – for the ω or ω^{∘} variants, we have to assume that G is isotrivial and S quasi-compact, or that the Γ-graduations or Γ-filtrations (over T) extend to V or V^{∘}. The S-scheme ℂ^{Γ}(G) is a coarse moduli scheme for either Γ-graduations or Γ-filtrations (on the various fiber functors): two such objects (over T) are fpqc locally (on T) isomorphic if and only if the induced morphisms T → ℂ^{Γ}(G) are equal.

3.11.3. From this perspective, we may either deduce non-trivial properties of the S-schemes constructed in chapter 2 from easier properties of Γ-graduations and Γ-filtrations, or non-trivial properties of the latter from already established properties of the former. For instance, theorem 34 implies that Γ-filtrations split over affine bases, a strengthening of the splitting results that we have established:

Corollary 63. Suppose that S is affine. Then every Γ-filtration F on V_{S} or V_{S}^{∘} splits over S, and so do the Γ-filtrations on ω_{S} or ω_{S}^{∘} if G is isotrivial.

Proof. This follows from [21, XXVI 2.2] as in section 2.2.7. □

3.11.4. In the other direction, we obtain the expected functoriality.

Corollary 64. The fundamental sequence of section 2.2.6

𝔾^{Γ}(G) ⟶^{Fil} 𝔽^{Γ}(G) ⟶^{t} ℂ^{Γ}(G)

is covariantly functorial on the fibered category of reductive groups over schemes and covariantly functorial in the totally ordered commutative group Γ.

Proof. We have to show that for a morphism φ : G_{1} → f^{∗}G_{2} over f : T_{1} → T_{2} in the former category, there is a canonical commutative diagram of schemes

\xymatrix{𝔾^{Γ}(G_{1}) \ar[r]^{Fil} \ar[d]^{φ} & 𝔽^{Γ}(G_{1}) \ar[r]^{t} \ar[d]^{φ} & ℂ^{Γ}(G_{1}) \ar[r]^{struct} \ar[d]^{φ} & T_{1} \ar[d]^{f} \\ 𝔾^{Γ}(G_{2}) \ar[r]^{Fil} & 𝔽^{Γ}(G_{2}) \ar[r]^{t} & ℂ^{Γ}(G_{2}) \ar[r]^{struct} & T_{2}}

In the Tannakian point of view, the first two vertical morphisms are induced by pre-composition with the restriction functor Rep(f^{∗}G_{2}) → Rep(G_{1}) which maps τ to τ ∘ φ. For the third one: if T is a T_{1}-scheme and x is a T -valued point of ℂ^{Γ}(G_{1}), it lifts to a Γ-filtration over an fpqc covering {T_{i} → T } of T, and two such lifts become isomorphic over a common refinement of the corresponding fpqc coverings. The image of these lifts in 𝔽^{Γ}(G_{2}) thus yield a well-defined morphism φ(x) : T → ℂ^{Γ}(G_{2}), and this defines the morphism φ : ℂ^{Γ}(G_{1}) → ℂ^{Γ}(G_{2}). The proof of the covariance in Γ is similar, using post-composition with the morphisms

\xymatrix{Gr^{Γ_{1}}QCoh \ar[r]^{Fil} \ar[d]_{f} & Fil^{Γ_{1}}QCoh \ar[d]_{f} \\ _{Gr_{Γ_{2}}QCoh} \ar[r]^{Fil} & _{Fil_{Γ_{2}}QCoh}}

of fpqc stacks induced by f : (Γ_{1}, +, ≤) → (Γ_{2}, +, ≤), which are given by

f(G)_{γ_{2}} = ⊕_{f(γ_{1})=γ_{2}}G_{γ_{1}} and f(F)^{γ_{2}} = _{∑f(γ_{1})≥γ_{2}}F^{γ_{1}}

for T over S, G ∈ Gr^{Γ_{1}}QCoh(T), F ∈ Fil^{Γ_{1}}QCoh(T) and γ_{2} ∈ Γ_{2}. □

Corollary 65. If G = GL(V) for some V ∈ LF(S) of rank r ∈ ℕ^{×}, evaluation at the tautological representation τ of G on V identifies

\xymatrix{𝔾^{Γ}(G) \ar[r]^{Fil} & 𝔽^{Γ}(G) \ar[r]^{t} & ℂ^{Γ}(G)}

with

\xymatrix{𝔾^{Γ}(V) \ar[r]^{Fil} & 𝔽^{Γ}(V) \ar[r]^{t} & ℂ^{Γ}(V)}

where for any S-scheme T,

𝔾^{Γ}(V)(T) = {Γ − graduations on V_{T} }
𝔽^{Γ}(V)(T) = {Γ − filtrations on V_{T} }
ℂ^{Γ}(V)(T) = {locally constant functions f : T → Γ_{≥}^{r}}

where Γ_{≥}^{r} = {(γ_{1} ≥ ⋯ ≥ γ_{r}) ∈ Γ^{r}} and t sends a Γ-filtration F on V_{T} to the function which maps x ∈ T to the r-tuple with dim_{k(x)} Gr_{F}^{γ}(x) copies of γ ∈ Γ.

Proof. Evaluation at τ gives the morphisms τ_{g}, τ_{f} of the diagram

\xymatrix{𝔾^{Γ}(G) \ar[r]^{Fil} \ar[d]_{τ_{g}} & 𝔽^{Γ}(G) \ar[r]^{t} \ar[d]_{τ_{f}} & ℂ^{Γ}(G) \ar[d]_{τ_{c}} \\ 𝔾^{Γ}(V) \ar[r]^{Fil} & 𝔽^{Γ}(V) \ar[r]^{t} & ℂ^{Γ}(V)}

and the remaining morphism τ_{c} comes along by noting that t ∘ τ_{f} is G-invariant. Plainly, τ_{g} is an isomorphism: a morphism 𝔻_{T} (Γ) → G_{T} is nothing but a representation of 𝔻_{T} (Γ) on V_{T}, i.e. a Γ-graduation on V_{T}. Since every Γ-filtration on V_{T} splits locally for the fpqc topology on T by definition (and locally for the Zariski topology by proposition 39), Fil : 𝔾^{Γ}(V) → 𝔽^{Γ}(V) is an epimorphism of fpqc sheaves on Sch/S, and so is therefore also τ_{f}. If F_{1}, F_{2} ∈ 𝔽^{Γ}(G)(T) induce the same filtration F_{1}(τ) = F_{2}(τ) on V_{T} = V_{T} (τ), they also have the same image at det τ (a quotient of τ^{⊗r}) and τ′ = τ ⊕ det(τ)^{−1}. Arguing as in section 3.10.7, we obtain that both filtrations agree on ρ_{reg}, thus actually F_{1} = F_{2}. It follows that τ_{f} is also a monomorphism, i.e. it is an isomorphism. One checks easily that t : 𝔽^{Γ}(V) → ℂ^{Γ}(V) is an epimorphism of fpqc sheaves on Sch/S, and so is therefore also ℂ^{Γ}(G) → ℂ^{Γ}(V). If x, y ∈ ℂ^{Γ}(G)(T) have the same image in ℂ^{Γ}(V)(T) and X , Y are chosen lifts of x and y to 𝔾^{Γ}(G)(T′) for some fpqc cover T′ → T, then, locally on T′, X_{γ}(τ) and Y_{γ}(τ) are free with the same rank, thus isomorphic. Gluing these isomorphisms, we obtain a g ∈ G(T′) which maps X to Y, thus x_{T′} = t ∘ Fil(X) = t ∘ Fil(Y) = y_{T′} in ℂ^{Γ}(G)(T′) by section 2.2.7. But then x = y in ℂ^{Γ}(G)(T), therefore τ_{c} is also a monomorphism, i.e. it is an isomorphism. □

Remark 66. For G = GL(V) as above, the weak and strong dominance orders on ℂ^{Γ}(G) are equal. They correspond to the following order on ℂ^{Γ}(V): for an S-scheme T and locally constant functions f_{1}, f_{2} : T → Γ_{≥}^{r}, we have

f_{1} ≤ f_{2} in ℂ^{Γ}(V)(T) ⟺ ∀t ∈ T : f_{1}(t) ≤ f_{2}(t) in Γ_{≥}^{r}

for the usual partial dominance order on Γ_{≥}^{r}, given by

(γ_{1} ≥ ⋯ ≥ γ_{r}) ≤ (γ′_{1} ≥ ⋯ ≥ γ′_{r})

⟺ \begin{cases}∀1 ≤ i ≤ r − 1 : & γ_{1} + ⋯ + γ_{i} ≤ γ′_{1} + ⋯ + γ′_{i}, \\ and & γ_{1} + ⋯ + γ_{r} = γ′_{1} + ⋯ + γ′_{r}. \end{cases}

For a connected T, we will usually identify ℂ^{Γ}(V)(T) and Γ_{≥}^{r}.

3.11.5. We have already mentioned that the monoid structure on ℂ^{Γ}(G) is not functorial in G. On the other hand, the weak dominance partial order ≤ on ℂ^{Γ}(G) defined in section 2.2.12 is functorial in G.

Proposition 67. Let φ : G → H be a morphism of reductive group over S, let T be an S-scheme. Then for any t_{1}, t_{2} ∈ ℂ^{Γ}(G)(T),

t_{1} ≤ t_{2} in ℂ^{Γ}(G)(T) ⟹ φ(t_{1}) ≤ φ(t_{2}) in ℂ^{Γ}(H)(T).

In particular for any τ : G → GL(V) in Rep^{∘}(G)(S), with V = V (τ) ∈ LF(S),

t_{1} ≤ t_{2} in ℂ^{Γ}(G)(T) ⟹ t_{1}(τ) ≤ t_{2}(τ) in ℂ^{Γ}(V)(T).

Proof. Since ≤ is open in ℂ^{Γ}(H), we may assume that T is a geometric point. The proposition then follows from the stronger proposition 68 below. □

3.11.6. Suppose that G is isotrivial over a connected S. Then G is split by a single finite étale cover π : S′ → S, and we may assume that S′ is connected and Galois over S with Galois group Θ = Aut(S′/S). Let ℛ = ℛ(G) = (M, R, M^{∗}, R^{∗}) be the constant type of G [21, XXII 6.8] with Weyl group W = W (ℛ) and fix a system of positive roots R_{+} ⊂ R, giving rise to a based root data ℛ_{+} = (M, R, M^{∗}, R^{∗}; Δ). Let G_{0} = G_{Spec(ℤ)}^{Ep}(ℛ_{+}) be the corresponding pinned Chevalley group over Spec(ℤ) [21, XXV 1.2] and pick an isomorphism γ : G_{0,S′} ≃ G_{S′}. It exists by [21, XXIII 1.1] and corresponds to a pinning ℰ = (T, ι : 𝔻_{S′}(M) →^{≃} T, (X_{α})_{α∈Δ}) of type ℛ_{+} of G_{S′} [21, XXIV 1.0] by [21, XXIV 1.20]. For any θ in Θ, the pull-back θ^{∗}γ is another isomorphism G_{0,S′} ≃ G_{S′}, corresponding to the pinning θ^{∗}ℰ = (θ^{∗}T, θ^{∗}ι, (θ^{∗}X_{α})_{α∈Δ}) of type ℛ_{+} of G_{S′}. By [21, XXIV 1.5], there is a unique inner automorphism u_{θ} of G_{S′} mapping θ^{∗}ℰ back to ℰ, inducing an automorphism v_{θ} of ℛ. This defines an action of Θ on ℛ_{+}, analogous to the twisted action considered in section 2.4.17, which itself induces compatible actions of Θ on M_{d}, Hom^{+}(M, Γ), ℕ[M]^{W}…

For τ ∈ Rep^{∘}(G)(S′), we may use our fixed pinning ℰ to view the restriction of τ to the maximal torus T of G_{S′} as a representation of 𝔻_{S′}(M). We denote by ch_{ℰ} (τ) the corresponding element of ℕ[M]^{W}, i.e. ch_{ℰ} (τ) = ∑ rankV (τ)_{m} · e^{m} where e^{m} is the basis element of ℤ[M] corresponding to m ∈ M and V (τ)_{m} is the m-th eigenspace of τ|T ∘ ι. For any θ in Θ, the pull-back θ^{∗}τ also belongs to Rep^{∘}(G)(S′) and plainly ch_{θ^{∗}ℰ} (θ^{∗}τ) = ch_{ℰ} (τ). On the other hand

ch_{ℰ} (τ) = ch_{ℰ} (τ ∘ u_{θ}) = v_{θ} (ch_{σ^{∗}ℰ} (τ)) in ℕ[M]^{W} ,

for every τ, thus ch_{ℰ} (θ^{∗}τ) = θ · ch_{ℰ} (τ) in ℕ[M]^{W}. In particular ch_{ℰ} (τ) is fixed by Θ if θ^{∗}τ|T ≃ τ|T, for instance if τ comes from a representation in Rep^{∘}(G)(S).

Let τ_{0,λ,ℚ} ∈ Rep^{∘}(G_{0})(ℚ) be the irreducible representation of G_{0,ℚ} with highest weight λ ∈ M_{d} [44, Lemme 5], let τ_{0,λ} ∈ Rep^{∘}(G_{0})(ℤ) be any extension of τ_{0,λ,ℚ} to a representation of G_{0} [44, Lemme 2], let τ′_{λ} ∈ Rep^{∘}(G)(S′) be the corresponding representation of G_{S′} and set τ_{λ} = π_{∗}τ′_{λ} ∈ Rep^{∘}(G)(S). Then

τ_{λ,S′} = π^{∗}τ_{λ} ≃ ⊕_{θ∈Θ}θ^{∗}τ′_{λ} in Rep^{∘}(G)(S′).

Thus ch_{ℰ} (τ_{λ,S′}) = ∑_{θ∈Θ} θ · ch_{ℰ} (τ′_{λ}) with ch_{ℰ} (τ′_{λ}) = ch_{ℰ_{0}} (τ_{0,λ}) = ch_{ℰ_{0,ℚ}} (τ_{0,λ,ℚ}) in ℕ[M]^{W}, where ℰ_{0} = (T_{0}, ι_{0}, ⋯) is the pinning of G_{0}. Since the other weights of τ_{0,λ,ℚ} are contained in λ − ℕ · R_{+}, it follows that for any f ∈ Hom^{+}(M, Γ),

max f (ch_{ℰ} (τ_{λ,S′})) = max {f(θ · λ) : θ ∈ Θ} = max {(θ · f)(λ) : θ ∈ Θ} .

Our fixed pinning ℰ also induces an isomorphism of partially ordered commutative S′-monoid between ℂ^{Γ}(G_{S′}) and Hom^{+}(M, Γ)_{S′}, and the resulting isomorphism ℂ^{Γ}(G)(S′) ≃ Hom^{+}(M, Γ) is Θ-equivariant, cf. section 2.2.11. If t ∈ ℂ^{Γ}(G)(S′) maps to t_{ℰ} : M → Γ, then for every τ ∈ Rep^{∘}(G)(S′), we have

t(τ) = t_{ℰ} (ch_{ℰ} (τ)) in Γ_{≥}^{r(τ)} ⊂ ℕ[Γ]

under the natural identification of Γ_{≥}^{r(τ)} with the subset of ℕ[Γ] made of those

elements which have degree r(τ) = rankV (τ) (if τ = 0, we set Γ_{≥}^{r(τ)} = 0), thus also

max t(τ) = max t_{ℰ} (ch_{ℰ} (τ)) in Γ.

For τ = τ_{λ,S′} as above we therefore obtain

max t(τ_{λ}) = max t(τ_{λ,S′}) = max {t_{ℰ} (θ · λ) : θ ∈ Θ} = max {(θ · t_{ℰ} )(λ) : θ ∈ Θ} .

If moreover t belongs to ℂ^{Γ}(G)(S), θ · t_{ℰ} = t_{ℰ} for all θ ∈ Θ, thus

max t(τ_{λ}) = t_{ℰ} (λ) in Γ.

3.11.7. We may now prove the following strenghtening of Proposition 67.

Proposition 68. Suppose that G is isotrivial over a connected base scheme S. Then for every t_{1}, t_{2} ∈ ℂ^{Γ}(G)(S), the following conditions are equivalent:

(1) t_{1} ≤ t_{2} in ℂ^{Γ}(G)(S).

(2) For every τ ∈ Rep^{∘}(G)(S), t_{1}(τ) ≤ t_{2}(τ) in Γ_{≥}^{r(τ)}.
(3) For every τ ∈ Rep^{∘}(G)(S), max t_{1}(τ) ≤ max t_{2}(τ) in Γ.

In (2), r(τ) is the constant rank of V (τ). In (3), max t(τ) = 0 if τ = 0.

Proof. Let t_{ℰ ,i} be the Θ-invariant morphism in Hom^{+}(M, Γ) corresponding to the base change t_{i,S′} ∈ ℂ^{Γ}(G)(S′) of t_{i} ∈ ℂ^{Γ}(G)(S). Then

t_{1} ≤ t_{2} in ℂ^{Γ}(G)(S) ⟺ t_{1,S′} ≤ t_{2,S′} in ℂ^{Γ}(G)(S′)
⟺ t_{ℰ ,1} ≤ t_{ℰ ,2} in Hom^{+}(M, Γ)
⟺ ∀λ ∈ M_{d} : t_{ℰ ,1}(λ) ≤ t_{ℰ ,2}(λ) in Γ
⟺ ∀x ∈ ℕ[M]^{W} : max t_{ℰ ,1}(x) ≤ max t_{ℰ ,2}(x) in Γ

using lemma 30 for the last equivalence. Thus (1) ⇒ (3) with x = ch_{ℰ} (τ_{S′}) and (3) ⇒ (1) with τ = τ_{λ}. Plainly (2) ⇒ (3). Moreover, the equivalence (1) ⇔ (3) already implies Proposition 67, from which (1) ⇒ (2) immediately follows. □

Remark 69. For Γ-filtrations F_{1}, F_{2} ∈ 𝔽^{Γ}(G)(S), the proposition implies:

t(F_{1}) ≤ t(F_{2}) in ℂ^{Γ}(G)(S) ⟺
∀τ ∈ Rep^{∘}(G)(S), ∀γ ∈ Γ : F_{2}^{γ}(τ) = 0 ⇒ F_{1}^{γ}(τ) = 0

Indeed max t(F_{i})(τ) = max t(F_{i}(τ)) = max{γ : F_{i}^{γ}(τ) ≠ 0} if τ ≠ 0.

3.11.8. Still assuming that G is isotrivial over a connected base scheme S, suppose moreover that Γ is divisible. Let T be any connected S-scheme. We claim that the monomorphism ℂ^{Γ}(G)(S) ↪ ℂ^{Γ}(G)(T) then has a canonical retraction

♯ : ℂ^{Γ}(G)(T) ↠ ℂ^{Γ}(G)(S)

in the category of partially ordered commutative monoids, which is also functorial in T. To see this, we first fix a geometric point s → T, giving rise to a morphism

π_{1}(T, s) → π_{1}(S, s)

between the profinite étale fundamental group which classify the finite étale covers of T and S. Since ℂ^{Γ}(G) becomes constant over the Galois cover S′/S, it is itself a disjoint union of finite étale covers of S (indexed by the orbits of Θ in Hom^{+}(M, Γ)). Thus π_{1}(S, s) (resp. π_{1}(T, s)) acts on ℂ^{Γ}(G)(s) with finite orbits and fixed point set ℂ^{Γ}(G)(S) (resp. ℂ^{Γ}(G)(T)). These actions respect the auxilliary structures, and averaging over the π_{1}(S, s)-orbits thus yields the desired retraction. If s′ → T is another geometric point, there is a non-canonical equivariant diagram whose vertical maps are isomorphisms [2, V 7]

\begin{aligned}π_{1}(T, s′) & → & π_{1}(S, s′) \\ ≃↓ & & ↓≃ \\ π_{1}(T, s) & → & π_{1}(S, s) \end{aligned} acting on \begin{matrix}ℂ^{Γ}(G, s′) \\ ≃↓ \\ ℂ^{Γ}(G, s) \end{matrix}

Our retraction is therefore independent of s, and thus also functorial in T.

Proposition 70. Suppose that Γ is divisible and G is isotrivial over a connected base scheme S. For every connected S-scheme T and t_{1}, t_{2} ∈ ℂ^{Γ}(G)(T), consider the following conditions:

(1) t_{1}^{♯} ≤ t_{2}^{♯} in ℂ^{Γ}(G)(S).

(2) For every τ ∈ Rep^{∘}(G)(S), t_{1}(τ_{T}) ≤ t_{2}(τ_{T}) in Γ_{≥}^{r(τ)}.
(3) For every τ ∈ Rep^{∘}(G)(S), max t_{1}(τ_{T}) ≤ max t_{2}(τ_{T}) in Γ.

Then (2) ⟺ (3) ⟹ (1) and (1) ⟺ (2) ⟺ (3) if t_{1}^{♯} = t_{1}.

Proof. We may assume that T = s is a geometric point of the connected finite étale Galois cover S′ of S splitting G, realizing Θ = Aut(S′/S) as a quotient of π_{1}(S, s) through which all of the above actions factor. Let t_{ℰ ,i} be the image of t_{i} under ℂ^{Γ}(G)(s) ≃ ℂ^{Γ}(G, S′) ≃ Hom^{+}(M, Γ). Then t_{i}^{♯} maps to the average of the Θ-orbit of t_{ℰ ,i}. Plainly (2) ⇒ (3) and conversely (3) ⇒ (2) since

t_{1}(τ_{s}) ≤ t_{2}(τ_{s}) ⟺ \begin{cases}∀1 ≤ i ≤ r(τ) & max t_{1}(Λ^{i}τ_{s}) ≤ max t_{2}(Λ^{i}τ_{s}), \\ and & max t_{1}(Λ^{r(τ)}τ_{s}^{∨}) ≤ max t_{2}(Λ^{r(τ)}τ_{s}^{∨}). \end{cases}

For the remaining implications, note that

t_{1}^{♯} ≤ t_{2}^{♯} in ℂ^{Γ}(G)(S) ⟺ t_{1,S′}^{♯} ≤ t_{2,S′}^{♯} in ℂ^{Γ}(G)(S′)
⟺ t_{ℰ ,1}^{♯} ≤ t_{ℰ ,2}^{♯} in Hom^{+}(M, Γ)
⟺ ∀λ ∈ M_{d} : t_{ℰ ,1}^{♯}(λ) ≤ t_{ℰ ,2}^{♯}(λ) in Γ
⟺ ∀λ ∈ M_{d} : t_{ℰ ,1}(λ^{♯}) ≤ t_{ℰ ,2}(λ^{♯}) in Γ

where λ^{♯} ∈ M ⊗ ℚ is the average of the Θ-orbit of λ, thus also

t_{1}^{♯} ≤ t_{2}^{♯} in ℂ^{Γ}(G)(S) ⟺ ∀λ ∈ M_{d}^{Θ} : t_{ℰ ,1}(λ) ≤ t_{ℰ ,2}(λ) in Γ

since M_{d}^{Θ} ⊂ M_{d}^{♯} ⊂ ℚ_{≥}M_{d}^{Θ}. Thus (3) ⇒ (1) with τ = τ_{λ} for λ ∈ M_{d}^{Θ}, since

max t_{i}(τ_{λ}) = max {t_{ℰ ,i}(θ · λ) : θ ∈ Θ} = t_{ℰ ,i}(λ) in Γ.

Suppose finally that t_{1}^{♯} = t_{1}. Then using lemma 30, we have

t_{1} ≤ t_{2}^{♯} in ℂ^{Γ}(G)(S) ⟺ ∀x ∈ ℕ[M]^{W} : max t_{ℰ ,1}(x) ≤ max t_{ℰ ,2}^{♯}(x) in Γ
⟹ ∀x ∈ ℕ[M]^{W,Θ} : max t_{ℰ ,1}(x) ≤ max t_{ℰ ,2}(x) in Γ

since indeed for any x ∈ ℕ[M]^{W} we have

max t_{ℰ ,2}^{♯}(x) ≤ \frac{1}{♯Θ} ∑_{θ∈Θ} max(θ · t_{ℰ ,2})(x) = \frac{1}{♯Θ} ∑_{θ∈Θ} max t_{ℰ ,2}(θ · x).

Thus (1) ⇒ (3) if t_{1}^{♯} = t_{1}, with x = ch_{ℰ} (τ_{s}) ∈ ℕ[M]^{W,Θ} for τ ∈ Rep^{∘}(G)(S). □

3.11.9. The results of sections 3.11.5-3.11.8 were inspired by propositions 6.3.9 and 9.4.2 of [17]. However, the latter is contradicted by the following example, which shows that usually (1) does not imply (2) in proposition 70. Take

Γ = ℚ, S = SpecK, T = SpecL and G = Res_{L/K}𝔾_{m,L}

where L is a quadratic extension of a field K. Then ℂ^{Γ}(G)(L) = ℚ^{2} with the trivial partial order. The non-trivial element ι of Gal(L/K) acts by (x, y) ↦ (y, x), thus (x, y)^{♯} ≤ (x′, y′)^{♯} if and only if x + y = x′ + y′. For n, m ∈ ℤ, the formula z ↦ z^{n}(ιz)^{m} defines a 2-dimensional representation τ_{n,m} : G → GL_{K}(L) which is irreducible if m ≠ n. It maps (x, y) ∈ ℚ^{2} to

(x, y)(τ_{n,m}) = (max, min){xn + ym, yn + xm} ∈ ℚ_{≥}^{2}

Thus for t_{1} = (1, −1) and t_{2} = (0, 0), t_{1}^{♯} = t_{2}^{♯} = 0 in ℂ^{Γ}(G)(K) = ℚ but for every n, m ∈ ℤ with n ≠ m, t_{1}(τ_{n,m}) = (|n − m| , − |n − m|) > (0, 0) = t_{2}(τ_{n,m}) in ℚ_{≥}^{2}.

3.11.10. The addition map of section 2.3.2 has the following Tannakian description. For an S-scheme T, (F, G) ∈ 𝕊𝕋𝔻^{Γ}(G)(T) and any ρ ∈ Rep(G)(T),

(F + G)^{γ}(ρ) = ∑_{γ_{1}+γ_{2}=γ} F^{γ_{1}}(ρ) ∩ G^{γ_{2}}(ρ).

Indeed, the question is local on T for the Zariski topology, thus by definition of 𝕊𝕋𝔻^{Γ}(G), we may assume that P_{F} ∩ P_{G} contains a maximal subtorus H of G_{T}. Then F, G lift to f, g : 𝔻_{T} (Γ) → H and (F + G) = Fil(f + g). Let V (ρ)_{γ_{1},γ_{2}} be the subsheaf of V (ρ|H) where 𝔻_{T} (Γ) acts by γ_{1} through f and γ_{2} through g. Then

F^{γ_{1}}(ρ) = ⊕_{η≥γ_{1}} ⊕_{η′} V (ρ)_{η,η′}
G^{γ_{2}}(ρ) = ⊕_{η} ⊕_{η′≥γ_{2}} V (ρ)_{η,η′}
(F + G)^{γ}(ρ) = ⊕_{η+η′≥γ}V (ρ)_{η,η′}

thus indeed (F + G)^{γ}(ρ) = ∑_{γ_{1}+γ_{2}=γ} F^{γ_{1}}(ρ) ∩ G^{γ_{2}}(ρ).

Corollary 71. Let φ : G → H be a morphism of reductive groups over S. Then for any S-scheme T and t_{1}, t_{2} ∈ ℂ^{Γ}(G)(T),

φ(t_{1} + t_{2}) ⪯ φ(t_{1}) + φ(t_{2}) in ℂ^{Γ}(H)(T).

Proof. We may assume that T = s is a geometric point, and lift (t_{1}, t_{2}) to a pair of Γ-filtrations (F_{1}, F_{2}) ∈ 𝔽^{Γ}(G)(s) in osculatory relative position. Then φ(F_{1}) and φ(F_{2}) also are in standard relative position (cf. Remark 22) and the above formula shows that φ(F_{1} + F_{2}) = φ(F_{1}) + φ(F_{2}) in 𝔽^{Γ}(H)(s). Thus

φ (t(F_{1}) + t(F_{2})) = φ (t(F_{1} + F_{2}))
= t (φ(F_{1} + F_{2}))
= t (φ(F_{1}) + φ(F_{2})) ⪯ t (φ(F_{1})) + t (φ(F_{2}))

by proposition 24, i.e. φ(t_{1} + t_{2}) ⪯ φ(t_{1}) + φ(t_{2}) in ℂ^{Γ}(H)(s). □

3.11.11. The morphism defined in section 2.3.4 has the following Tannakian description. Let P be a parabolic subgroup of G with unipotent radical U, and suppose that P = P_{F} for some Γ-filtration F on ω_{S}. For every ρ ∈ Rep(G)(S) and γ ∈ Γ, we may view Gr_{F}^{γ}(ρ) = F^{γ}(ρ)/F_{+}^{γ}(ρ) as a representation of P/U. Then for every S-scheme T and every Γ-filtration G on ω_{T} such that P_{T} and P_{G} are in standard relative position (i.e. P_{T} ∩ P_{G} is a smooth subscheme of G_{T}),

Gr_{P} (G) (Gr_{F}^{γ}(ρ)) = Gr_{F}^{γ}(G, ρ)

where Gr_{F}^{γ}(G, ρ) is the Γ-filtration on Gr_{F}^{γ}(ρ)_{T} = F^{γ}(ρ)_{T} /F_{+}^{γ}(ρ)_{T} induced by the Γ-filtration G(ρ) on V (ρ)_{T}, so that for every θ ∈ Γ,

Gr_{F}^{γ}(G, ρ)^{θ} = (F^{γ}(ρ)_{T} ∩ G^{θ}(ρ) + F_{+}^{γ}(ρ)_{T} ) /F_{+}^{γ}(ρ)_{T} .

This follows from the explicit description of Gr_{P} in the proof of proposition 26.

3.11.12. The functors 𝔾^{Γ}(−) and 𝔽^{Γ}(−) preserve closed immersions.

Proposition 72. Let H ↪ G be a closed immersion of reductive group schemes over S. Then the induced morphisms

𝔾^{Γ}(H) → 𝔾^{Γ}(G) and 𝔽^{Γ}(H) → 𝔽^{Γ}(G)

are finitely presented closed immersions.

Proof. Plainly, 𝔾^{Γ}(H) → 𝔾^{Γ}(G) is a monomorphism. Let x : T → 𝔾^{Γ}(G) be a morphism corresponding to f : 𝔻(Γ)_{T} → G_{T}. Put Z = f^{−1}(H_{T}), a closed subgroup scheme of Y = 𝔻(Γ)_{T}. For every morphism a : T′ → T, we have:

x ∘ a : T′ → 𝔾^{Γ}(G) factors through 𝔾^{Γ}(H)
⟺ f_{T′} : 𝔻(Γ)_{T′} → G_{T′} factors through H_{T′}
⟺ Z_{T′} = Y_{T′}.

This last condition is represented by a closed subscheme of T by [20, VIII 6.3 & 6.4]. Thus 𝔾^{Γ}(H) → 𝔾^{Γ}(G) is relatively representable by closed immersions, i.e. itself a closed immersion. Since 𝔾^{Γ}(H) → S and 𝔾^{Γ}(G) → S are locally of finite presentation (by theorem 1), so is 𝔾^{Γ}(H) ↪ 𝔾^{Γ}(G) by [25, 1.4.3.v], which therefore is a finitely presented closed immersion.

The second morphism 𝔽^{Γ}(H) → 𝔽^{Γ}(G) is a monomorphism: we have seen that Γ-filtrations on ω_{H} are uniquely determined by their value on the regular reprentation of H, which is a quotient of the restriction to H of the regular representation of G. Since 𝔾^{Γ}(H) → 𝔾^{Γ}(G) → 𝔽^{Γ}(G) is quasi-compact and 𝔾^{Γ}(H) → 𝔽^{Γ}(H) is surjective, 𝔽^{Γ}(H) → 𝔽^{Γ}(G) is quasi-compact [25, 1.1.3]. Since 𝔽^{Γ}(H) → S and 𝔽^{Γ}(G) → S are separated and locally of finite presentation, so is 𝔽^{Γ}(H) → 𝔽^{Γ}(G) by [23, 5.5.1.v] and [25, 1.4.3.v]. Since moreover 𝔽^{Γ}(H) → S satisfies the valuative criterion of properness, so does 𝔽^{Γ}(H) → 𝔽^{Γ}(G), which thus is a proper morphism by [24, 7.3.8] and a (finitely presented) closed immersion by [28, 18.12.6]. □

3.12. Ranks and relative positions

Let G be a reductive group over S.

3.12.1. Recall from section 3.8 that for every S-scheme T and F ∈ 𝔽^{Γ}(G)(T), the exact ⊗-functor Gr_{F}^{•} : Rep^{∘}(G)(T) → Gr^{Γ}LF(T) yields a ring homomorphism

K_{0}(G_{T}) → C(T, ℤ[Γ])

mapping the class of τ ∈ Rep^{∘}(G)(T) in K_{0}(G_{T}) to the function

t ↦ ∑_{γ} dim_{k(t)} (Gr_{F}^{γ}(τ) ⊗ k(t)) · e^{γ}

where e^{γ} is the basis element of ℤ[Γ] corresponding to γ ∈ Γ. This construction is functorial in T and invariant under the action of G on 𝔽^{Γ}(G)(T). It therefore induces a morphism of fpqc sheaves of commutative rings (Sch/S)^{∘} → Ring,

κ : \underline{K}_{0}(G) → \underline{Mor} (ℂ^{Γ}(G), ℤ[Γ]_{S})

where \underline{K}_{0}(G) is the fpqc sheaf associated to the presheaf T ↦ K_{0}(G_{T}) while \underline{Mor} (ℂ^{Γ}(G), ℤ[Γ]_{S}) is the fpqc sheaf of morphisms of S-schemes from the cone ℂ^{Γ}(G) to the constant sheaf of rings ℤ[Γ]_{S}.

3.12.2. Let now (F_{1}, F_{2}) ∈ 𝕊𝕋𝔻^{Γ}(G)(T) be a pair of Γ-filtrations in standard relative position (cf. 2.3). Then the formula

Gr_{F_{1},F_{2}}^{γ_{1},γ_{2}}(τ) = \frac{F_{1}^{γ_{1}}(τ) ∩ F_{2}^{γ_{2}}(τ)}{F_{1,+}^{γ_{1}}(τ) ∩ F_{2}^{γ_{2}}(τ) + F_{1}^{γ_{1}}(τ) ∩ F_{2,+}^{γ_{2}}(τ)}

also defines an exact ⊗-functor

Gr_{F_{1},F_{2}}^{•,•} : Rep^{∘}(G)(T) ⟶ Gr^{Γ×Γ}LF(T).

Indeed, we have to show that Gr_{F_{1},F_{2}}^{γ_{1},γ_{2}}(τ) is locally free of finite rank, exact in τ, and such that for every τ′, τ′′ ∈ Rep^{∘}(G)(T) and γ_{1}, γ_{2} ∈ Γ, the natural map

⊕_{(γ′_{1},γ′_{2})+(γ′′_{1},γ′′_{2})=(γ_{1},γ_{2})}Gr_{F_{1},F_{2}}^{γ′_{1},γ′_{2}}(τ′) ⊗ Gr_{F_{1},F_{2}}^{γ′′_{1},γ′′_{2}}(τ′′) → Gr_{F_{1},F_{2}}^{γ_{1},γ_{2}}(τ′ ⊗ τ′′)

is an isomorphism. All this is local in the fpqc topology on T. We may thus assume that P_{F_{1}} ∩ P_{F_{2}} contains a maximal torus H of G which is split, i.e. H = 𝔻_{T} (M) for some finitely generated free abelian group M, in which case F_{1} and F_{2} are split by morphisms G_{1} and G_{2} : 𝔻_{T} (Γ) → 𝔻_{T} (M). If V (τ) = ⊕_{m∈M}V (τ)_{m} is the H-eigenspace decomposition of τ|_{H}, we then have a canonical isomorphism

Gr_{F_{1},F_{2}}^{γ_{1},γ_{2}}(τ) ≃ ⊕_{m∈M:(m∘G_{1},m∘G_{2})=(γ_{1},γ_{2})}V (τ)_{m}

and our claim easily follows. We thus obtain a ring homomorphism

K_{0}(G_{T}) → C(T, ℤ[Γ × Γ])

which maps the class of τ ∈ Rep^{∘}(G)(T) in K_{0}(G_{T}) to the function

t ↦ ∑_{γ_{1},γ_{2}} dim_{k(t)} (Gr_{F_{1},F_{2}}^{γ_{1},γ_{2}}(τ) ⊗ k(t)) · e^{γ_{1}} ⊗ e^{γ_{2}}

where e^{γ_{1}} ⊗ e^{γ_{2}} is the basis element of ℤ[Γ × Γ] = ℤ[Γ] ⊗ ℤ[Γ] corresponding to the element (γ_{1}, γ_{2}) of Γ × Γ.

3.12.3. The above construction is again functorial in T and invariant under the diagonal action of G on 𝕊𝕋𝔻^{Γ}(G). It therefore induces a morphism of fpqc sheaves of commutative rings (Sch/S)^{∘} → Ring,

κ : \underline{K}_{0}(G) → \underline{Mor} (𝕋𝕊𝕋𝔻^{Γ}(G), ℤ[Γ × Γ]_{S}) .

3.12.4. If now f : ℤ[Γ × Γ]_{S} → X is a morphism of S-schemes, we denote by

⟨−, −⟩_{f} : \underline{K}_{0}(G) → \underline{Mor} (𝕊𝕋𝔻^{Γ}(G), X)
⟨−, −⟩_{f}^{os} : \underline{K}_{0}(G) → \underline{Mor} (ℂ^{Γ}(G)^{2}, X)
⟨−, −⟩_{f}^{tr} : \underline{K}_{0}(G) → \underline{Mor} (ℂ^{Γ}(G)^{2}, X)

the morphisms of fpqc sheaves on S which are obtained by post-composition of κ with the obvious morphisms induced by f and, respectively: the quotient map

t_{2} : 𝕊𝕋𝔻^{Γ}(G) ↠ 𝕋𝕊𝕋𝔻^{Γ}(G)

and the osculatory and transverse sections

os and tr : ℂ^{Γ}(G)^{2} ↪ 𝕋𝕊𝕋𝔻^{Γ}(G)

of section 2.3. For τ ∈ \underline{K}_{0}(G)(S), we thus obtain morphisms of S-schemes

\begin{aligned}⟨−, −⟩_{f,τ} : & 𝕊𝕋𝔻^{Γ}(G) & → & X \\ ⟨−, −⟩_{f,τ}^{os} : & ℂ^{Γ}(G)^{2} & → & X \\ ⟨−, −⟩_{f,τ}^{tr} : & ℂ^{Γ}(G)^{2} & → & X \end{aligned}

By construction, for every S-scheme T and (F_{1}, F_{2}) ∈ 𝔾𝔼ℕ^{Γ}(G)(T),

⟨F_{1}, F_{2}⟩_{f,τ} = ⟨t(F_{1}), t(F_{2})⟩_{f,τ}^{tr} in X(T).

3.12.5. We will only consider these constructions in the following situation: Γ is a subgroup of ℝ, X is the constant scheme ℝ_{S}, f is induced by the bilinear form Γ × Γ ∋ (γ_{1}, γ_{2}) ↦ γ_{1}γ_{2} ∈ ℝ and τ is a genuine representation in Rep^{∘}(G)(S). Then for any S-scheme T and (F_{1}, F_{2}) ∈ 𝕊𝕋𝔻^{Γ}(G)(T), ⟨F_{1}, F_{2}⟩_{τ} = ⟨F_{1}, F_{2}⟩_{f,τ} is the locally constant function T → ℝ given by

t ↦ ∑_{γ_{1},γ_{2}} dim_{k(t)} (Gr_{F_{1},F_{2}}^{γ_{1},γ_{2}}(τ) ⊗ k(t)) · γ_{1}γ_{2}.

3.13. Appendix: pure subsheaves

Let X be a scheme.

Lemma 73. For A → B → C in QCoh(X), consider the following conditions:

(1) For every quasi-coherent sheaf F on X,

0 → A ⊗ F → B ⊗ F → C ⊗ F → 0 is exact in QCoh(X).

(2) For every morphism f : Y → X,

0 → f^{∗}A → f^{∗}B → f^{∗}C → 0 is exact in QCoh(Y).

(3) For every morphism f : Y → X and quasi-coherent sheaf F on Y,

0 → f^{∗}A ⊗ F → f^{∗}B ⊗ F → f^{∗}C ⊗ F → 0 is exact in QCoh(Y).

Then (1) ⇐ (2) ⇔ (3) and (1) ⇔ (2) ⇔ (3) if X is quasi-separated.

Proof. Obviously (3) ⇒ (1) and (2). Suppose (2) holds. Let f : Y → X be a morphism, F a quasi-coherent sheaf on Y, g : Z → Y the structural morphism of Z = Spec(O_{Y} [F]) where O_{Y} [F] = O_{Y} ⊕ F is the quasi-coherent O_{Y} -algebra defined by F · F = 0. By assumption, 0 → h^{∗}A → h^{∗}B → h^{∗}C → 0 is an exact sequence of quasi-coherent sheaves on Z, where h = f ∘ g. Since g is affine,

0 → g_{∗}h^{∗}A → g_{∗}h^{∗}B → g_{∗}h^{∗}C → 0

is an exact sequence of quasi-coherent sheaves on Y. But

g_{∗}h^{∗}X = g_{∗}g^{∗}f^{∗}X = f^{∗}X ⊕ f^{∗}X ⊗ F

for any X in QCoh(X), therefore

0 → f^{∗}A ⊗ F → f^{∗}B ⊗ F → f^{∗}C ⊗ F → 0

is exact and (2) ⇒ (3). Suppose now that X is quasi-separated and (1) holds. Let f : Y → X be any morphism. Let {X_{i}} and {Y_{i,j}} be open coverings of X and Y by affine schemes such that f(Y_{i,j}) ⊂ X_{i} and let f_{i,j} : Y_{i,j} → X_{i} be the induced morphism. Since (f^{∗}X)|_{Y_{i,j}} = f_{i,j}^{∗}(X |_{X_{i}}) for every X ∈ QCoh(X), we have to show that 0 → f_{i,j}^{∗}(A_{i}) → f_{i,j}^{∗}(B_{i}) → f_{i,j}^{∗}(C_{i}) → 0 is exact on Y_{i,j} for every i, j, with X_{i} = X |_{X_{i}}. Since Y_{i,j} and X_{i} are affine, this amounts to showing that

0 → A_{i} ⊗ O_{i,j} → B_{i} ⊗ O_{i,j} → C_{i} ⊗ O_{i,j} → 0

is exact on X_{i} for every i, j, for the quasi-coherent sheaf O_{i,j} = (f_{i,j})_{∗}O_{Y_{i,j}} on X_{i}. Since X is quasi-separated, the immersion ι_{i} : X_{i} ↪ X is quasi-compact and quasi-separated by [25, 1.2.2.i & 1.2.7.b], thus F_{i,j} = (ι_{i})_{∗}O_{i,j} is a quasi-coherent sheaf on X by [25, 1.7.4] and 0 → A ⊗ F_{i,j} → B ⊗ F_{i,j} → C ⊗ F_{i,j} → 0 is an exact sequence on X by assumption. Pulling back through the exact restriction functor ι_{i}^{∗} : QCoh(X) → QCoh(X_{i}) yields the desired result. □

Definition 74. We say that the sequence 0 → A →^{ι} B → C → 0 is pure exact, or that ι is a pure monomorphism, or that ι(A) is a pure (quasi-coherent) subsheaf of B if the above condition (2) holds.

Lemma 75. Let B be a quasi-coherent sheaf on X. Then

P : (Sch/X)^{∘} → Set T ↦ {pure quasi-coherent subsheaves A of B_{T} }

is an fpqc sheaf on Sch/X.

Proof. It is a functor: if A ∈ P(T) and α : T′ → T is an X-morphism, the monomorphism α^{∗}(A ↪ B_{T}) identifies α^{∗}(A) with a quasi-coherent subsheaf of α^{∗}(B_{T}) = B_{T′}, which is pure since for any morphism f′ : Y → T′, if f = α ∘ f′, then f^{′∗} ∘ α^{∗}(A ↪ B_{T}) = f^{∗}(A ↪ B_{T}) is a monomorphism of quasi-coherent sheaves on Y since A is pure in B_{T}. It is an fpqc sheaf: if {T_{i} → T } is an fpqc cover and A_{i} ∈ P(T_{i}) have the same image A_{i,j} ∈ P(T_{i} ×_{T} T_{j}), then the quasi-coherent subsheaves A_{i} of B_{T_{i}} glue to a quasi-coherent subsheaf A of B_{T} which is pure since for any f : Y → T, f^{∗}(A ↪ B_{T}) is a monomorphism of quasi-coherent sheaves on Y as it becomes so in the fpqc cover {Y ×_{T} T_{i} → Y } of Y. □

Lemma 76. Let A be a quasi-coherent subsheaf of B.

(1) Suppose that locally on X for the fpqc topology, A is a direct factor of B. Then A is a pure subsheaf of B.

(2) Suppose that A is a pure subsheaf of B and C = B/A is finitely presented. Then locally on X for the Zariski topology, A is a direct factor of B.

Proof. (1) A direct factor being obviously pure, this follows from the previous lemma. As for (2): the assumptions are local in the Zariski topology by the previous lemma, we may thus assume that X = Spec(R) for some ring R. Then A = Γ(X, A) is a pure R-submodule of B = Γ(X, B) in the sense of [34, Appendix to §7] by (2) ⇒ (1) of lemma 73, and C = B/A is a finitely presented R-module. Therefore A is a direct factor of B by [34, Theorem 7.14], i.e. A is a direct factor of B. □

CHAPTER 4 The vectorial Tits building F^{Γ}(G)

Let O be a local ring, G a reductive group over Spec(O). We shall here take a closer look at the set F^{Γ}(G) = 𝔽^{Γ}(G)(O) of sections of 𝔽^{Γ}(G) over Spec(O).

4.1. Combinatorial structures

4.1.1. We say that a morphism of posets f : (X, ≤) → (Y, ≤) is nice if

∀x, y ∈ X × Y with f(x) ≤ y, there is a unique x′ ∈ f^{−1}(y) with x ≤ x′.

We say that it is very nice if also

∀x, y ∈ X × Y with f(x) ≥ y, there is an x′ ∈ f^{−1}(y) with x ≥ x′.

4.1.2. We will define below an Aut(G)-equivariant sequence of nice surjective morphisms of posets

\xymatrix{SBP(G) \ar@{->>}[r]^{a} & SP(G) \ar@{->>}[r]^{b} & OPP(G) \ar@{->>}[r]^{p_{1}} & P(G) \ar@{->>}[r]^{t} & O(G)}

The group G = G(O) acts on it through Int : G → Aut(G), and we will see that

G\SBP(G) = G\SP(G) = G\OPP(G) = G\P(G) = O(G).

4.1.3. We first define our posets. We will use the following notations:

S(G) = {S : maximal split torus of G}
B(G) = {B : minimal parabolic subgroup of G}
P(G) = {P : parabolic subgroup of G}
SP(G) = {(S, P) : Z_{G}(S) ⊂ P }
SBP(G) = {(S, B, P) : Z_{G}(S) ⊂ B ⊂ P }
OPP(G) = {(P, P′) : opposed parabolic subgroups of G}

Thus P(G) = ℙ(G)(O) and OPP(G) = 𝕆ℙℙ(G)(O). In addition, we set

O(G) = image of t : ℙ(G)(O) → 𝕆(G)(O).

We endow P(G) and O(G) with their natural partial orders and the remaining three sets SBP(G), SP(G) and OPP(G) with the following ones:

(S_{1}, B_{1}, P_{1}) ≤ (S_{2}, B_{2}, P_{2}) ⟺ S_{1} = S_{2}, B_{1} = B_{2} and P_{1} ⊂ P_{2}
(S_{1}, P_{1}) ≤ (S_{2}, P_{2}) ⟺ S_{1} = S_{2} and P_{1} ⊂ P_{2}
(P_{1}, P′_{1}) ≤ (P_{2}, P′_{2}) ⟺ P_{1} ⊂ P_{2} and P′_{1} ⊂ P′_{2}

4.1.4. The morphism t : P(G) → O(G) maps P to its type t(P). It is plainly a morphism of posets. It is surjective by definition of O(G), nice by [21, XXVI 3.8] and even very nice by [21, XXVI 5.5]. The group G acts trivially on O(G), and G · P = t^{−1}t(P) by [21, XXVI 5.2], thus G\P(G) = O(G).

4.1.5. The morphism p_{1} : OPP(G) → P(G) maps (P, P′) to P. It is plainly a morphism of posets, and it is surjective by [21, XXVI 2.3 & 4.3.2]. Consider now (P, P′) ∈ OPP(G), Q ∈ P(G) and suppose first that P ⊂ Q. Since t is nice, there is a unique Q′ ∈ P(G) with P′ ⊂ Q′ and t(Q′) = ιt(Q), where ι is the opposition involution of O(G). We have (Q, Q′) ∈ OPP(G) by [21, XXVI 4.3.2 & 4.2.1], thus p_{1} is nice. If Q ⊂ P, then Q_{L} = Q ∩ L is a parabolic subgroup of L = P ∩ P′ and its Levi subgroups are the Levi subgroups of Q contained in L by [21, XXVI 1.20]. Since p_{1} : OPP(L) → P(L) is surjective, there is a parabolic subgroup Q′_{L} of L opposed to Q_{L}. Then Q′_{L} = Q′ ∩ L for a unique parabolic subgroup Q′ of G contained in P′, and (Q, Q′) ∈ OPP(G) since Q ∩ Q′ = Q_{L} ∩ Q′_{L} is a Levi subgroup of Q_{L} and Q′_{L}, thus also of Q and Q′. Therefore p_{1} is very nice. Finally, the stabilizer of P in G is P = P (O) by [21, XXVI 1.2], and P · (P, P′) = p_{1}^{−1}(P) by [21, XXVI 1.8 & 4.3.2], thus G\OPP(G) = G\P(G).

4.1.6. The morphism b : SP(G) → OPP(G) maps (S, P) to (P, ι_{S}P), where ι_{S}P is defined in the next lemma, which also says that b is a morphism of posets.

Lemma 77. For S ∈ S(G) and P ∈ P(G) with Z_{G}(S) ⊂ P, there exists a unique Levi subgroup L of P and a unique parabolic subgroup ι_{S}P of G opposed to P with Z_{G}(S) ⊂ L, ι_{S}P. Moreover L = P ∩ι_{S}P and P ↦ ι_{S}P preserves inclusions.

Proof. By [1, XIV 3.20], there is a maximal torus T in Z_{G}(S). It is also maximal in G and P. By [21, XXVI 1.6], there is a unique Levi subgroup L of P with T ⊂ L. We have to show that Z_{G}(S) ⊂ L. By [21, XXVI 6.11], this is equivalent to R_{sp}(L) ⊂ S, where R_{sp}(L) is the split radical of L, i.e. the maximal split subtorus R(L)_{sp} of the radical R(L) of L. Since T is a maximal torus in L, R(L) is contained in T, thus R_{sp}(L) is contained in the maximal split subtorus T_{sp} of T, which obviously contains S and in fact equals S by maximality of S. This proves the existence and uniqueness of L. That of ι_{S}P follows from [21, XXVI 4.3.2] which also shows that L = P ∩ι_{S}P. If P ⊂ Q, there is a unique (Q, Q′) ∈ OPP(G) with ι_{S}P ⊂ Q′ because p_{1} is nice, and obviously ι_{S}Q = Q′, thus ι_{S}P ⊂ ι_{S}Q. □

Starting with (P, P′) ∈ OPP(G) put L = P ∩ P′ and let S be a maximal split torus in G containing the split radical R_{sp}(L) of L. Then Z_{G}(S) is contained in Z_{G}(R_{sp}(L)) which equals L by [21, XXVI 6.11], thus (S, P) ∈ SP(G) and b(S, P) equals (P, P′), i.e. b is surjective. It is obviously nice, although not very nice. The stabilizer of b(S, P) in G is L = L(O) where L = P ∩ι_{S}P, and L·(S, P) = b^{−1}b(S, P) by [21, XXVI 6.16], thus G\SP(G) = G\OPP(G). The opposition involution ι(P_{1}, P_{2}) = (P_{2}, P_{1}) of OPP(G) lifts to the involution ι(S, P) = (S, ι_{S}P) of SP(G).

4.1.7. The morphism a : SBP(G) → SP(G) maps (S, B, P) to (S, P). It is plainly a nice morphism of poset, although not very nice. Fix (S, P) ∈ SP(G), let L = P ∩ι_{S}P. Then [21, XXVI 1.20] sets up a bijection between: the set of minimal parabolic subgroup B of G with Z_{G}(S) ⊂ B ⊂ P (the fiber a^{−1}(S, P)) and the set of minimal parabolic subgroups B_{L} = B ∩ L of L with Z_{G}(S) ⊂ B_{L}. The latter set is not empty by [21, XXVI 6.16], thus a is surjective. The stabilizer of (S, P) in G equals N_{L}(S) = N_{L}(S)(O) and N_{L}(S) · (S, B, P) = a^{−1}(S, P) by [21, XXVI 7.2] applied to Z_{G}(S) ⊂ L, thus G\SBP(G) = G\SP(G). The stabilizer of (S, B, P) in G is the stabilizer of (S, B), namely Z_{G}(S) = Z_{G}(S)(O) since Z_{G}(S) = B ∩ ι_{S}B.

4.1.8. By [21, XXVI 5.7], there is a smallest element ∘ in O(G). For

X in {SBP(G), SP(G), OPP(G), P(G)},

the morphism f : X → O(G) is very nice. We’ve proved it already in the last two cases. Since f is nice, our assertion is equivalent to: X_{min} = f^{−1}(∘) where X_{min} is the set of minimal elements in X. This is obvious for SBP(G), and also for SP(G) since a is surjective. For any x ∈ X_{min} = f^{−1}(∘), there is then a unique section

\xymatrix{(X, ≤) \ar@{->>}[r]_{f} & (O(G), ≤) \ar[l]_{s_{x}}}

with s_{x}(∘) = x, and these sections cover X.

4.1.9. Let now Γ = (Γ, +, ≤) be a non-trivial totally ordered commutative group and form the Aut(G)-equivariant cartesian diagram of sets:

\xymatrix{ACF^{Γ}(G) \ar@{->>}[r]^{a} \ar[d]^{F} & AF^{Γ}(G) \ar@{->>}[r]^{b} \ar[d]^{F} & G^{Γ}(G) \ar@{->>}[r]^{Fil} \ar[d]^{F} & F^{Γ}(G) \ar@{->>}[r]^{t} \ar[d]^{F} & C^{Γ}(G) \ar[d]^{F} \\ SBP(G) \ar@{->>}[r]^{a} & SP(G) \ar@{->>}[r]^{b} & OPP(G) \ar@{->>}[r]^{p_{1}} & P(G) \ar@{->>}[r]^{t} & O(G)}

where C^{Γ}(G) is the inverse image of O(G) under F : ℂ^{Γ}(G)(O) → 𝕆(G)(O). Of course we may and do identify F^{Γ}(G) with 𝔽^{Γ}(G)(O) and G^{Γ}(G) with 𝔾^{Γ}(G)(O), see section 2.2.6. With these identifications, we find:

AF^{Γ}(G) = {(S, F) ∈ S(G) × F^{Γ}(G) with Z_{G}(S) ⊂ P_{F}}
= {(S, G) ∈ S(G) × G^{Γ}(G) with Z_{G}(S) ⊂ L_{G}}
= {(S, G) : S ∈ S(G), G ∈ G^{Γ}(S)}
ACF^{Γ}(G) = {(S, B, F) : S(G) × B(G) × F^{Γ}(G) with Z_{G}(S) ⊂ B ⊂ P_{F}}
{(S, B, G) : S ∈ S(G), G ∈ G^{Γ}(S) with Z_{G}(S) ⊂ B ⊂ P_{G}} .

The opposition involution ι of G^{Γ}(G) lifts to an involution of AF^{Γ}(G), given by

ι(S, F) = (S, ι_{S}F) or ι(S, G) = (S, ιG).

Here ιG = G^{−1} in G^{Γ}(S) and (Fil(G), Fil(ιG)) = (F, ι_{S}F).

4.1.10. Fix S ∈ S(G). Let M be its group of characters, R ⊂ M the roots of S in 𝔤 = Lie(G) and 𝔤 = 𝔤_{0} ⊕ ⊕_{α∈R}𝔤_{α} the corresponding decomposition of 𝔤. Put

W = (N_{G}(S)/Z_{G}(S))(O) = N_{G}(S)(O)/Z_{G}(S)(O).

By [21, XXVI 7.4], there exists a unique root datum ℛ = (M, R, M^{∗}, R^{∗}) with Weyl group W and a W -equivariant bijection B ↔ R_{+} between the set of all B ∈ B(G) with Z_{G}(S) ⊂ B and the set of all systems of positive roots R_{+} ⊂ R, given by

Lie(B) = 𝔤_{0} ⊕ ⊕_{α∈R_{+}}𝔤_{α}.

Fix one such B and let Δ ⊂ R_{+} be the corresponding set of simple roots. By [21, XXVI 7.7], there is an inclusion preserving bijection P ↔ A between the set of all P ∈ P(G) with B ⊂ P and the set of all subsets A of Δ, given by

Lie(P) = 𝔤_{0} ⊕ ⊕_{α∈R_{A}}𝔤_{α}

where R_{A} = R_{+} ∐(ℤA ∩ R_{−}) is the set of roots in R = R_{+} ∐ R_{−} which are either positive or in the group spanned by A. We write P_{A} for the parabolic associated to A. Since f : SBP(G) → O(G) is (very) nice, we obtain a poset bijection

f_{S,B} : ({A ⊂ Δ}, ⊂) → (O(G), ≤), A ↦ t(P_{A}).

Fix one such P = P_{A}. Then the fiber of F : ACF^{Γ}(G) → SBP(G) above (S, B, P) is the set of all (S, B, G) with G ∈ G^{Γ}(S) = Hom(M, Γ) such that

∀α ∈ Δ : \begin{cases}G(α) = 0 & if α ∈ A, \\ G(α) > 0 & if α ∉ A. \end{cases}

Since the elements of Δ are linearly independent and Γ is non-trivial, this fiber is not empty and F : ACF^{Γ}(G) → SBP(G) is surjective.

4.1.11. It follows that the five F ’s in our diagram are surjective. Their fibers are called facets, the type of a facet is its image in O(G), and all facets of the same type are canonically isomorphic. The facets of type ∘ are called chambers. For any f′ : X′ → C^{Γ}(G) over f : X → O(G) in our diagram, the closed facet of x ∈ X is F^{−1}(\overline{x}) ⊂ X′ where \overline{x} = \{y ≥ x\}. It is a disjoint union of finitely many facets. Since x = min F F^{−1}(\overline{x}), closed facets have a well-defined type and those of the same type are canonically isomorphic. We equip the set of closed facets with the partial order given by inclusion, which is opposite to the partial order on X. A closed chamber is a maximal closed facet, and the set of all closed chambers equals X_{min} = f^{−1}(∘). Since f is nice, every x ∈ X_{min} defines compatible sections

\xymatrix{X′ \ar@{->>}[r]_{f′} \ar[d]_{F} & C^{Γ}(G) \ar[l]_{s_{x}} \ar[d]^{F} \\ X \ar@{->>}[r]_{f} & O(G) \ar[l]_{s_{x}}}

and the closed chamber F^{−1}(\overline{x}) is the image of s_{x} : C^{Γ}(G) → X′. Since f is very nice, any x′ ∈ X′ belongs to some closed chamber. Since G\X′ = C^{Γ}(G), any closed chamber is a fundamental domain for the action of G on X′.

4.1.12. The facets which are minimal among the set of non-minimal facets are called panels. A panel F^{−1}(x) bounds a chamber F^{−1}(y) if F^{−1}(x) ⊂ F^{−1}(\overline{y}), i.e. y ≤ x. Any panel bounds at least 3 chambers. Indeed, this means that a non-minimal parabolic subgroup P of G contains at least 3 minimal parabolic subgroups. To establish this, fix a Levi subgroup L of P – which exists by [21, XXVI 2.3] or the surjectivity of p_{1}. Then Q ↦ L ∩ Q yields a bijection between the parabolic subgroups Q of G contained in P and the parabolic subgroups of L, by [21, XXVI 1.20]. Since P is non-minimal, L is not a minimal parabolic subgroup of itself. By [21, XXVI 5.11], it contains at least 3 such subgroups, and so does P.

4.1.13. The apartment attached to S ∈ S(G) is the subset F^{Γ}(S) of all F’s in F^{Γ}(G) such that Z_{G}(S) ⊂ P_{F}. It is canonically isomorphic to G^{Γ}(S) by the map which sends G : 𝔻_{O}(Γ) → S to Fil(G). Our notations are thus consistent since

F^{Γ}(S) = G^{Γ}(S) = 𝔾^{Γ}(S)(O) = 𝔽^{Γ}(S)(O).

Since F : AF^{Γ}(G) → SP is surjective, F^{Γ}(S) is the disjoint union of the facets F^{−1}(P) with Z_{G}(S) ⊂ P. Since Z_{G}(S) = B ∩ B′ for some pair of opposed minimal parabolic subgroups of G, F^{Γ}(S) determines Z_{G}(S) = ∩_{F^{−1}(P)⊂F^{Γ}(S)}P and its split radical S. Thus S ↦ F^{Γ}(S) is an Aut(G)-equivariant bijection from S(G) onto the set A(G) of apartments in F^{Γ}(G). In particular,

AF^{Γ}(G) = {(A, F) : A ∈ A(G), F ∈ A}
ACF^{Γ}(G) = {(A, C, F) : F ∈ C = closed chamber of A ∈ A(G)} .

Since AF^{Γ}(G) → F^{Γ}(G) is surjective, every F ∈ F^{Γ}(G) belongs to some A ∈ A(G). The stabilizer of F^{Γ}(S) in G = G(O) equals N_{G}(S) = N_{G}(S)(O) and its pointwise stabilizer equals Z_{G}(S) = Z_{G}(S)(O). Thus W_{G}(S) = N_{G}(S)/Z_{G}(S) acts on F^{Γ}(S), and this gives the usual action of W_{G}(S) on F^{Γ}(S) = G^{Γ}(S) = Hom(𝔻_{O}(Γ), S).

4.1.14. A panel bounds exactly two chambers in any apartment which contains it. Indeed, let F^{−1}(Q) be a panel in F^{Γ}(S). Given [21, XXVI 1.20], we have to show that there are exactly two minimal parabolic subgroups of L = Q ∩ ι_{S}Q containing Z_{G}(S). By assumption, O(L) = {∘, t(L)}. Our claim then follows from 4.1.10.

4.1.15. For any F_{1}, F_{2} ∈ F^{Γ}(G), there is an apartment A ∈ A(G) containing F_{1} and F_{2} if and only if P_{F_{1}} and P_{F_{2}} are in standard position [21, XXVI 4.5]. Indeed if F_{1}, F_{2} ∈ F^{Γ}(S) for some S ∈ S(G), then Z_{G}(S) contains a maximal torus T by [1, XIV 3.20], thus T ⊂ Z_{G}(S) ⊂ P_{F_{1}} ∩ P_{F_{2}}. If conversely T ⊂ P_{F_{1}} ∩ P_{F_{2}} for some maximal torus T of G, then F_{1}, F_{2} ∈ F^{Γ}(S) for any S ∈ S(G) containing the maximal split torus T_{sp} of T: if R_{i} is the split radical of the unique Levi subgroup L_{i} of P_{F_{i}} containing T [21, XXVI 1.6], then R_{i} ⊂ T_{sp} ⊂ S, therefore Z_{G}(S) ⊂ Z_{G}(R_{i}) = L_{i} ⊂ P_{F_{i}} by [21, XXVI 6.11]. We will denote by

Std(G) = 𝕊𝕋𝔻(G)(O) and Std^{Γ}(G) = 𝕊𝕋𝔻^{Γ}(G)(O)

the corresponding subsets of P(G)^{2} and F^{Γ}(G)^{2}, so that

Std^{Γ}(G) = F^{−1}(Std(G)) = ∪_{S∈S(G)}F^{Γ}(S) × F^{Γ}(S) ⊂ F^{Γ}(G)^{2}.

For any S ∈ S(G), the map + : Std^{Γ}(G) → F^{Γ}(G) of section 2.3.2 induces the natural commutative group structure on F^{Γ}(S) = G^{Γ}(S) = Hom(𝔻_{O}(Γ), S).

4.1.16. For P ∈ P(G) with unipotent radical U and Levi L, we also define

Std^{Γ}(P) = {F ∈ F^{Γ}(G) : (P, P_{F}) ∈ Std(G)} = ∪_{Z_{G}(S)⊂P} F^{Γ}(S).

As explained in sections 2.3.3 and 2.3.4, the functorial map F^{Γ}(L) → F^{Γ}(G) lands in Std^{Γ}(P) and actually defines a section of a P-equivariant map

Gr_{P} : Std^{Γ}(P) ↠ F^{Γ}(P/U)

which may be computed as follows: starting with F ∈ Std^{Γ}(P), pick S ∈ S(G) such that Z_{G}(S) ⊂ P ∩ P_{F}, let G ∈ G^{Γ}(S) be the corresponding splitting of F and let \overline{G} be the image of G in G^{Γ}(P/U). Then Gr_{P} (F) = Fil(\overline{G}) in F^{Γ}(P/U). Thus for F ∈ F^{−1}(P), F ∈ Std^{Γ}(P) and Gr_{P} (F) = \overline{F} with \overline{F} ∈ G^{Γ}(\overline{R}(P)) as in 2.2.8.

Theorem. [21, XXVI 4.1.1] If O = K is a field, then Std(G) = P(G)^{2}, thus also Std^{Γ}(G) = F^{Γ}(G)^{2} and Gr_{P} is defined on the whole of F^{Γ}(G) = Std^{Γ}(P):

Gr_{P} : F^{Γ}(G) ↠ F^{Γ}(P/U)

4.1.17. Suppose now that O is a Henselian local ring with residue field k.

Proposition 78. The specialization from O to k induces a map from the diagram of section 4.1.9 for G to the similar diagram for G_{k}. In the resulting commutative diagram, all the specialization maps X(G) → X(G_{k}) are surjective, all the squares involving two F ’s are cartesian, and O(G) ≃ O(G_{k}), C^{Γ}(G) ≃ C^{Γ}(G_{k}).

Proof. Since 𝔾^{Γ}, 𝔽^{Γ}, ℂ^{Γ}, 𝕆ℙℙ, ℙ and 𝕆 are smooth over Spec(O), the specialization from O to k induces a map from the last two squares of our diagram for G to the last two squares of the analogous diagram for G_{k}, in which all specialization maps X(G) → X(G_{k}) are surjective by [28, 18.5.17]. Since 𝕆 is finite étale over Spec(O), O(G) → O(G_{k}) is also injective by [28, 18.5.4-5], i.e. O(G) = O(G_{k}). It follows that P(G) ↠ P(G_{k}) induces B(G) ↠ B(G_{k}). If S is a maximal split torus in G, then Z_{G}(S) is a Levi subgroup of a minimal parabolic subgroup B of G, S is the maximal split subtorus of the radical R of Z_{G}(S), thus R/S is an anisotropic torus, i.e. Hom(𝔾_{m,O}, R/S) = 0. Then by proposition 3, lemma 4 and [28, 18.5.4-5], also Hom(𝔾_{m,k}, R_{k}/S_{k}) = 0, thus S_{k} is the maximal split subtorus of the radical R_{k} of the Levi subgroup Z_{G}(S)_{k} = Z_{G_{k}}(S_{k}) of the minimal parabolic subgroup B_{k} of G_{k}, in particular S_{k} is a maximal split subtorus of G_{k} and the specialization map S(G) → S(G_{k}) is well-defined. It is surjective: starting with \overline{S} in S(G_{k}), choose \overline{B} ∈ B(G_{k}) containing Z_{G_{k}}(\overline{S}), lift \overline{B} to some B ∈ B(G), choose S′ ∈ S(G) with Z_{G}(S′) ⊂ B, write \overline{S} = Int(\overline{b})(S′_{k}) for some b ∈ B(k), lift \overline{b} to some b ∈ B(O) using [28, 18.5.17] and set S = Int(b)(S′). Then S ∈ S(G) and S_{k} = \overline{S}. The same argument shows that SBP(G) → SBP(G_{k}) and SP(G) → SP(G_{k}) are well-defined and surjective, from which follows that also ACF^{Γ}(G) → ACF^{Γ}(G_{k}) and AF^{Γ}(G) → AF^{Γ}(G_{k}) are well-defined. To establish all of the remaining claims, it is sufficient to show that C^{Γ}(G) ↠ C^{Γ}(G_{k}) is also injective, which again follows from [28, 18.5.4-5] since ℂ^{Γ}(G) is separated and étale over O. Alternatively, fix (S, B) as above and let s : C^{Γ}(G) ↪ F^{Γ}(G) and s_{k} : C^{Γ}(G_{k}) ↪ F^{Γ}(G_{k}) be the corresponding sections. They are compatible with the specialization maps and there images are respectively contained in the apartments F^{Γ}(S) of F^{Γ}(G) and F^{Γ}(S_{k}) of F^{Γ}(G_{k}). Since G^{Γ}(S) ≃ G^{Γ}(S_{k}), the specialization map F^{Γ}(G) → F^{Γ}(G_{k}) restricts to a bijection F^{Γ}(S) ≃ F^{Γ}(S_{k}), therefore C^{Γ}(G) ↠ C^{Γ}(G_{k}) is indeed injective. □

4.1.18. Suppose now that O is a valuation ring with fraction field K.

Proposition 79. The generization from O to K induces a map from the diagram of section 4.1.9 for G to the similar diagram for G_{K}. In the resulting commutative diagram, all the generization maps X(G) → X(G_{K}) are injective, they are bijective for X ∈ {F^{Γ}, C^{Γ}, P, O} and all the squares involving two F ’s are cartesian.

Proof. Since 𝔾^{Γ}, 𝔽^{Γ}, ℂ^{Γ}, 𝕆ℙℙ, ℙ and 𝕆 are separated over Spec(O), the generization from O to K induces a map from the last two squares of our diagram for G to the last two squares of the analogous diagram for G_{K}, in which all generization maps X(G) → X(G_{K}) are injective. Since 𝕆 and ℙ are proper over Spec(O), the maps P(G) → P(G_{K}) and O(G) → O(G_{K}) are in fact bijective. It follows that P(G) ≃ P(G_{K}) induces B(G) ≃ B(G_{K}). If S is a maximal split torus in G, then Z_{G}(S) is a Levi subgroup of a minimal parabolic subgroup B of G, S is the maximal split subtorus of the radical R of Z_{G}(S), thus R/S is an anisotropic torus, i.e. Hom(𝔾_{m,O}, R/S) = 0. Then by proposition 3 and lemma 4, also Hom(𝔾_{m,K}, R_{K}/S_{K}) = 0, thus S_{K} is the maximal split subtorus of the radical R_{K} of the Levi subgroup Z_{G}(S)_{K} = Z_{G_{K}} (S_{K}) of the minimal parabolic subgroup B_{K} of G_{K}, in particular S_{K} is a maximal split subtorus of G_{K} and the specialization map S(G) → S(G_{K}) is well-defined. It is injective by [21, XXII 5.8.3], so are SBP(G) → SBP(G_{K}) and SP(G) → SP(G_{K}), while ACF^{Γ}(G) → ACF^{Γ}(G_{K}) and AF^{Γ}(G) → AF^{Γ}(G_{K}) are well-defined. To establish the remaining claims, it is sufficient to show that C^{Γ}(G) ↪ C^{Γ}(G_{K}) is also surjective, which again follows from lemma 4 since ℂ^{Γ}(G) is a quasi-isotrivial twisted constant scheme over O. Alternatively, fix (S, B) as above and let s : C^{Γ}(G) ↪ F^{Γ}(G) and s_{K} : C^{Γ}(G_{k}) ↪ F^{Γ}(G_{k}) be the corresponding sections. They are compatible with the generization maps and there images are respectively contained in the apartments F^{Γ}(S) of F^{Γ}(G) and F^{Γ}(S_{K}) of F^{Γ}(G_{K}). Since G^{Γ}(S) ≃ G^{Γ}(S_{K}), the generization map F^{Γ}(G) → F^{Γ}(G_{K}) restricts to a bijection F^{Γ}(S) ≃ F^{Γ}(S_{K}), therefore C^{Γ}(G) ↪ C^{Γ}(G_{K}) is indeed surjective. □

Remark 80. Under the identifications of theorem 34 (note that G is isotrivial over the valuation ring O by proposition 48), the inverse of F^{Γ}(G) → F^{Γ}(G_{K}) maps a Γ-filtration F_{K} on ω_{K}^{∘} to the Γ-filtration F on ω^{∘} defined by

∀τ ∈ Rep^{∘}(G)(O), γ ∈ Γ : F^{γ}(τ) = F_{K}^{γ}(τ) ∩ V (τ) in V_{K}(τ).

It is not at all obvious that this formula indeed defines a right exact functor!

4.1.19. Set Gen(G) = 𝔾𝔼ℕ(G)(O), Gen^{Γ}(G) = 𝔾𝔼ℕ^{Γ}(G)(O) and define

Gen(Y) = {P ∈ P(G) : {P } × Y ⊂ Gen(G)}
Gen^{Γ}(X) = {F ∈ F^{Γ}(G) : {F} × X ⊂ Gen^{Γ}(G)}

for Y ⊂ P(G) and X ⊂ F^{Γ}(G). We say that Y (resp. X) is thin if

t(Gen(Y)) = O(G) (resp. t(Gen^{Γ}(X)) = C^{Γ}(G)).

Plainly, F^{−1}(Gen(Y)) = Gen^{Γ}(F^{−1}(Y)) and F (Gen^{Γ}(X)) = Gen(F (X)), thus

(Y is thin ⇔ F^{−1}(Y) is thin) and (X is thin ⇔ F (X) is thin) .

Moreover Y is thin ⇔ ∘ ∈ t(Gen(Y)) and a thin subset of a thin set is thin.

Lemma 81. Suppose that O is a strictly Henselian valuation ring with residue field k and fraction field K. Then any subset Y_{K} of P(G_{K}) (resp. X_{K} of F^{Γ}(G_{K})) whose image in P(G_{k}) is finite is a thin subset of P(G_{K}) (resp. F^{Γ}(G_{K})).

Proof. It is sufficient to treat the case of a subset Y_{K} of P(G_{K}). Let Y ≃ Y_{K} be its pre-image in P(G) with finite image Y_{k} in P(G_{k}). For every y ∈ Y_{k},

U_{y} = t^{−1}(∘_{k}) × {y} ∩ 𝔾𝔼ℕ(G_{k})

is a non-empty open subscheme of the (geometrically) irreducible k-scheme t^{−1}(∘_{k}) by [21, XXVI 4.2.4.iii], thus (∩_{y∈Y_{k}}U_{y})(k) = B(G_{k}) ∩ Gen(Y_{k}) is not empty since k is algebraically closed and t^{−1}(∘_{k}) is of finite type over k. If B ∈ B(G) lifts B_{k} ∈ B(G_{k}) ∩ Gen(Y_{k}), then {B} × Y ⊂ Gen(G) since 𝔾𝔼ℕ(G) is open in ℙ(G)^{2}, thus also {B_{K}} × Y_{K} ⊂ Gen(G_{K}), i.e. B_{K} ∈ Gen(Y_{K}). Since also B_{K} ∈ B(G_{K}), Y_{K} is indeed a thin subset of P(G_{K}). □

4.2. Distances and angles

Suppose from now on that Γ is a subring of ℝ with the induced total order on the underlying commutative group.

4.2.1. Recall from theorem 34 that for τ ∈ Rep^{∘}(G)(O), any F ∈ F^{Γ}(G) defines a Γ-filtration F(τ) on the (free) O-module V (τ). For any (F_{1}, F_{2}) ∈ Std^{Γ}(G) and γ_{1}, γ_{2} ∈ Γ, the O-module

Gr_{F_{1},F_{2}}^{γ_{1},γ_{2}}(τ) = \frac{F_{1}^{γ_{1}}(τ) ∩ F_{2}^{γ_{2}}(τ)}{F_{1,+}^{γ_{1}}(τ) ∩ F_{2}^{γ_{2}}(τ) + F_{1}^{γ_{1}}(τ) ∩ F_{2,+}^{γ_{2}}(τ)}

is free of finite rank: if F_{i} = Fil(G_{i}) with G_{i} ∈ G^{Γ}(S) = Hom(M, Γ) for some S in S(G) with M = Hom(S, 𝔾_{m,O}), then F_{i}(τ)^{γ} = ⊕_{G_{i}(m)≥γ}V (τ)_{m} for any i ∈ {1, 2} and γ ∈ Γ where V (τ) = ⊕_{m∈M}V (τ)_{m} is the eigenspace decomposition of τ|_{S}, thus

Gr_{F_{1},F_{2}}^{γ_{1},γ_{2}}(τ) = ⊕_{m:G_{i}(m)=γ_{i}}V (τ)_{m}.

4.2.2. Since Γ is a subring of ℝ:

• Any apartment is endowed with a canonical structure of free Γ-module, and these structures are preserved by the action of Aut(G) on F^{Γ}(G). Indeed,

F^{Γ}(S) = G^{Γ}(S) = Hom(M(S), Γ) with M(S) = Hom(S, 𝔾_{m,O}).

• Any τ ∈ Rep^{∘}(G)(O) defines a G-invariant function

⟨−, −⟩_{τ} : Std^{Γ}(G) → Γ, ⟨F_{1}, F_{2}⟩_{τ} = _{∑γ_{1},γ_{2}}rank_{O} (Gr_{F_{1},F_{2}}^{γ_{1},γ_{2}}(τ)) · γ_{1}γ_{2}

whose restriction to F^{Γ}(S) is bilinear, symmetric and non-negative, given by

⟨F_{1}, F_{2}⟩_{τ} = _{∑m∈M(S)}rank_{O} (V (τ)_{m}) · G_{1}(m)G_{2}(m)

if F_{i} ∈ F^{Γ}(S) corresponds to G_{i} ∈ Hom(M(S), Γ). Its kernel equals G^{Γ}(ker(τ|S)), thus ⟨−, −⟩_{τ} is positive definite when τ is a faithful representation of G.

• Write ∥F∥_{τ} = ⟨F, F⟩_{τ}^{1/2}. Thus ∥−∥_{τ} : F^{Γ}(G) → ℝ_{+} is a G-invariant function. It descends to a G-invariant function ∥−∥_{τ} : C^{Γ}(G) → ℝ_{+} with ∥F∥_{τ} = ∥t(F)∥_{τ}. We have the Cauchy-Schwartz inequality

∀(F_{1}, F_{2}) ∈ Std^{Γ}(G) : |⟨F_{1}, F_{2}⟩_{τ} | ≤ ∥F_{1}∥_{τ} ∥F_{2}∥_{τ} .

• We may thus also define a G-invariant angle

∡_{τ} (−, −) : Std^{Γ}(G) → [0, π], ⟨F_{1}, F_{2}⟩_{τ} = cos (∡_{τ} (F_{1}, F_{2})) · ∥F_{1}∥_{τ} ∥F_{2}∥_{τ}

and a G-invariant function

d_{τ} (−, −) : Std^{Γ}(G) → ℝ_{+}, d_{τ} (F_{1}, F_{2}) = \sqrt{∥F_{1}∥_{τ}^{2} + ∥F_{2}∥_{τ}^{2} − 2 ⟨F_{1}, F_{2}⟩_{τ}}

inducing the distance d_{τ} (F_{1}, F_{2}) = ∥F_{2} − F_{1}∥_{τ} on any apartment.

• If τ is faithful, then d_{τ} (F_{1}, F_{2}) = 0 if and only if F_{1} = F_{2} and the map G ↦ (Fil(G), Fil(ιG)) induces a G-equivariant bijection

G^{Γ}(G) ≃ {(F_{1}, F_{2}) ∈ Std^{Γ}(G) : ∥F_{1}∥_{τ} = ∥F_{2}∥_{τ} and ∡_{τ} (F_{1}, F_{2}) = π} .

4.2.3. For (F_{1}, F_{2}) ∈ Std^{Γ}(G), we also have the following formula

⟨F_{1}, F_{2}⟩_{τ} = ∑_{γ} γ · deg Gr_{F_{1}}^{γ}(F_{2}, τ) = ∑_{γ} γ · deg Gr_{F_{2}}^{γ}(F_{1}, τ)

where Gr_{F}^{γ}(G, τ) is the filtration induced by G(τ) on Gr_{F}^{γ}(τ) = F^{γ}(τ)/F_{+}^{γ}(τ), i.e.

∀θ ∈ Γ, Gr_{F}^{γ}(G, τ)^{θ} = F^{γ}(τ) ∩ G^{θ}(τ) + F_{+}^{γ}(τ)/F_{+}^{γ}(τ)

and the degree of an ℝ-filtration F on a finite free O-module V is given by

deg(F) = _{∑γ}γ · rank_{O}(Gr_{F}^{γ}).

Choosing a splitting of F, one checks easily that also

deg(F) = deg(det F)

where det F is the ℝ-filtration on det V = Λ_{O}^{r}V, r = rank_{O}V defined by

(det F)^{γ} = span of {v_{1} ∧ ⋯ ∧ v_{r} : v_{i} ∈ F^{γ_{i}}, ∑γ_{i} = γ} .

4.2.4. If Γ is a ℚ-vector space, the decomposition

𝔽^{Γ}(G) = 𝔽^{Γ}(G^{der}) × 𝔾^{Γ}(Z(G))

of section 2.2.13 induces an analogous decomposition

F^{Γ}(G) = F^{Γ}(G^{der}) × G^{Γ}(Z(G))

which is orthogonal in the following sense: for F_{1}, F_{2} ∈ Std^{Γ}(G) and F ∈ F^{Γ}(G),

⟨F_{1}, F_{2}⟩_{τ} = ⟨F_{1}^{r}, F_{2}^{r}⟩_{τ} + ⟨F_{1}^{c}, F_{2}^{c}⟩_{τ}
d_{τ} (F_{1}, F_{2})^{2} = d_{τ} (F_{1}^{r}, F_{2}^{r})^{2} + d_{τ} (F_{1}^{c}, F_{2}^{c})^{2}
∥F∥_{τ}^{2} = ∥F^{r}∥_{τ}^{2} + ∥F^{c}∥_{τ}^{2}

where F^{r} ∈ F^{Γ}(G^{der}) ⊂ F^{Γ}(G) and F^{c} ∈ G^{Γ}(Z(G)) ⊂ F^{Γ}(G) are the components of F. We prove the first formula. Pick S ∈ S(G) with F_{1}, F_{2} ∈ F^{Γ}(S). Then also F_{i}^{r}, F_{i}^{c} ∈ F^{Γ}(S) with F_{i} = F_{i}^{r} + F_{i}^{c} in the apartment F^{Γ}(S), thus

⟨F_{1}, F_{2}⟩_{τ} = ⟨F_{1}^{r}, F_{2}^{r}⟩_{τ} + ⟨F_{1}^{c}, F_{2}^{c}⟩_{τ} + ⟨F_{1}^{r}, F_{2}^{c}⟩_{τ} + ⟨F_{1}^{c}, F_{2}^{r}⟩_{τ}

since ⟨−, −⟩_{τ} is a bilinear form on F^{Γ}(S). It is therefore sufficient to show that

⟨F, G⟩_{τ} = ⟨G, F⟩_{τ} = _{∑γ}γ · deg Gr_{G}^{γ}(F, τ) = 0

for any F ∈ F^{Γ}(G^{der}) and G ∈ G^{Γ}(Z(G)). Since G : 𝔻_{O}(Γ) → Z(G) is central in G, τ = ⊕τ_{γ} with V (τ_{γ}) = G_{γ}(τ) and Gr_{G}^{γ}(F, τ) ≃ F(τ_{γ}) on Gr_{G}^{γ}(τ) ≃ τ_{γ}, thus

deg Gr_{G}^{γ}(F, τ) = deg F(τ_{γ}) = deg (det F(τ_{γ})) = deg (F(det τ_{γ})) = 0

because the restriction of det τ_{γ} to G^{der} is trivial.

4.2.5. For x, y ∈ O(G), there is a single G-orbit of (P, Q)’s in t^{−1}(x)×t^{−1}(y) such that P and Q are in osculatory (resp. transverse) position [21, XXVI 5.3-5], and this orbit is contained in Std(G). Thus for any x, y ∈ C^{Γ}(G), there is a single G-orbit of (F_{1}, F_{2})’s in t^{−1}(x) × t^{−1}(y) with the property that P_{F_{1}} and P_{F_{2}} are in osculatory (resp. transverse) position, and it is contained in Std^{Γ}(G). We set ∡_{τ}^{os}(x, y) = ∡_{τ} (F_{1}, F_{2}) and ⟨x, y⟩_{τ}^{os} = ⟨F_{1}, F_{2}⟩_{τ} (resp. ∡_{τ}^{tr}(x, y) = ∡_{τ} (F_{1}, F_{2}) and ⟨x, y⟩_{τ}^{tr} = ⟨F_{1}, F_{2}⟩_{τ}), thus obtaining two other pairs of symmetric functions

∡_{τ}^{os}(−, −) : C^{Γ}(G) × C^{Γ}(G) → [0, π] and ⟨−, −⟩_{τ}^{os} : C^{Γ}(G) × C^{Γ}(G) → ℝ,
∡_{τ}^{tr}(−, −) : C^{Γ}(G) × C^{Γ}(G) → [0, π] and ⟨−, −⟩_{τ}^{tr} : C^{Γ}(G) × C^{Γ}(G) → ℝ.

They are of course related by the formulas

⟨x, y⟩_{τ}^{os} = cos (∡_{τ}^{os}(x, y)) · ∥x∥_{τ} ∥y∥_{τ} ,
⟨x, y⟩_{τ}^{tr} = cos (∡_{τ}^{tr}(x, y)) · ∥x∥_{τ} ∥y∥_{τ} .

We also define yet another symmetric function

d_{τ} : C^{Γ}(G) × C^{Γ}(G) → ℝ_{+} d_{τ} (x, y) = \sqrt{∥x∥_{τ}^{2} + ∥y∥_{τ}^{2} − 2 ⟨x, y⟩_{τ}^{os}}.

By construction, for (F_{1}, F_{2}) in Gen^{Γ}(G) = 𝔾𝔼ℕ^{Γ}(G)(O),

∡_{τ} (F_{1}, F_{2}) = ∡_{τ}^{tr} (t(F_{1}), t(F_{2})) and ⟨F_{1}, F_{2}⟩_{τ} = ⟨F_{1}, F_{2}⟩_{τ}^{tr} .

4.2.6. The above constructions are merely special cases of those of section 3.12. In particular, our functions are induced by morphisms of schemes over O, and some of them (the “bilinear forms”) still make sense for an arbitrary τ in the Grothendieck ring K_{0}(G) of Rep^{∘}(G)(O), or even for a global section of the sheafified version \underline{K}_{0}(G) of that ring. However, the positivity of these forms requires some sort of effectiveness/faithfulness of the initial τ, which we have not tried to axiomatize. From now on until 4.2.11, we fix a faithful τ in Rep^{∘}(G)(O).

4.2.7. Fix (S, B) ∈ S(G) × B(G) with Z_{G}(S) ⊂ B, let B′ = ι_{S}B be the minimal parabolic subgroup of G containing Z_{G}(S) opposed to B, and denote by

s, s′ : C^{Γ}(G) ↪ F^{Γ}(G)

the corresponding sections of t : F^{Γ}(G) ↠ C^{Γ}(G). Then for x, y ∈ C^{Γ}(G),

B ⊂ P_{s(x)} ∩ P_{s(y)} and B′ ⊂ P_{s′(x)} ∩ P_{s′(y)},

thus P_{s(x)} and P_{s(y)} are in osculatory position while P_{s(x)} and P_{s′(y)} are in transverse position. It follows that for every x, y ∈ C^{Γ}(G),

\begin{aligned}∡_{τ}^{os}(x, y) & = & ∡_{τ} (s(x), s(y)) & and & ⟨x, y⟩_{τ}^{os} & = & ⟨s(x), s(y)⟩_{τ} , \\ ∡_{τ}^{tr}(x, y) & = & ∡_{τ} (s(x), s′(y)) & and & ⟨x, y⟩_{τ}^{tr} & = & ⟨s(x), s′(y)⟩_{τ} . \end{aligned}

In particular, the “scalar products” are compatible with the monoid structure:

\begin{aligned}⟨x_{1} + x_{2}, y⟩_{τ}^{os} & = & ⟨x_{1}, y⟩_{τ}^{os} + ⟨x_{2}, y⟩_{τ}^{os} & and & ⟨x, y_{1} + y_{2}⟩_{τ}^{os} & = & ⟨x, y_{1}⟩_{τ}^{os} + ⟨x, y_{2}⟩_{τ}^{os} , \\ ⟨x_{1} + x_{2}, y⟩_{τ}^{tr} & = & ⟨x_{1}, y⟩_{τ}^{tr} + ⟨x_{2}, y⟩_{τ}^{tr} & and & ⟨x, y_{1} + y_{2}⟩_{τ}^{tr} & = & ⟨x, y_{1}⟩_{τ}^{tr} + ⟨x, y_{2}⟩_{τ}^{tr} . \end{aligned}

Moreover, d_{τ} is a distance on C^{Γ}(G).

4.2.8. The following lemma is related to the angle rigidity axiom of [29, 4.1.2].

Lemma 82. For any x, y ∈ C^{Γ}(G), the set

D_{τ} (x, y) = {∡_{τ} (F_{1}, F_{2}) : (F_{1}, F_{2}) ∈ Std^{Γ}(G) ∩ t^{−1}(x) × t^{−1}(y)}

is finite with

min D_{τ} (x, y) = ∡_{τ}^{os}(x, y) and max D_{τ} (x, y) = ∡_{τ}^{tr}(x, y).
Proof. Fix (S, B) and s, s′ : C^{Γ}(G) ↪ F^{Γ}(G) as above. Then any pair
(F_{1}, F_{2}) ∈ Std^{Γ}(G) ∩ t^{−1}(x) × t^{−1}(y)

is G-conjugated to some pair in W_{G}(S) · s(x) × W_{G}(S) · s(y) ⊂ F^{Γ}(S)^{2}, thus

D_{τ} (x, y) = {∡_{τ} (w_{1} · s(x), w_{2} · s(y)) : (w_{1}, w_{2}) ∈ W_{G}(S)^{2}}
= {∡_{τ} (s(x), w · s(y)) : w ∈ W_{G}(S)}

is finite. To establish our final claim, we have to show that

⟨s(x), s(y)⟩_{τ} ≥ ⟨s(x), w · s(y)⟩_{τ} ≥ ⟨s(x), s′(y)⟩_{τ}

for every w ∈ W_{G}(S), which follows from [7, Proposition 18]. □

Corollary 83. The type map t : F^{Γ}(G) ↠ C^{Γ}(G) is compatible with the d_{τ} ’s:

∀(F_{1}, F_{2}) ∈ Std^{Γ}(G) : d_{τ} (t(F_{1}), t(F_{2})) ≤ d_{τ} (F_{1}, F_{2}).

4.2.9. Let us use the above notions to show that

Proposition 84. For a facet F, a chamber C and apartments A_{1}, A_{2} in F^{Γ}(G) with F ∪ C ⊂ A_{1} ∩ A_{2}, there exists g ∈ G with gA_{1} = A_{2} and g ≡ 1 on \overline{F} ∪ \overline{C}.

Proof. In group theoretical terms, this means that for P ∈ P(G), B ∈ B(G) and S_{1}, S_{2} ∈ S(G) with Z_{G}(S_{i}) ⊂ B ∩ P, there is a g ∈ G such that Int(g)(S_{1}) = S_{2} and g ∈ B ∩ P with B = B(O), P = P (O). This does not depend upon Γ, and we may thus assume that Γ = ℝ. Since (S_{1}, B) and (S_{2}, B) ∈ SP(G) have the same image in O(G), there exists an element g ∈ G with g(S_{1}, B) = (S_{2}, B), i.e. Int(g)(S_{1}) = S_{2} and g ∈ B. We will show that also g ∈ P, i.e. gF = F for any F ∈ F^{−1}(P) ⊂ F^{ℝ}(G). Note that F, gF and the chamber C = F^{−1}(B) are all contained in the apartment F^{ℝ}(S_{2}). Fix a faithful τ ∈ Rep^{∘}(G)(O). Then

⟨F, F′⟩_{τ} = ⟨gF, gF′⟩_{τ} = ⟨gF, F′⟩_{τ}

for all F′ ∈ F^{−1}(B), thus F = gF because F^{−1}(B) is a non-empty open subset of the Euclidean space (F^{ℝ}(S_{2}), ⟨−, −⟩_{τ} ) by 4.1.10. □

4.2.10. Suppose for this and the next subsection that our local ring O = k is a field. Then every pair (F_{1}, F_{2}) ∈ F^{Γ}(G) is contained in some apartment since

Theorem 85. [21, XXVI 4.1.1] Std(G) = P(G)^{2} and Std^{Γ}(G) = F^{Γ}(G)^{2}.

Corollary 86. For any apartments A_{1}, A_{2} in F^{Γ}(G) and facets F_{1}, F_{2} in A_{1} ∩ A_{2}, there exists g ∈ G mapping A_{1} to A_{2} with g ≡ 1 on \overline{F}_{1} ∪ \overline{F}_{2}.

Proof. Fix closed chambers F_{1} ⊂ \overline{C}_{1} ⊂ A_{1} and F_{2} ⊂ \overline{C}_{2} ⊂ A_{2} and choose an apartment A_{3} containing C_{1} and C_{2}. The previous proposition shows that there exists elements g_{1}, g_{2} ∈ G such that g_{1}A_{1} = A_{3} = g_{2}A_{2}, g_{1} ≡ 1 on \overline{C}_{1} ∪ \overline{F}_{2} and g_{2} ≡ 1 on \overline{C}_{2} ∪ \overline{F}_{1}. Then g = g_{2}^{−1}g_{1} maps A_{1} to A_{2} and g ≡ 1 on \overline{F}_{1} ∪ \overline{F}_{2}. □

Corollary 87. For a monomorphism f : G_{1} → G_{2} of reductive groups over k, the induced map f : F^{Γ}(G_{1}) → F^{Γ}(G_{2}) is injective.

Proof. Fix a faithful τ ∈ Rep^{∘}(G_{2})(k). Then f^{∗}τ = τ ∘ f ∈ Rep^{∘}(G_{1})(k) is also faithful and for every F, F′ ∈ F^{Γ}(G_{1}),

⟨f(F), f(F′)⟩_{τ} = ⟨F, F′⟩_{f^{∗}τ} and d_{τ} (f(F), f(F′)) = d_{f^{∗}τ} (F, F′).

Therefore f(F) = f(F′) implies F = F′. □

Corollary 88. Let P be a parabolic subgroup of G with unipotent radical U and Levi subgroup L. Then F^{Γ}(L) is a fundamental domain for the action of U(k) on F^{Γ}(G). Let r = r_{P,L} : F^{Γ}(G) ↠ F^{Γ}(L) be the corresponding retraction. Then

∀x, y ∈ F^{Γ}(G) : d_{τ} (rx, ry) ≤ d_{τ} (x, y).
Corollary 89. The function d_{τ} : F^{Γ}(G) × F^{Γ}(G) → ℝ_{+} is a distance:
∀x, y, z ∈ F^{Γ}(G) : d_{τ} (x, y) ≤ d_{τ} (x, z) + d_{τ} (z, y).

Proof. Fix S_{0} ∈ S(L). The P = P (k) and L = L(k) orbits of S_{0} in S(G) are respectively equal to S(G, P) = {S ∈ S(G) : Z_{G}(S) ⊂ P } and S(L). Since any F ∈ F^{Γ}(G) belongs to F^{Γ}(S) for some S ∈ S(G, P), we find that with U = U(k),

F^{Γ}(G) = ∪_{S∈S(G,P)}F^{Γ}(S) = P · F^{Γ}(S_{0}) = U · ∪_{S∈S(L)}F^{Γ}(S) = U · F^{Γ}(L).

Suppose that F, uF ∈ F^{Γ}(L) for some u ∈ U, and choose an S ∈ S(L) with F, uF ∈ F^{Γ}(S). Since Z_{G}(S) ⊂ L ⊂ P, there is a B ∈ B(G) with Z_{G}(S) ⊂ B ⊂ P. Let C = F^{−1}(B) be the corresponding (G-)chamber in A = F^{Γ}(S). Since U ⊂ B, uC = C and F, C ∈ A ∩ u^{−1}A. Choose g ∈ G with gu^{−1}A = A, gF = F and gC = C. Then g belongs to B = B(k), thus gu^{−1} belongs to B ∩ N_{G}(S) = Z_{G}(S) which acts trivially on A. Therefore uF = gu^{−1}uF = gF = F and F^{Γ}(L) is a fundamental domain for the action of U on F^{Γ}(G).

For A ∈ A(G) containing F^{−1}(P), there is a unique A_{L} ∈ A(L) ∩ U · {A} such that r(x) = ux for any x ∈ A and u ∈ U such that uA = A_{L}. Indeed, there is a p = lu in P = LU such that pA is an apartment of F^{Γ}(L), then uA = l^{−1}pA ⊂ F^{Γ}(L) and r(x) = ux for every x ∈ A. Thus for x, y ∈ A, d_{τ} (rx, ry) = d_{τ} (x, y).

For the remaining claims, we may assume that Γ = ℝ and use induction on the semi-simple rank s of G. If s = 0 everything is obvious. If s > 0 but G = L, then r is the identity thus d_{τ} (rx, ry) = d_{τ} (x, y) for every x, y ∈ F^{ℝ}(G). If G ≠ L, choose an apartment A in F^{ℝ}(G) containing x and y, let [x, y] be the corresponding segment of A, and write [x, y] = ∪_{i=0}^{n−1}[x_{i}, x_{i+1}] for consecutive points x_{i} ∈ [x, y] with x_{0} = x, x_{n} = y and ]x_{i}, x_{i+1}[ contained in a facet F_{i} ⊂ A. Then there is an apartment containing F^{−1}(P) and \{x_{i}, x_{i+1}\} ⊂ \overline{F}_{i}, thus d_{τ} (rx_{i}, rx_{i+1}) = d_{τ} (x_{i}, x_{i+1}) for every i ∈ {0, ⋯, n − 1}. Since d_{τ} is a distance on F^{ℝ}(L) by our induction hypothesis,

d_{τ} (rx, ry) ≤ ∑_{i=0}^{n−1}d_{τ} (rx_{i}, rx_{i+1}) = ∑_{i=0}^{n−1}d_{τ} (x_{i}, x_{i+1}) = d_{τ} (x, y).

Finally for x, y, z ∈ F^{ℝ}(G), choose an apartment F^{ℝ}(S) containing x, y and a chamber F^{−1}(B), let r = r_{B,Z_{G}(S)} be the corresponding retraction. Then

d_{τ} (x, y) = d_{τ} (rx, ry) ≤ d_{τ} (rx, rz) + d_{τ} (rz, ry) ≤ d_{τ} (x, z) + d_{τ} (z, y).

This finishes the proof of corollaries 88 and 89. □

Corollary 90. If Γ = ℝ, then (F^{ℝ}(G), d_{τ}) is a complete CAT(0)-space.

Proof. Plainly, (C^{ℝ}(G), d_{τ}) is a complete metric space. Let (x_{n}) be a Cauchy sequence in (F^{ℝ}(G), d_{τ} ). Then t(x_{n}) is a Cauchy sequence in (C^{ℝ}(G), d_{τ}) by corollary 83, it thus converges to some y ∈ C^{ℝ}(G). Now choose for each n a minimal pair (S_{n}, B_{n}) ∈ SP(G) corresponding to a section s_{n} : C^{ℝ}(G) → F^{ℝ}(G) passing through x_{n}, and let y_{n} = s_{n}(y). Then d_{τ} (x_{n}, y_{n}) = d_{τ} (t(x_{n}), y) converges to 0 and y_{n} is also a Cauchy sequence in F^{ℝ}(G). But d_{τ} (y_{n}, y_{m}) takes finitely many values by lemma 82, therefore y_{n} is stationary and x_{n} converges to its limit. Thus (F^{ℝ}(G), d_{τ}) is complete, and a geodesic space by theorem 85. Finally, for any triple x, y, z in F^{ℝ}(G), choose a minimal pair (S, B) ∈ SP(G) such that x, y ∈ F^{ℝ}(S) and the middle point m of the segment [x, y] of F^{ℝ}(S) belongs to F^{−1}(B). Let r = r_{B,Z_{G}(S)} : F^{ℝ}(G) → F^{ℝ}(S) be the corresponding retraction and pick u in U = U(k) with uz = r(z), where U is the unipotent radical of B. Then

d_{τ} (z, m)^{2} = d_{τ} (uz, um)^{2} = d_{τ} (r(z), m)^{2} = \frac{1}{2} d_{τ} (r(z), x)^{2} + \frac{1}{2} d_{τ} (r(z), y)^{2} − \frac{1}{4} d_{τ} (x, y)^{2}
≤ \frac{1}{2} d_{τ} (z, x)^{2} + \frac{1}{2} d_{τ} (z, y)^{2} − \frac{1}{4} d_{τ} (x, y)^{2}

thus F^{ℝ}(G) is a CAT(0)-space by [8, II.1.9]. □

Corollary 91. For any F ∈ F^{ℝ}(G), the function G ↦ ⟨F, G⟩_{τ} from F^{ℝ}(G) to ℝ is homogeneous, concave and ∥F∥_{τ} -Lipschitzian.

Proof. Homogeneity means that ⟨F, tG⟩_{τ} = t ⟨F, G⟩_{τ} for all t ∈ ℝ_{+}, which is obvious from the definitions. Concavity means that for any G_{0}, G_{1} ∈ F^{ℝ}(G) and t ∈ [0, 1], if G_{t} is the unique point at distance td_{τ} (G_{0}, G_{1}) from G_{0} on the geodesic segment [G_{0}, G_{1}] of the uniquely geodesic space (F^{ℝ}(G), d_{τ}) [8, II.1.4], then

⟨F, G_{t}⟩_{τ} ≥ t ⟨F, G_{1}⟩_{τ} + (1 − t) ⟨F, G_{0}⟩_{τ} .

Let (0, f, g_{0}, g_{1}) be a comparison tetrahedron for (0, F, G_{0}, G_{1}) in the Euclidean space ℝ^{3}, by which we mean that the lengths of the edges containing 0 and the angles between them are the same for both tetrahedron. Then the lengths of the other three edges are also the same for both tetrahedron, since every triangle (0, X , Y) in F^{ℝ}(G) is flat by theorem 85. In particular, (f, g_{0}, g_{1}) is a comparison triangle for (F, G_{0}, G_{1}), thus d_{τ} (F, G_{t}) ≤ d(f, g_{t}) where g_{t} = tg_{1} + (1 − t)g_{0} in ℝ^{3} by the previous corollary. Since ∥G_{t}∥ = ∥g_{t}∥ (because (0, G_{0}, G_{1}) is flat), it follows that

⟨F, G_{t}⟩_{τ} ≥ ⟨f, g_{t}⟩ = t ⟨f, g_{1}⟩ + (1 − t) ⟨f, g_{0}⟩ = t ⟨F, G_{1}⟩_{τ} + (1 − t) ⟨F, G_{0}⟩_{τ} .

Similarly, we find that

|⟨F, G_{1}⟩_{τ} − ⟨F, G_{0}⟩_{τ} | = |⟨f, g_{1}⟩ − ⟨f, g_{0}⟩| ≤ ∥f∥ ∥g_{1} − g_{0}∥ = ∥F∥_{τ} · d_{τ} (G_{0}, G_{1})

thus G ↦ ⟨F, G⟩_{τ} is indeed ∥F∥_{τ} -Lipschitzian. □

Corollary 92. For any F, G, H ∈ F^{ℝ}(G),

⟨F, G + H⟩_{τ} ≥ ⟨F, G⟩_{τ} + ⟨F, H⟩_{τ} .

Proof. Apply the previous lemma to the middle point G + H of [2G, 2H]. □

Remark 93. We could pursue here with many further corollaries, but our knowledgeable readers will recognize that already with corollary 86, we have established that F^{ℝ}(G), together with its collections of apartments and facets (and the function d_{τ} for some choice of a faithful τ), is a (discrete) Euclidean building in the sense of [40, 6.1]. It is the vectorial (Tits) building defined in [40, 10.6]. But the construction given there singles out a pair Z_{G}(S) ⊂ B and uses more of the finest results from [5]: F^{ℝ}(G) is the building associated to the saturated Tits system (G, B, N) = (G, B, N_{G}(S))(k). By contrast, we may retrieve some of the results of [5] using the strongly transitive and strongly type-preserving action of G on our globally constructed building F^{ℝ}(G), for instance the fact that (G, B, N) is indeed a saturated Tits system [40, 8.6]. The main advantage of our construction is however that it is plainly functorial in G and k.

Remark 94. Corollary 87 also immediately follows from proposition 72 together with [2, VIB 1.4.2].

4.2.11. If τ′ is another faithful representation of G, the distances d_{τ′} and d_{τ} are equivalent. One checks it first on a fixed apartment A, thus obtaining constants c, C > 0 such that cd_{τ} (x, y) ≤ d_{τ′}(x, y) ≤ Cd_{τ} (x, y) for x, y ∈ A. Then this holds true for every x, y ∈ F^{Γ}(G), since any such pair is G-conjugated to one in A. We thus obtain a canonical metrizable G-invariant topology on F^{Γ}(G). The G-invariant functions of section 4.2.2 are continuous with respect to the canonical topology. The apartments and the “closed facets” of section 4.1.11 are topologically closed, being complete for the induced metrics. The canonical topology on C^{Γ}(G) is the quotient topology of the canonical topology on G^{Γ}(G), it is compatible with the monoid structure on C^{Γ}(G), the sections defined by the “closed chambers” are homeomorphisms and the functions defined in section 4.2.5 are continuous.

4.2.12. Suppose now that our local ring O is an integral domain with fraction field K and residue field k, giving rise to morphisms of cartesian squares

[figure: F^{Γ}(G_{K}); t; F; F^{Γ}(G); t; F; F^{Γ}(G_{k}); t; F; C^{Γ}(G_{K}); F; C^{Γ}(G); F; C^{Γ}(G_{k}); F; P(G_{K}); t; P(G); t; P(G_{k}); t; O(G_{K}); O(G); O(G_{k})]

We write x ↦ x_{K} for the generization maps, x ↦ x_{k} for the specialization maps.

Proposition 95. For any faithful τ ∈ Rep^{∘}(G)(O) and x, y ∈ F^{Γ}(G),

⟨x_{k}, y_{k}⟩_{τ_{k}} ≥ ⟨x_{K}, y_{K}⟩_{τ_{K}}
∡_{τ_{k}} (x_{k}, y_{k}) ≤ ∡_{τ_{K}} (x_{K}, y_{K})
d_{τ_{k}} (x_{k}, y_{k}) ≤ d_{τ_{K}} (x_{K}, y_{K})
∥x_{k}∥_{τ_{k}} = ∥x_{K}∥_{τ_{K}}

Proof. We may assume that Γ = ℝ. For (x, y) ∈ Std^{ℝ}(G), one checks easily that all of the above inequalities are in fact equalities. In particular,

∥x_{k}∥_{τ_{k}} = ∥x_{K}∥_{τ_{K}}

for all x ∈ F^{Γ}(G). For an arbitrary pair (x, y) in F^{ℝ}(G), the facet decomposition of F^{ℝ}(G) induces a decomposition of the segment [x, y] = ∪_{i=0}^{n−1}[x_{i}, x_{i+1}] as in the proof of corollary 88, with (x_{i}, x_{i+1}) ∈ Std^{ℝ}(G) for every i. Thus

d_{τ_{K}} (x_{K}, y_{K}) = ∑_{i=0}^{n−1}d_{τ_{K}} (x_{i,K}, x_{i+1,K}) = ∑_{i=0}^{n−1}d_{τ_{k}}(x_{i,k}, x_{i+1,k}) ≥ d_{τ_{k}}(x_{k}, y_{k})

and the other two inequalities easily follow. □

Remark 96. On the other hand for any x, y ∈ C^{Γ}(G),

\begin{aligned}⟨x_{k}, y_{k}⟩_{τ_{k}}^{os} & = & ⟨x_{K}, y_{K}⟩_{τ_{K}}^{os} & and & ∡_{τ_{k}}^{os}(x_{k}, y_{k}) & = & ∡_{τ_{K}}^{os} (x_{K}, y_{K}), \\ ⟨x_{k}, y_{k}⟩_{τ_{k}}^{tr} & = & ⟨x_{K}, y_{K}⟩_{τ_{K}}^{tr} & and & ∡_{τ_{k}}^{tr}(x_{k}, y_{k}) & = & ∡_{τ_{K}}^{tr} (x_{K}, y_{K}). \end{aligned}

However, it does happen that D_{τ_{k}}(x_{k}, y_{k}) ≠ D_{τ_{K}} (x_{K}, y_{K}).

CHAPTER 5 Affine F(G)-buildings

Let G be a reductive group over a field K. From now on, Γ = (ℝ, +, ≤) and we drop it from our notations. We also fix a faithful finite dimensional representation τ of G and drop it from the notations of section 4.2.2. We use sans-serif fonts to denote the set of K-valued points of a K-scheme, as in G = G(K), P = P (K) etc…

5.1. The dominance order

5.1.1. Since Γ = ℝ is divisible, the weak and strong partial dominance order on ℂ^{Γ}(G) agree. We denote by ≤ the induced partial order on the commutative monoid C(G) = ℂ(G)(K) or its submonoid C(G) = t(F(G)). It is compatible with the monoid structure on C(G) and related to the decomposition

C(G) = C^{r}(G) × G(Z) with C^{r}(G) = ℂ(G)^{r}(K) and G(Z) = ℂ(G)^{c}(K)

of section 2.2.13 as follows: for x = (x^{r}, x^{c}) and y = (y^{r}, y^{c}) in C^{r}(G) × G(Z),

x ≤ y ⟺ x^{r} ≤ y^{r} and x^{c} = y^{c}.

The poset (C(G), ≤) is a lattice and G(Z) ⊂ C(G) is its subset of minimal elements.

5.1.2. Choose a minimal pair (S, B) in SP(G), giving rise to the relative based root data ℛ_{+} = (M, R, M^{∗}, R^{∗}; R_{+}) with Weyl group W_{G}(S) = W_{G}(S)(K), and to the partial dominance order ≤ on Hom^{+}(M, ℝ) as defined in section 2.4.13. Let also s : C(G) ↪ F(S) be the corresponding section of t : G(G) ↠ C(G), whose image C = s(C(G)) equals Hom^{+}(M, ℝ) inside Hom(M, ℝ) = F(S) by section 4.1.10. Let finally C^{∗} be the dual cone of C in F(S) with respect to the scalar product ⟨−, −⟩ on F(S) which is attached to our chosen τ, so that

C^{∗} = {t ∈ F(S) : ∀c ∈ C, ⟨t, c⟩ ≥ 0} .

Then for every x, y ∈ C(G),

x ≤ y ⟺ s(x) ≤ s(y) in Hom^{+}(M, ℝ),
⟺ s(x) belongs to the convex hull of W_{G}(S) · s(y),
⟺ s(y) − s(x) belongs to the dual cone C^{∗},
⟺ ∀z ∈ C(G) : ⟨x, z⟩^{os} ≤ ⟨y, z⟩^{os} ,
⟺ ∀z ∈ C(G) : ⟨x, z⟩^{tr} ≥ ⟨y, z⟩^{tr} .

The first equivalence follows from Proposition 31, the second from section 2.4.11, the third one from [4, 12.14] and the last two from the formulas of section 4.2.7. The equivalence of the first and third line on the right is actually a tautology, since in fact C^{∗} = ℝ_{+}R_{+}^{∗} in F(S) = Hom(M, ℝ). Indeed for α ∈ R, let α_{v} be the unique element of F(S) such that ⟨x, α_{v}⟩ = x(α) for every x ∈ F(S). Then s_{α} is the orthogonal reflection of F(S) with respect to the hyperplane α_{v}^{⊥}, thus α^{∗} = \frac{α_{v}}{⟨α_{v},α_{v}⟩} in F(S). Since C = {x ∈ F(S) : ∀α ∈ R_{+}, ⟨x, α_{v}⟩ ≥ 0}, its dual cone C^{∗} is spanned by the α_{v}’s for α ∈ R_{+} and thus C^{∗} = ℝ_{+}R_{+}^{∗}.

5.1.3. For every x, y ∈ C(G), we have

x ≤ y ⟹ ∥x∥^{2} ≤ ⟨x, y⟩^{os} ≤ ∥y∥^{2}
x = y ⟺ x ≤ y and ∥x∥ = ∥y∥ .

Indeed the first implication follows from the equivalence

x ≤ y ⟺ ∀z ∈ C(G) : ⟨x, z⟩^{os} ≤ ⟨y, z⟩^{os}

with z = x or y, and with s as above, it says that

x ≤ y ⟹ ∥s(x)∥^{2} ≤ ∥s(x)∥ ∥s(y)∥ cos ∡(s(x), s(y)) ≤ ∥s(y)∥^{2} .

Thus if x ≤ y and ∥x∥ = ∥y∥, s(x) = s(y) and x = y.

5.1.4. The next proposition slightly refines proposition 24.

Proposition 97. For every F, G ∈ F(G), we have

t(F + G) ≤ t(F) + t(G) in (C(G), ≤)

with equality if and only if F, G ∈ C for some closed chamber C of F(G).

Proof. With notations as above, we may choose (S, B) such that F, G ∈ F(S) with F + G ∈ C, C = s(C(G)). Set F′ = s ∘ t(F) and G′ = s ∘ t(G), so that

F + G = s (t(F + G)) and F′ + G′ = s (t(F) + t(G)) .

The (acute) dual cone C^{∗} defines a partial order ≤ on F(S), given by

x ≤ y ⟺ y − x ∈ C^{∗}.

Since F′ ∈ (W_{G}(S) · F) ∩ C and G′ ∈ (W_{G}(S) · G) ∩ C, we have

F ≤ F′ and G ≤ G′

by [7, VI §1 Proposition 18] (or lemma 29). Thus F + G ≤ F′ + G′ with equality if and only if F = F′ and G = G′, i.e. F and G belong to C. □

5.1.5. The above inequality can also be established and somehow refined as follows. For every z ∈ C(G), there is an H ∈ F(G) with t(H) = z such that H and F + G are in (relative) transverse position, see 4.2.5. For any such H, 4.2.7, lemma 82 and corollary 92 together imply that

⟨t(F) + t(G), z⟩^{tr} = ⟨t(F), z⟩^{tr} + ⟨t(G), z⟩^{tr}
≤ ⟨F, H⟩ + ⟨G, H⟩
≤ ⟨F + G, H⟩
= ⟨t(F + G), z⟩^{tr} .

Thus indeed t(F + G) ≤ t(F) + t(G) in (C(G), ≤).

5.2. AFFINE F(G)-SPACES AND BUILDINGS 92

5.2. Affine F(G)-spaces and buildings

5.2.1. Affine F(G)-spaces interact with the vectorial Tits building F(G) in the same way as affine spaces do with their underlying vector space.

Definition 98. An affine F(G)-space is a set X(G) equipped with: • a left action G × X(G) → X(G), written (g, x) ↦ g · x or gx,

• a G-equivariant pull map X(G) × F(G) → X(G), written (x, F) ↦ x + F,

• a G-equivariant apartment map S(G) → P(X(G)), written S ↦ X(S), such that for (one or) every S ∈ S(G), the pull map sends X(S) × F(S) to X(S) and induces a structure of affine G(S)-space (in the usual sense) on X(S).

5.2.2. The group N_{G}(S) thus acts on X(S) by affine morphisms, the vectorial part of this action equals ν_{S}^{v} : N_{G}(S) ↠ W_{G}(S) ⊂ Aut(G(S)) and the kernel Z_{G}(S) of ν_{S}^{v} acts on X(S) by translations, through a W_{G}(S)-equivariant morphism

ν_{X,S} : Z_{G}(S) → G(S).

For any other S′ ∈ S(G), there is commutative diagram

\xymatrix{Z_{G}(S) \ar[r]^{ν_{X,S}} \ar[d]_{≃} & G(S) \ar[d]^{≃} \\ Z_{G}(S′) \ar[r]^{ν_{X,S′}} & G(S′)}

where the vertical maps are induced by Int(g) for any g ∈ G with Int(g)(S) = S′. Set W_{G} = \underleftarrow{lim}W_{G}(S). The type of X(G) is the W_{G}-equivariant morphism

ν_{X} = \underleftarrow{lim}ν_{X,S} : \underleftarrow{lim}Z_{G}(S) → \underleftarrow{lim}G(S)

which is obtained from these diagrams by taking the limits over all S ∈ S(G). We say that X(G) is discrete when the image of ν_{X} is a discrete subgroup of the real vector space \underleftarrow{lim}G(S). Equivalently: X(G) is discrete when the image of ν_{X,S} is a discrete subgroup of G(S) for one or every S ∈ S(G).

5.2.3. Affine F(G)-buildings are affine F(G)-spaces satisfying a long list of axioms, which shall be gradually introduced below. The following definition picks up an (hopefully minimal) subset of these axioms, from which all others will be derived in due time, along with various properties.

Definition 99. An affine F(G)-building is an affine F(G)-space which satisfies the axioms L(s), R(s), R(i), C^{∘}, NE, UN, CO and UG listed below.

5.2.4. Example. The Tits building F(G) itself, equipped with its left action of G, the addition map of section 2.3.2 and the apartment map of section 4.1.13 is a discrete affine F(G)-space with trivial type ν_{F} = 0. We will see that it satisfies all of the required axioms, thus F(G) = (F(G), +, F(−)) is an affine F(G)-building.

5.2.5. Many apartments. An affine F(G)-building X(G) satisfies L(s) For every x ∈ X(G) and F ∈ F(G),

S(x, F) = {S ∈ S(G) : x ∈ X(S) and F ∈ F(S)}

is not empty.

R(s) For every x, y ∈ X(G),

S(x, y) = {S ∈ S(G) : x, y ∈ X(S)}

is not empty. T (s) For every x, y ∈ X(G),

F(x, y) = {F ∈ F(G) : y = x + F}

is not empty. Note that R(s) implies T (s) while L(s) is equivalent to L′(s) For every x ∈ X(G) and every closed chamber C of F(G),

S(x, C) = {S ∈ S(G) : x ∈ X(S) and C ⊂ F(S)} is not empty.

This in turn implies that the pull map is well-behaved: AC (Action) For every closed chamber C of F(G), the map

X(G) × C ↪ X(G) × F(G) ⟶^{+} X(G)

defines an action of the commutative monoid (C, +) on X(G). Thus for any x ∈ X(G) and F, G ∈ F(G), x + 0 = x and

(x + F) + G = x + (F + G) = (x + G) + F

if P_{F} and P_{G} are in osculatory position. In particular,

(x + λF) + μF = x + (λ + μ)F

for every λ, μ ≥ 0 and F ∈ F(G).

5.2.6. Strong transitivity. An affine F(G)-building X(G) satisfies L(i) For every x ∈ X(G) and F ∈ F(G),

G_{x,F} = {g ∈ G : gx = x and gF = F}

acts transitively on S(x, F). R(i) For every x, y ∈ X(G),

G_{x,y} = {g ∈ G : gx = x and gy = y}

acts transitively on S(x, y). T (i) For every x, y ∈ X(G), G_{x,y} acts transitively on F(x, y).

5.2.7. The labels of the L, R, or T -axioms reflect their equivalence with the surjectivity or injectivity of the relevant maps in the commutative diagram

\xymatrix{G\ (X(G) × F(G)) \ar[r]^{T} & G\ (X(G) × X(G)) \\ N_{G}(S)\ (X(S) × F(S)) \ar[u]^{L} \ar[r] & N_{G}(S)\ (X(S) × X(S)) \ar[u]_{R}}

which is induced by the equivariant commutative diagram

\xymatrix{X(G) × F(G) \ar[r]^{(x,F)↦(x,x+F)} & X(G) × X(G) \\ X(S) × F(S) \ar@{^{(}->}[u] \ar[r] & X(S) × X(S) \ar@{^{(}->}[u]}

The bottom map in each diagram is always bijective since X(S) is an affine F(S)- space. Thus R(i) ⇒ T (i), and R(i) + L(s) + T (s) imply all of the above axioms.

5.2.8. The vectorial distance. It follows from the axioms already introduced that for an affine F(G)-building X(G), there is a unique G-invariant map

d : X(G) × X(G) → C(G)

such that for every x ∈ X(G) and F ∈ F(G),

d(x, x + F) = t(F) in C(G).

The following properties are easily established: for x, y ∈ X(G),

d(y, x) = d(x, y)^{ι} and d(x, y) = 0 ⟺ x = y.

Moreover for x ∈ X(G), F ∈ F(G) and 0 ≤ λ ≤ λ′,

d(x + λF, x + λ′F) = (λ′ − λ) · t(F).

This vectorial distance d may also satisfy the following properties – and it does for affine F(G)-buildings, by lemma 100 and proposition 101 below: ⋆ For every x, y ∈ X(G) and F, G ∈ F(G),

d(x + F, y + G) ≤ d(x, y) + d(F, G) in C(G).

T R (Triangle inequality) For every x, y, z ∈ X(G),

d(x, z) ≤ d(x, y) + d(y, z) in C(G).

T R′ For every y ∈ X(G) and F, G ∈ F(G),

d(y + F, y + G) ≤ d(F, G) in C(G).

NE (Non expanding) For every x, y ∈ X(G) and F ∈ F(G),

d(x + F, y + F) ≤ d(x, y) in C(G).

C^{∘} (Continuity) For every sequences (x_{n}), (y_{n}) and points x, y in X(G),

( \begin{matrix}d(x_{n}, x) → 0 \\ d(y_{n}, y) → 0 \end{matrix} in C(G)) ⟹ (d(x_{n}, y_{n}) → d(x, y) in C(G)) .

Note that ⋆ and T R′ also involve the vectorial distance d for F(G) – it follows from 4.2.10 that the affine F(G)-space F(G) indeed satisfies the required axioms for the existence of d: L(s) = R(s) is theorem 85 and L(i) = R(i) is its corollary 86.

Lemma 100. The above properties of d are related as follows:

⋆ ⟺ T R + T R′ + NE and T R ⟺ T R′ ⟹ C^{∘}.

Proof. (T R + T R′ + NE ⇒ ⋆). For x, y ∈ X(G) and F, G ∈ F(G), we find

d(x + F, y + G) ≤ d(x + F, y + F) + d(y + F, y + G) ≤ d(x, y) + d(F, G)

using T R for the first inequality, NE and T R′ for the second.

(⋆ ⇒ T R + T R′ + NE). Taking x = y (resp. F = G) in ⋆ yields T R′ (resp. NE). Taking F = 0 and G ∈ F(y, z) (using T (s)) yields T R.

(T R ⇒ T R′). For x ∈ X(G), F, G ∈ F(G) and λ ∈ [0, 1], set

F(λ) = (1 − λ)F + λG in F(G) and x(λ) = x + F(λ) in X(G).

Pick S ∈ S(G) with F, G ∈ F(S). There is a subdivision 0 = λ_{0} < ⋯ < λ_{n} = 1 of [0, 1] and for each i ∈ {1, ⋯, n}, a closed chamber C_{i} of F(S) such that F(λ) ∈ C_{i} for all λ ∈ [λ_{i−1}, λ_{i}]. By L′(s), there is an S_{i} ∈ S(G) such that x ∈ X(S_{i}) and C_{i} ⊂ F(S_{i}), thus also x(λ) ∈ X(S_{i}) for every λ ∈ [λ_{i−1}, λ_{i}]. Then

d(x + F, x + G) ≤ ∑_{i=1}^{n}d(x(λ_{i−1}), x(λ_{i})) = ∑_{i=1}^{n}d(F(λ_{i−1}), F(λ_{i})) = d(F, G)

using respectively T R in X(G) and trivial computations in X(S_{i}) and F(G). (T R′ ⇒ T R). For x, y, z ∈ X(G), pick F, G ∈ F(G) with

x = y + F and z = y + G

using T (s). Choose S ∈ S(G) such that F, G ∈ F(S), and set F′ = ι_{S}F. Then

d(x, z) ≤ d(F, G) = t(F′ + G) ≤ t(F′) + t(G) = d(x, y) + d(y, z)

using respectively T R′ in X(G), proposition 97 and

t(F′) = t(F)^{ι} = d(y, x)^{ι} = d(x, y), t(G) = d(y, z).

(T R ⇒ C^{∘}). Suppose that d(x_{n}, x) → 0 and d(y_{n}, y) → 0 for sequences (x_{n}), (y_{n}) and points x, y in X(G). Then also d(x, x_{n}) → 0 and d(y, y_{n}) → 0 in C(G). Let c be a limit point of d(x_{n}, y_{n}) in the Alexandrov compactification C(G) ∪ {∞} of the locally compact space C(G). We have to show that c = d(x, y), for then d(x_{n}, y_{n}) → d(x, y) in C(G). By the triangle inequality T R,

d(x, y) ≤ d(x, x_{n}) + d(x_{n}, y_{n}) + d(y_{n}, y)
and d(x_{n}, y_{n}) ≤ d(x_{n}, x) + d(x, y) + d(y, y_{n})

for every n ≥ 0, thus c ∈ C(G) and d(x, y) ≤ c ≤ d(x, y), i.e. c = d(x, y). □

5.2.9. The classical distance. These axioms imply that the composition

d : X(G) × X(G) → ℝ_{+}, x ↦ ∥d(x)∥

of the vectorial distance d with the norm ∥−∥ : C(G) → ℝ_{+} attached to our chosen τ is a genuine G-invariant distance on X(G). Its restriction to any appartment is Euclidean and (X(G), d) is a geodesic space: for x, y ∈ X(G) and any apartment X(S) containing x and y, the unique geodesic from x to y in X(S) is a geodesic from x to y in X(G). For any sequence (x_{n}) in X(G) and x in X(G), we have

x_{n} → x in (X(G), d) ⟺ d(x_{n}, x) → 0 in ℝ_{+} ⟺ d(x_{n}, x) → 0 in C(G).

The induced metrizable topology on X(G) thus does not depend upon τ (see also 4.2.11). We call it the canonical topology of X(G). The apartments are closed, being complete for the induced metric. The vectorial distance and pull map

d : X(G) × X(G) → C(G) and + : X(G) × F(G) → X(G)

are continuous for the canonical topologies on X(G), C(G) and F(G) by C^{∘} and ⋆.

5.2.10. The retractions. For an affine F(G)-building, we also require: UN (Unipotent) For every x ∈ X(G), F ∈ F(G) and u ∈ U_{F},

lim_{s→+∞} d(x + sF, ux + sF) = 0.

For F ∈ F(G) and any Levi subgroup L of P_{F}, we denote by F_{L}^{ι} the unique filtration opposed to F with P_{F} ∩ P_{F_{L}^{ι}} = L. Thus F_{L}^{ι} = Fil(ιG) where G ∈ G(G) is the unique splitting of F with L_{G} = L. We have F_{L}^{ι} = ι_{S}F for any S ∈ S(L).

Proposition 101. Let X(G) be an affine F(G)-space satisfying the axioms of sections 5.2.5 and 5.2.6, together with C^{∘}, NE and UN. Then it also satisfy T R. Moreover, for every parabolic subgroup P of G with Levi decomposition P = U ⋊ L,

X(L) = ∪_{S∈S(L)}X(S)

is a fundamental domain for the action of U on X(G) and the induced retraction

r_{P,L} : X(G) ↠ X(L)

is non-expanding for d: for every x, y ∈ X(G),

d(r_{P,L}(x), r_{P,L}(y)) ≤ d(x, y) in C(G).

Finally, for any F ∈ F^{−1}(P), if F′ = F_{L}^{ι}, then for all x ∈ X(G),

r_{P,L}(x) = lim_{s→∞} (x + sF) + sF′ in (X(G), d) .

Proof. Fix P, L, F and F′ = F_{L}^{ι} as above. For any x ∈ X(G), there is by L(s) an S′ ∈ S(G) such that x ∈ X(S′) and F ∈ F(S′), i.e. Z_{G}(S′) ⊂ P_{F}. Let L′ be the unique levi subgroup of P_{F} containing Z_{G}(S′) and let u be the unique element of U such that Int(u)(L′) = L. Then S = Int(u)(S′) belongs to S(L) and ux belongs to X(S) ⊂ X(L), thus U · X(L) = X(G). For s ≥ 0 and x ∈ X(G), set

r_{s}(x) = (x + sF) + sF′ in X(G).

Then x ↦ r_{s}(x) is non-expanding for d by NE and for any u ∈ U,

lim_{s→∞} d(r_{s}(x), r_{s}(u · x)) = 0 in C(G)

by UN and NE. If x belongs to X(L), say x ∈ X(S) for some S ∈ S(L), then F, F′ ∈ F(S) with F + F′ = 0 in F(S), thus r_{s}(x) = x for all s ≥ 0 since X(S) is an affine F(S)-space. If x and u · x belong to X(L), d(x, u · x) = d(r_{s}(x), r_{s}(u · x)) for all s ≥ 0, thus d(x, u · x) = 0 and x = u · x. In particular, X(L) is indeed a fundamental domain for the action of U on X(G). Let r : X(G) ↠ X(L) be the corresponding retraction. For x ∈ X(G), pick u ∈ U such that r(x) = u · x. Then

d(r_{s}(x), r(x)) = d(r_{s}(x), u · x) = d(r_{s}(x), r_{s}(u · x)) → 0.

Applying this to x, y ∈ X(G) and using C^{∘}, we find that

lim_{s→∞} d(r_{s}(x), r_{s}(y)) = d(r(x), r(y)),

thus d(r(x), r(y)) ≤ d(x, y) since d(r_{s}(x), r_{s}(y)) ≤ d(x, y) for all s ≥ 0.

Turning now to the proof of T R, first note that by proposition 97, the triangle inequality holds whenever x, y, z belong to X(S) for some S ∈ S(G). For a general triple x, y, z in X(G), choose S ∈ S(G) with x, z ∈ X(S) using R(s), pick a minimal parabolic subgroup B of G with Levi subgroup L = Z_{G}(S) and let r : X(G) ↠ X(S) be the corresponding retraction. Then r(x) = x and r(z) = z, thus indeed

d(x, z) = d(r(x), r(z)) ≤ d(r(x), r(y)) + d(r(y), r(z)) ≤ d(x, y) + d(y, z)

since the triangle inequality holds on X(S) and r is non-expanding for d. □

Corollary 102. The apartment map S ↦ X(S) is then uniquely determined by the pull map + : X(G) × F(G) → X(G): for every S ∈ S(G),

X(S) = {x ∈ X(G) : ∀F, F′ ∈ F(S), (x + F) + F′ = x + (F + F′)} .

Proof. Let X′(S) be the right hand side. Plainly, X(S) ⊂ X′(S). Conversely, pick a minimal parabolic subgroup B of G with Levi Z_{G}(S), let r : X(G) ↠ X(S) be the corresponding retraction, choose F ∈ F^{−1}(B) and set F′ = ι_{S}F. For x in X′(S), (x + λF) + λF′ = x for all λ ≥ 0, thus r(x) = x belongs to X(S). □

5.2.11. Standard geodesics. For x ∈ X(G) and F ∈ F(G), the function

[0, 1] → X(G) (resp. ℝ_{+} → X(G)) t ↦ x + tF

is a geodesic segment (resp. geodesic ray) in (X(G), d). We refer to these geodesics as the standard ones. Thus a geodesic (segment or ray) is standard precisely when it is contained in some apartment, and the set of all standard geodesics does not depend upon the choice of τ. If (X(G), d) is uniquely geodesic, then every geodesic segment is standard, but there might still be some non-standard geodesic rays.

5.2.12. Convexity. An affine F(G)-building satisfies all of the above axioms, together with the following convexity axiom: CO^{+} For every pair of geodesics x, y : [0, 1] → X(G) in (X(G), d), the function

f : [0, 1] → C(G), f(t) = d(x(t), y(t))
is convex, i.e. for every λ and t_{1} ≤ t_{2} in [0, 1],
f ((1 − λ)t_{1} + λt_{2}) ≤ (1 − λ)f(t_{1}) + λf(t_{2}) in C(G).

This implies that the metric space (X(G), d) itself is convex in the sense of [8, II.1.3]. In particular, it is uniquely geodesic and for every x ∈ X(G) and F, G ∈ F(G),

x + F = x + G ⟹ ∀t ∈ [0, 1] : x + tF = x + tG.

Proposition 103. Let X(G) be an affine F(G)-building. Let (P, P′) be a pair of opposed parabolic subgroups of G with common Levi subgroup L = P ∩ P′. Let

r, r′ : X(G) ↠ X(L)

be the corresponding retractions, as in proposition 101. Then

X(L) = {x ∈ X(G) : r(x) = r′(x)} .

Proof. For x ∈ X(L), r(x) = x = r′(x), thus x belongs to

X′(L) = {x ∈ X(G) : r(x) = r′(x)} .

Suppose conversely that x ∈ X′(L) and set y = r(x) = r′(x). Pick a pair of opposed filtrations (F, F′) with P_{F} = P, P_{F′} = P′ and ∥F∥ = ∥F′∥ = 1. For t ∈ ℝ, set

X(t) = \begin{cases}x + |t| F & if t ≥ 0, \\ x + |t| F′ & if t ≤ 0, \end{cases} and Y (t) = \begin{cases}y + |t| F & if t ≥ 0, \\ y + |t| F′ & if t ≤ 0. \end{cases}

Plainly, Y : ℝ → X(G) is a geodesic line and d(X(t), Y (t)) → 0 when |t| → ∞. Note that for any 0 ≤ t_{1}, t_{2} ≤ t, d (Y (−t), Y (t)) = 2t is not greater than

d (Y (−t), X(−t)) + d (X(−t), X(−t_{1})) + d (X(−t_{1}), X(t_{2})) + d (X(t_{2}), X(t)) + d (X(t), Y (t)) .

The second and fourth term sum to 2t − (t_{1} + t_{2}), thus t_{1} + t_{2} ≤ d (X(−t_{1}), X(t_{2})). Since also d (X(−t_{1}), X(t_{2})) ≤ t_{1} + t_{2}, it follows that X : ℝ → X(G) is a geodesic line as well. Since the metric d is convex, the function t ↦ d(X(t), Y (t)) is convex. Since it is also bounded, it must be constant, thus actually trivial. In particular, d(x, y) = d(X(0), Y (0)) = 0, therefore x = y belongs to X(L). □

Definition 104. The enclosure of x, z ∈ X(G) is defined by

♢(x, z) = {y ∈ X(G) : d(x, z) = d(x, y) + d(y, z)}
= {y ∈ X(G) : d(x, z) ≥ d(x, y) + d(y, z)}

Corollary 105. For any S ∈ S(G) and x, x′ ∈ X(S), let F and F′ be the pair of opposed facets in F(S) such that x′ ∈ x + F and x ∈ x′ + F′. Then

♢(x, x′) = (x + \overline{F}) ∩ (x′ + \overline{F}′).

In particular for any x, z ∈ X(G), the enclosure ♢(x, z) is a closed and convex subset of X(G) which is contained in any apartment containing x and z.

Proof. For y ∈ X(S), write y = x + a and x′ = y + b with a, b ∈ F(S), so that

d(x, x′) = t(a + b), d(x, y) = t(a) and d(y, x′) = t(b).

Thus y belongs to ♢(x, x′) if and only if there exists a closed chamber C in F(S) containing a and b by proposition 97, which occurs precisely when a and b both belong to the closure \overline{F} of the facet F of F(S) which contains c = a + b. Hence

♢(x, x′) ∩ X(S) = (x + \overline{F}) ∩ (x′ + \overline{F}′).

In particular, the function y ↦ d(x, y) is injective on ♢(x, x′) ∩ X(S). Now pick a pair of opposed minimal parabolic subgroups (B, B′) of G with B ∩ B′ = Z_{G}(S). Let r, r′ : X(G) ↠ X(S) be the corresponding retractions. For any y ∈ ♢(x, x′),

d(x, r(y)) = d(r(x), r(y)) ≤ d(x, y) and d(r(y), x′) = d(r(y), r(x′)) ≤ d(y, x′)

since r is non-expanding for d, therefore

d(x, x′) ≤ d(x, r(y)) + d(r(y), x′) ≤ d(x, y) + d(y, x′) = d(x, x′).

Thus r(y) belongs to ♢(x, x′) ∩ X(S) and d(x, r(y)) = d(x, y). Since the same conclusion holds for r′(y), we obtain r(y) = r′(y). Hence y belongs to X(S) and indeed ♢(x, x′) = (x + \overline{F}) ∩ (x′ + \overline{F}′). The remaining assertions easily follow. □

5.2.13. Unique Geodesics. It may seem that the validity of the axiom CO^{+} for a given affine F(G)-space X(G) depends upon the chosen τ, but it does not. In fact, CO^{+} is plainly equivalent to the conjonction of the following two axioms: CO For any pair of standard geodesics x, y : [0, 1] → X(G) in X(G), the function

f : [0, 1] → C(G), f(t) = d(x(t), y(t))
is convex, i.e. for every λ and t_{1} ≤ t_{2} in [0, 1],
f ((1 − λ)t_{1} + λt_{2}) ≤ (1 − λ)f(t_{1}) + λf(t_{2}) in C(G).

UG The metric space (X(G), d) is uniquely geodesic. Now CO plainly does not depend upon the choice of τ, and UG also does not. Indeed, suppose that X(G) satisfies all of the above axioms (using τ in UG) and let τ′ be another faithful representation of G, giving rise to a distance d′ on X(G). We have to show that every geodesic segment c : [0, 1] → X(G) in (X(G), d′) is standard, for then CO implies UG for (X(G), d′). Now for all t ∈ [0, 1], we have

d(c(0), c(1)) ≤ d(c(0), c(t)) + d(c(t), c(1)) in C(G)
and d′(c(0), c(1)) = d′(c(0), c(t)) + d′(c(t), c(1)) in ℝ_{+}

from which easily follows that actually

d(c(0), c(1)) = d(c(0), c(t)) + d(c(t), c(1)) in C(G).

Thus c(t) belongs to ♢(c(0), c(1)) for all t ∈ [0, 1] and c is standard, being indeed contained in any apartment which contains c(0) and c(1) by corollary 105.

5.2.14. By the usual dyadic, reparametrization and triangulation tricks, it is sufficient to test the inequalities in CO or CO^{+} for (t_{1}, t_{2}, λ) = (0, 1, \frac{1}{2}), for pairs of geodesics issuing from the same point. Thus CO is equivalent to either one of

CO′ For every x ∈ X(G), F, G ∈ F(G) and λ ∈ [0, 1],

d(x + λF, x + λG) ≤ λd(x + F, x + G) in C(G).

CO′′ For every x ∈ X(G) and F, G ∈ F(G),

d(x + \frac{1}{2}F, y + \frac{1}{2}G) ≤ \frac{1}{2}d(x + F, x + G) in C(G).

5.2.15. There is a unique, G-equivariant and continuous map

X(G) × X(G) × [0, 1] → X(G), (x, y, λ) ↦ (1 − λ)x + λy

such that (1 − λ)x + λy = x + λF for any F ∈ F(x, y). We set

[x, y] = {(1 − λ)x + λy : λ ∈ [0, 1]}

and call it the segment between x and y. It is contained in the enclosure ♢(x, y), thus also contained in any apartment X(S) which contains x and y. In particular, the intersection of two apartments is a convex subset of both apartments. A subdivision of [x, y] is a finite collection x = x_{0}, ⋯, x_{n} = y of points in [x, y] such that

x_{i} = (1 − λ_{i})x + λ_{i}y, 0 ≤ λ_{0} ≤ ⋯ ≤ λ_{n} = 1.

Thus [x, y] = ∪_{i=1}^{n}[x_{i−1}, x_{i}] and

d(x, y) = ∑_{i=1}^{n}d(x_{i−1}, x_{i}).

5.2.16. For an affine F(G)-building X(G), we denote by

d^{r} : X(G) × X(G) → C^{r}(G) and d^{c} : X(G) × X(G) → G(Z)

the components of d. These are G-invariant functions. For x, y, z ∈ X(G),

d^{r}(x, z) ≤ d^{r}(x, y) + d^{r}(y, z) and d^{c}(x, z) = d^{c}(x, y) + d^{c}(y, z).

The function g ↦ d^{c}(x, gx) thus does not depend upon x and defines a morphism

ν_{X}^{c} : G → G(Z).

5.2.17. A morphism of affine F(G)-spaces f : X(G) → Y(G) is a G-equivariant map between the underlying sets which is compatible with their structure maps:

f(X(S)) ⊂ Y(S) and f(x + F) = f(x) + F

for every S ∈ S(G), x ∈ X(G) and F ∈ F(G). If Y(G) is an F(G)-building, it is sufficent to require the second condition. A morphism of affine F(G)-buildings is a morphism of the underlying affine F(G)-spaces. Any such morphism is an automorphism: it is bijective on any appartment, thus globally bijective by R(s). It is compatible with the d-maps, and an isometry of the underlying metric spaces.

5.2.18. An automorphism θ of an affine F(G)-building X(G) acts on the apartment X(S) by an N_{G}(S)-equivariant translation, which is thus given by a vector θ_{S} in G(Z) = G(S)^{W_{G}(S)}, where Z = Z(G). The G-equivariance of θ then implies that S ↦ θ_{S} is also G-equivariant, thus constant. It follows that

Aut(X(G)) = G(Z)

with G ∈ G(Z) acting on X(G) by x ↦ x + G.

5.2.19. For an affine F(G)-building X(G), we define

X^{r}(G) = X(G)/G(Z) and X^{e}(G) = X^{r}(G) × G(Z).

The group G acts: on the quotient X^{r}(G) of X(G), on G(Z) by translations through the morphism ν_{X}^{c} : G → G(Z), and on X^{e}(G) diagonally. Then, the formulas

X^{r}(S) = X(S)/G(Z) and X^{e}(S) = X^{r}(S) × G(Z)

yield G-equivariant maps X^{r} : S(G) → P(X^{r}(G)) and X^{e} : S(G) → P(X^{e}(G)), the pull map on X(G) descends to a G-equivariant map + : X^{r}(G) × F^{r}(G) → X^{r}(G), which together with the addition map on G(Z) yields a G-equivariant map

+ : X^{e}(G) × F(G) → X^{e}(G) ([x], θ) + F = ([x] + F^{r}, θ + F^{c}).

The resulting triple X^{e}(G) is yet another affine F(G)-building, with ν_{X^{e}} = ν_{X}. In fact, any point x_{0} ∈ X(G) defines an isomorphism of affine F(G)-buildings

X(G) ≃ X^{e}(G) x ↦ ([x], d^{c}(x_{0}, x)).

Thus X^{e}(G) appears as a rigidified version of X(G): there are no non-trivial automorphisms of X^{e}(G) preserving the subspace X^{r}(G) ≃ X^{r}(G) × {0} of X^{e}(G). The decomposition X^{e}(G) = X^{r}(G) × G(Z) is orthogonal in the following sense:

∀(x, θ), (x′, θ′) ∈ X^{e}(G) : d((x, θ), (x′, θ′))^{2} = d(x, x′)^{2} + d(θ, θ′)^{2}.

This follows from the analogous result for F(G), see 4.2.4.

5.2.20. If X(G) = (X(G), X(−), +) is an affine F(G)-space or building, then so is X_{λ}(G) = (X(G), X(−), +_{λ}) for any λ > 0 in ℝ, where x +_{λ} F = x + λF. The types ν_{X} of X(G) and ν_{X_{λ}} of X_{λ}(G) are related by ν_{X} = λ · ν_{X_{λ}}.

5.3. Further axioms

Let X(G) be an affine F(G)-space.

5.3.1. The axiom L(s)^{+}. The following is a sharp strengthening of L(s): L(s)^{+} For any x ∈ X(G) and F, G ∈ F(G), there exists S ∈ S(G) and ϵ > 0 such

that F ∈ F(S) and x + λG ∈ X(S) for every λ ∈ [0, ϵ].

Proposition 106. If X(G) satisfies L(s)^{+}, R(s), R(i) and UN, then it is an affine F(G)-building and (X(G), d) is a CAT (0)-space.

Suppose that X(G) satisfies L(s)^{+}, R(s), R(i) and UN. Then it already satisfies all the axioms of section 5.2.5 and 5.2.6, giving rise to the vectorial distance d which is the subject of the remaining axioms. We do not yet know that d satisfies T R, thus d = ∥d∥ may not be a distance on X(G). But for any apartment X(S), the restriction of d to X(S) satisfies T R and d is a Euclidean distance on X(S).

Lemma 107. For x ∈ X(G) and F, G ∈ F(G), there exists S ∈ S(G), G^{∗} ∈ F(G) and ϵ > 0 such that x ∈ X(S), F, G^{∗} ∈ F(S) and

∀λ ∈ [0, ϵ] : x + λG = x + λG^{∗} ∈ X(S).

Proof. By L(s)^{+}, there exists S ∈ S(G) and ϵ > 0 such that F ∈ F(S) and x(λ) = x+λG ∈ X(S) for λ ∈ [0, ϵ]. For any 0 ≤ λ ≤ λ′, x(λ′) = x(λ)+(λ′ −λ)G by AC, thus d(x(λ), x(λ′)) = (λ′ − λ) · t(G) in C(G) and d(x(λ), x(λ′)) = (λ′ − λ) · ∥G∥ in ℝ_{+}. In particular, x(−) : [0, ϵ] → X(S) is a geodesic segment in (X(S), d|_{X(S)}). There is thus a unique G^{∗} ∈ F(S) such that x(λ) = x + λG^{∗} for λ ∈ [0, ϵ]. □

Lemma 108. For any x, y ∈ X(G) and G ∈ F(G), the function

c : ℝ_{+} → C(G), c(λ) = d(y, x + λG)

is continuous for the canonical topologies on ℝ_{+} and C(G).

Proof. Pick F ∈ F(G) with y = x + F using T (s). By the previous lemma, there exists S ∈ S(G), G^{∗} ∈ F(S) and ϵ > 0 such that x ∈ X(S), F ∈ F(S) and x + λG = x + λG^{∗} for λ ∈ [0, ϵ]. Since x, y ∈ X(S) and G^{∗} ∈ F(S), the function λ ↦ d(y, x + λG^{∗}) is plainly continuous on ℝ_{+}, thus c is continuous on [0, ϵ]. Changing x to x+λG, we find that c is right continous on ℝ_{+}. By L(s), there is an S′ ∈ S(G) with x ∈ X(S′), G ∈ F(S′). Set G′ = ι_{S}G. Then for λ′ ≥ λ ≥ 0,

x(λ) = x(λ′) + (λ′ − λ) · G′  in X(G),
thus c(λ) = d (y, x(λ′) + (λ′ − λ) · G′) in C(G).

It follows that c is also left continuous on ℝ_{+}. □

Lemma 109. For any x ∈ X(G) and F, G ∈ F(G),

x + F = x + G ⟹ ∀λ ∈ [0, 1] : x + λF = x + λG.

Proof. Suppose x + F = x + G, put x(λ) = x + λF, y(λ) = x + λG and define

λ_{0} = inf{1, λ ∈ [0, 1] such that x(λ) ≠ y(λ)}.

Suppose that λ_{0} ∈ [0, 1[. If λ_{0} ≠ 0, then since x(λ) = y(λ) for all λ ∈ [0, λ_{0}[,

d(x(λ_{0}), y(λ_{0})) = lim_{s→s_{0}^{−}} d(x(λ_{0}), y(λ)) = lim_{s→s_{0}^{−}} d(x(λ_{0}), x(λ)) = 0

by the previous lemma, thus x(λ_{0}) = y(λ_{0}). Changing (x, F, G) to

(x(λ_{0}), (1 − λ_{0})F, (1 − λ_{0})G) ,

we may assume that λ_{0} = 0. By lemma 107, there exists S ∈ S(G), G^{∗} ∈ F(S) and ϵ > 0 such that x ∈ X(S), F ∈ F(S) and y(λ) = x + λG^{∗} for λ ∈ [0, ϵ]. Since x + ϵG = x + ϵG^{∗} and x + G = x + F in X(G), t(G^{∗}) = t(G) = t(F) in C(G) with F, G^{∗} ∈ F(S), thus G^{∗} = wF for some w ∈ W_{G}(S). In the affine F(S)-space X(S),

x(1) = x + F = (x + λG^{∗}) + (F − λG^{∗}) = y(λ) + (F − λwF)

for all λ ∈ [0, ϵ]. Since x(1) = y(1), we thus find that for λ ∈ [0, ϵ],

t((1 − λ)F) = (1 − λ)t(G) = d(y(λ), y(1)) = d(y(λ), x(1)) = t(F − λwF).

Let C be a closed chamber in F(S) such that F − λwF ∈ C for all λ ∈ [0, ϵ] (shrinking ϵ if necessary). Since t is injective on C, (1 − λ)F = F − λwF in C ⊂ F(S) for all λ ∈ [0, ϵ], thus F = wF = G^{∗}. But then x(λ) = y(λ) for all λ ∈ [0, ϵ], a contradiction. Therefore λ_{0} = 1, i.e. x(λ) = y(λ) for all λ ∈ [0, 1]. □

Using R(s) and the previous lemma, we may now define segments in X(G) and their subdivisions as in section 5.2.15, with [x, y] ⊂ X(S) if x, y ∈ X(S).

Lemma 110. For every x, y ∈ X(G) and z ∈ X(G) (resp. F ∈ F(G)), there exists a subdivision x = x_{0}, ⋯, x_{n} = y of the segment [x, y] and for i ∈ {1, ⋯, n}, an S_{i} ∈ S(G) such that [x_{i−1}, x_{i}] ⊂ X(S_{i}) and z ∈ X(S_{i}) (resp. F ∈ F(S_{i})).

Proof. By R(s), there is an S ∈ S(G) such that x, y ∈ X(S), so that

y = x + G^{+} and x = y + G^{−} with G^{±} ∈ F(S), G^{+} + G^{−} = 0.

For λ ∈ [0, 1], set x(λ) = x + λG^{+} and choose F_{λ} ∈ F(G) such that z = x(λ) + F_{λ} using T (s). By L(s)^{+}, there exists ϵ_{λ} > 0 and S_{λ}^{±} ∈ S(G) such that F_{λ} ∈ F(S_{λ}^{±}) (resp. F ∈ F(S_{λ}^{±})) and x(λ) + μG^{±} ∈ X(S_{λ}^{±}) for all μ ∈ [0, ϵ_{λ}]. Pick a finite set S ⊂ [0, 1] such that [0, 1] ⊂ ∪_{λ∈S}]λ − ϵ_{λ}, λ + ϵ_{λ}[ and let x = x_{0}, ⋯, x_{n} = y be the subdivision of [x, y] defined by [x, y] ∩ {x, y, x(λ), x(λ) ± ϵ_{λ} : λ ∈ S}. Then for each i ∈ {1, ⋯, n}, there exists an S_{i} ∈ {S_{λ}^{±} : λ ∈ S} such that [x_{i−1}, x_{i}] ∈ X(S_{i}) and z ∈ X(S_{i}) (resp. F ∈ F(S_{i})). □

Lemma 111. For a minimal parabolic subgroup B = U ⋊ Z_{G}(S) of G, the apartment X(S) is a fundamental domain for the action of U on X(G) and the corresponding retraction r : X(G) ↠ X(S) is non-expanding for d.

Proof. First, X(G) = U · X(S) by L(s). For x, y ∈ X(S) and any F ∈ F(S), d(x, y) = d(x + F, y + F) in C(G). If y = ux with u ∈ U and P_{F} = B, then also

lim_{s→∞} d(x + sF, y + sF) = 0

by UN, thus d(x, y) = 0 and x = y, i.e. X(S) is a fundamental domain for the action of U on X(G). Let r : X(G) ↠ X(S) be the corresponding retraction. For x, y ∈ X(G), there exists by the previous lemma a subdivision x = x_{0}, ⋯, x_{n} = y of [x, y] and for each i ∈ {1, ⋯, n}, an S_{i} ∈ S(G) such that [x_{i−1}, x_{i}] ⊂ X(S_{i}) and Z_{G}(S_{i}) ⊂ B. Then, there is a unique u_{i} ∈ U such that Int(u_{i})(S_{i}) = S, in which case also u_{i} · X(S_{i}) = X(S) and r(z) = u_{i}z for all z ∈ X(S_{i}). We thus obtain

d(x, y) = ∑_{i=1}^{n}d(x_{i−1}, x_{i})
= ∑_{i=1}^{n}d(u_{i}x_{i−1}, u_{i}x_{i})
= ∑_{i=1}^{n}d (r(x_{i−1}), r(x_{i}))
≥ d (r(x), r(y))

in C(G) by the known triangle inequality for d in X(S). □

Lemma 112. The vectorial distance d satisfies T R, NE, CO and (X(G), d) is a CAT (0)-metric space – thus X(G) also satisfies UG.

Proof. For x, y, z ∈ X(G), choose S ∈ S(G) with x, z ∈ X(S) using R(s), pick a minimal parabolic subgroup B of G with Levi Z_{G}(S) and let r : X(G) ↠ X(S) be the corresponding retraction. Then

d(x, z) ≤ d (x, r(y)) + d (r(y), z) ≤ d(x, y) + d(y, z)

by the triangle inequality in X(S) and the previous lemma. This proves T R.

For x, y ∈ X(G) and F ∈ F(G), pick a subdivision x = x_{0}, ⋯, x_{n} = y of [x, y] and for each i ∈ {1, ⋯, n}, an S_{i} ∈ S(G) such that [x_{i−1}, x_{i}] ∈ X(S_{i}) and F ∈ F(S_{i}), using lemma 110. Then

d(x + F, y + F) ≤ ∑_{i=1}^{n}d(x_{i−1} + F, x_{i} + F) = ∑_{i=1}^{n}d(x_{i−1}, x_{i}) = d(x, y)

by the triangle inequality in X(G) that we have just proven and a trivial computation in the affine F(S_{i})-space X(S_{i}). This proves NE.

For x ∈ X(G) and F, G ∈ F(G), set y = x+F, z = x+G. By lemma 110, there is a subdivision y = x_{0}, ⋯, x_{n} = z of the segment [y, z] and for each i ∈ {1, ⋯, n}, an S_{i} ∈ S(G) such that [x_{i−1}, x_{i}] ∈ X(S_{i}) and x ∈ X(S_{i}). For i ∈ {0, ⋯, n} and λ ∈ [0, 1], set x_{i}(λ) = (1 − λ)x + λx_{i} in [x, x_{i}]. Then

d(x_{0}(λ), x_{n}(λ)) ≤ ∑_{i=1}^{n}d(x_{i−1}(λ), x_{i}(λ)) = ∑_{i=1}^{n}λd(x_{i−1}, x_{i}) = λd(y, z)

by the triangle inequality in X(G) and a trivial computation in the affine F(S_{i})- space X(S_{i}). This proves CO′, from which CO follows.

To establish the CAT (0)-property, imagine a rigid comparison triangle (x̃, ỹ, z̃) for (x, y, z), lying on a Euclidean 2-plane E. Add flex points (x̃_{1}, ⋯, x̃_{n−1}) on the segment [ỹ, z̃] corresponding to (x_{1}, ⋯, x_{n−1}), and push them (inward or outward) one by one, so that each (x̃, x̃_{i−1}, x̃_{i}) becomes a comparison triangle for (x, x_{i−1}, x_{i}) (with x̃_{0} = ỹ and x̃_{n} = z̃). If a last outward move occurs at the i-th step, then in the final configuration, the chord between x̃_{i−1} and x̃_{i+1} intersects the radius between x̃ and x̃_{i} at some point ỹ = (1 − ν)x̃ + νx̃_{i}, with ν ∈ [0, 1[. For the corresponding point y = (1 − ν)x + νx_{i} on the segment [x, x_{i}] ⊂ X(S_{i}) ∩ X(S_{i+1}), we would have:

d(x_{i−1}, y) + d(y, x_{i+1}) = d(x̃_{i−1}, ỹ) + d(ỹ, x̃_{i+1})
< d(x̃_{i−1}, x̃_{i}) + d(x̃_{i}, x̃_{i+1})
= d(x_{i−1}, x_{i}) + d(x_{i}, x_{i+1})
= d(x_{i−1}, x_{i+1})

which contradicts the triangle inequality for d in X(G). It follows that there is no last outward move, i.e. no outward move at all. Thus for any λ ∈ [0, 1], if x̃(λ) is the point corresponding to x(λ) = (1 − λ)y + λz ∈ [y, z] on the articulated segment [ỹ, z̃] of our comparison triangle, the distance between x̃ and x̃(λ) is not greater in the final configuration than it was initially. Since the final distance is the actual distance between x and x(λ) in (X(G), d), this proves the required CAT (0) inequality for x and the standard geodesic segment x(−) : [0, 1] → X(G) from y to z. However, we still have to check that our metric space (X(G), d) is unically geodesic in the usual sense. Suppose therefore that x′(−) : [0, 1] → X(G) is another geodesic segment between y and z. For λ ∈ [0, 1], the CAT (0)-inequality that we have just established for the point x′(λ) and the standard geodesic x(−) : [0, 1] → X(G) implies that x(λ) = x′(λ), thus indeed x(−) = x′(−). □

5.3.2. Discrete Buildings. The following axiom is a strengthening of R(i): R(i)^{+} For S, S′ ∈ S(G), there is a g ∈ G with

Int(g)(S) = S′ and g ≡ Id on X(S) ∩ X(S′).

Lemma 113. A discrete affine F(G)-building X(G) satisfies R(i)^{+}.

Proof. We may assume that Z = X(S) ∩ X(S′) ≠ ∅. Then Z is a non-empty closed convex subset of the affine F(S)-space X(S), therefore Z has non-empty interior as a subset of its affine span A in X(S). Let ∼ be the equivalence relation on Z defined by x ∼ y if and only if x and y have the same stabilizer in N_{G}(S). Since X(G) is discrete, there are countably many equivalence classes, thus one of them at least, say E ⊂ Z, has the property that the closure of E has a non-empty interior in A. Then A is also the affine span of \overline{E} or E in X(S). Let C ⊂ N_{G}(S) be the common stabilizer of the points of E. Now for any g_{1}, g_{2} ∈ G such that Int(g_{i})(S) = S′ and g_{1}x = g_{2}x for some x ∈ E, g_{2} = g_{1}c for some c ∈ C, thus g_{1} ≡ g_{2} on E, A and Z. Fix x ∈ E. Then for any y ∈ Z, there exists by R(i) some g_{y} ∈ G such that Int(g_{y})(S) = S′ and g_{y}x = x, g_{y}y = y. For y, z ∈ Z, we have just seen that g_{y} ≡ g_{z} on E, A and Z, thus g_{y}z = g_{z}z = z and g_{y} ≡ Id on Z. □

Lemma 114. The metric of a discrete affine F(G)-building is complete.

Proof. Suppose that X(G) is discrete and equip C = G\X(G) with

\overline{d}(α, β) = inf D(α, β) where D(α, β) = \{d(x, y) : x ∈ α, y ∈ β\} .

Fix S ∈ S(G). Then by R(i) and R(s), also C = N_{G}(S)\X(S) and

D(α, β) = {d(a, n · b) : n ∈ N_{G}(S)}

if (a, b) lifts (α, β) in X(S). Since N_{G}(S) · b is discrete in the Euclidean space X(S) by assumption, if follows that there is a constant ϵ(α, β) > 0 such that

∀(x, y) ∈ α × β : d(x, y) ≤ \overline{d}(α, β) + ϵ(α, β) ⟹ d(x, y) = \overline{d}(α, β).

In particular, \overline{d} is a distance on C. Moreover, (C, \overline{d}) is complete: if (α_{n}) is a Cauchy sequence in C, it lifts to a bounded sequence (a_{n}) in X(S), the latter has a subsequence (a_{φ(n)}) converging to some a in X(S), whose image in C is then a limit of (α_{n}). Let now (x_{n}) be a Cauchy sequence in (X(G), d). Its image (α_{n}) is a Cauchy sequence in (C, \overline{d}), which thus converges to some α in C. For each n, lift α to some y_{n} ∈ X(G) with d(x_{n}, y_{n}) = \overline{d}(α_{n}, α). Then (y_{n}) is also a Cauchy sequence in X(G), hence d(y_{n}, y_{m}) ≤ ϵ(α, α) for n, m ≫ 0, which implies that (y_{n}) is actually stationary and (x_{n}) converges to its limit: (X(G), d) is complete. □

5.4. Walls and tight buildings

Let again X(G) be an affine F(G)-space.

5.4.1. For S ∈ S(G), let Φ(S, G) be the set of roots of S in the Lie algebra

Lie(G) = 𝔤 = 𝔤_{0} ⊕ ⊕_{a∈Φ(S,G)}𝔤_{a}.

We denote by U_{a} ⊂ G the root subgroup corresponding to some a ∈ Φ(G, S): if S_{a} denotes the neutral component of the kernel of a : S → 𝔾_{m,K}, then U_{a} is the unipotent radical of the unique parabolic subgroup of Z_{G}(S_{a}) containing Z_{G}(S) with Lie algebra 𝔤_{0} ⊕ ⊕_{b∈ℕa∩Φ(S,G)}𝔤_{b}. If 2a ∈ Φ(S, G), then U_{2a} ⊂ U_{a}.

5.4.2. For any u ∈ U_{a} \ {1}, there exists a unique triple (u_{1}, u_{2}, m(u)) with

u_{1}uu_{2} = m(u), u_{1}, u_{2} ∈ U_{−a} \ {1} and m(u) ∈ N_{G}(S).

Moreover, ν_{S}^{v}(m(u)) is the symmetry s_{a} ∈ W_{G}(S) attached to a, given by

s_{a} : G(S) → G(S), s_{a}(x) = x − a(x)a^{∨}

where a^{∨} : 𝔾_{m,K} → S is the coroot corresponding to a and a(x) = a ∘ x in

ℝ = Hom(𝔻_{K}(ℝ), 𝔾_{m,K}).

This follows from [5, §5] by [9, 6.1.2.2 & 6.1.3.c]. Considering the action of m(u) on X(S), we thus obtain a unique affine hyperplane X(S, u) in X(S) which is preserved by m(u). The underlying vector space is the fixed point set of s_{a}, namely

G(S_{a}) = {x ∈ G(S) : s_{a}(x) = x} = {x ∈ G(S) : a(x) = 0}

and m(u) acts on X(S, u) by x ↦ x + ν_{X}(S, u) for some v_{X}(S, u) ∈ G(S_{a}).

Example 115. For the affine F(G)-space X(G) = F(G), X(S, u) = G(S_{a}) is the fixed point set of m(u) acting on G(S) = F(S) and ν_{X}(S, u) = 0.

5.4.3. Of course m(u) fixes X(S, u) if and only if ν_{X}(S, u) = 0, and this happens when m(u) already has finite order in N_{G}(S), which holds true for any u ∈ U_{a} \ {1} if 2a ∉ Φ(S, G). Indeed, set Φ′(S, G) = {b ∈ Φ(S, G) : 2b ∉ Φ(S, G)}. This is again a root system and U_{b} ≃ 𝔾_{a,K}^{n(b)} for some n(b) ≥ 1 for all b ∈ Φ′(S, G). Choose a set of simple roots Δ′ of Φ′(S, G) containing a and choose for each b ∈ Δ′ a 1-dimensional K-subspace U′_{b} in U_{b} ≃ K^{n(b)}, with u ∈ U′_{a}. Then by [5, 7.2], there is a unique split reductive subgroup G′ of G containing S with Φ(S, G′) = Φ′(S, G) such that the root subgroup U′_{b} of b ∈ Δ′ in G′ is the subgroup of U_{b} determined by U′_{b}, i.e. U′_{b} = U′_{b}(K). Then (Z_{G′}(S_{a}), S, a) is an elementary system in the sense of [21, XX 1.3] by [21, XIX 3.9]. Let f : S_{L} → Z_{G′}(S_{a}) be the corresponding morphism constructed in [21, XX 5.8] and let X ≠ 0 be the unique element of L = Lie(U′_{a}) with f( \begin{matrix}1 & X \\ 0 & 1 \end{matrix} ) = u. Since

( \begin{matrix}1 & 0 \\ −X^{−1} & 1 \end{matrix} ) ( \begin{matrix}1 & X \\ 0 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 \\ −X^{−1} & 1 \end{matrix} ) = ( \begin{matrix}0 & X \\ −X^{−1} & 0 \end{matrix} )

in S_{L}(K), we find that

m(u) = f ( \begin{matrix}0 & X \\ −X^{−1} & 0 \end{matrix} ) , m(u)^{2} = f ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) and m(u)^{4} = 1.

On the other hand if 2a ∈ Φ(G, S), then [47, 1.15] provides examples where m(u) has infinite order. Note also that m(u) fixes X(S, u) when there is a z ∈ Z_{G}(S) such that zuz^{−1} = u^{−1}: since m(u^{−1}) = m(u)^{−1} and m(zuz^{−1}) = zm(u)z^{−1},

\begin{aligned}X(S, u^{−1}) & = & X(S, u) \\ ν_{X}(S, u^{−1}) & = & −ν_{X}(S, u) \end{aligned}  and  \begin{aligned}X(S, zuz^{−1}) & = & X(S, u) + ν_{X,S}(z) \\ ν_{X}(S, zuz^{−1}) & = & ν_{X}(S, u) \end{aligned}

therefore zuz^{−1} = u^{−1} implies ν_{X}(S, u) = 0. Note also that since

(m(u)u_{2}m(u)^{−1}) u_{1}u = m(u) and uu_{2} (m(u)^{−1}u_{1}m(u)) = m(u)

we find that m(u_{1}) = m(u_{2}) = m(u), thus

X(S, u_{1}) = X(S, u_{2}) = X(S, u) and ν_{X}(S, u_{1}) = ν_{X}(S, u_{2}) = ν_{X}(S, u).

5.4.4. For a subset Ω ≠ ∅ of X(S), we denote by G_{Ω} the pointwise stabilizer of Ω in G and by G_{S,Ω} the subgroup of G spanned by N_{G}(S)_{Ω} = G_{Ω} ∩ N_{G}(S) and

{u ∈ U_{a} \ {1} : a ∈ Φ(G, S), Ω ⊂ X^{+}(S, u)}

where for any a ∈ Φ(G, S) and u ∈ U_{a} \ {1},

X^{+}(S, u) = X(S, u) + {F ∈ F(S) : a(F) ≥ 0} .

When Ω = {x}, we simply write G_{x} = G_{{x}} and G_{S,x} = G_{S,{x}}. Thus

G_{Ω} = ∩_{x∈Ω}G_{x} and G_{S,Ω} ⊂ ∩_{x∈X}G_{S,x}.

For ∅ ≠ Ω′ ⊂ Ω ⊂ X(S), G_{Ω} ⊂ G_{Ω′} and G_{S,Ω} ⊂ G_{S,Ω′}. Finally for g ∈ G and S′ = Int(g)(S), Ω′ = g · Ω, one checks easily that

Int(g)(G_{Ω}) = G_{Ω′} and Int(g)(G_{S,Ω}) = G_{S′,Ω′}.

5.4.5. Example. For the affine F(G)-space X(G) = F(G) and u ∈ U_{a} \ {1},

X^{+}(S, u) = {F ∈ F(S) : a(F) ≥ 0} .

For F ∈ F(S), G_{F} = P_{F} and G_{S,F} is therefore the group spanned by N_{G}(S) ∩ P_{F} and the U_{a}’s for a ∈ Φ(G, S), a(F) ≥ 0. The U_{a}’s with a(F) > 0 span U_{F} by [5, 3.11]. Moreover, the group N_{G}(S) ∩ P_{F} = N_{L}(S) and the U_{a}’s with a(F) = 0 together span L = L(K) where L is the Levi subgroup of P_{F} which contains Z_{G}(S), by the Bruhat decomposition of L, see [5, 5.15]. Therefore

G_{S,F} = P_{F} = G_{F}.

5.4.6. We next consider the following axioms: ST (Stabilizers) For some (or every) S ∈ S(G),

∀x ∈ X(S) : G_{S,x} = G_{x}.

ST^{−} For some (or every) S ∈ S(G),

∀x ∈ X(S) : G_{S,x} ⊂ G_{x}.

ST_{1}^{−} For some (or every) S ∈ S(G),

∀∅ ≠ Ω ∈ X(S) : G_{S,Ω} ⊂ G_{Ω}.

ST_{2}^{−} For some (or every) S ∈ S(G) and any a ∈ Φ(G, S), u ∈ U_{a} \ {1},

∃x ∈ X(S, u) : ux = x.

UN^{+} For x ∈ X(G), F ∈ F(G) and u ∈ U_{F},

∀t ≫ 0 : u(x + tF) = x + tF.

Lemma 116. These axioms are related as follows:

\begin{aligned}& ST ⟹ ST^{−} ⟺ ST_{1}^{−} ⟺ ST_{2}^{−} \\ and & ST^{−} + L(s) ⟹ UN^{+} ⟹ UN. \end{aligned}

Under ST^{−}, for every S ∈ S(G), a ∈ Φ(G, S) and u ∈ U_{a} \ {1}, ν_{X}(S, u) = 0 and

X^{+}(S, u) = {x ∈ X(S) : ux = x} = {x ∈ X(S) : ux ∈ X(S)} .

Proof. Plainly, ST ⇒ ST^{−}, ST^{−} ⇔ ST_{1}^{−} and UN^{+} ⇒ UN. Fix S ∈ S(G), a ∈ Φ(G, S), u ∈ U_{a} \ {1}. Since x ∈ X(S, u) implies u ∈ G_{S,x}, ST^{−} ⇒ ST_{2}^{−}. On the other hand if u fixes some x ∈ X(S), it also fixes x + F for every F ∈ F(S) with a(F) ≥ 0 because a(F) ≥ 0 ⟺ U_{a} ⊂ P_{F}, thus ST_{2}^{−} ⇒ ST^{−}. Under ST_{2}^{−},

X^{+}(S, u) ⊂ {x ∈ X(S) : ux = x} ⊂ {x ∈ X(S) : ux ∈ X(S)} .

Applying this to u_{1}, u_{2} ∈ U_{−a} \{1}, we obtain: u_{1} ≡ u_{2} ≡ Id on X(S)\X^{+}(S, u). If x and ux belong to X(S) but x does not belong to X^{+}(S, u), then ux also does not belong to X^{+}(S, u), however m(u)x = u_{1}uu_{2}x = u_{1}ux = ux does, a contradiction. This proves the required displayed equality, and ν_{X}(S, u) = 0 since m(u) = u_{1}uu_{2} with u, u_{1}, u_{2} ≡ Id on X(S, u) = X(S, u_{1}) = X(S, u_{2}). Suppose finally that ST^{−} and L(s) hold. For x ∈ X(G), F ∈ F(G) and any u ∈ U_{F} with u ≠ 1, pick S ∈ S(G) with x ∈ X(S) and F ∈ F(S) using L(s). Since U_{F} is spanned by the U_{a}’s with a ∈ Φ(G, S), a(F) > 0 (as in Example 5.4.5), we may write u = u_{1} ⋯ u_{n} with u_{i} ∈ U_{a_{i}} \ {1} for some a_{i} ∈ Φ(G, S), a_{i}(F) > 0. For any sufficiently large t ≥ 0, x + tF ∈ X(S) then belongs to X^{+}(S, u_{i}) for all i ∈ {1, ⋯, n}, thus u_{i}(x + tF) = x + tF by ST^{−} and u(x + tF) = x + tF, which proves UN^{+}. □

5.4.7. The next axiom is related to alcove-based retractions, see [37, 1.4]. HA (Half-apartments) For S_{1}, S_{2}, S_{3} ∈ S(G), if X(S_{i}) ∩ X(S_{j}) contains an half-

subspace of X(S_{i}) for every pair (i, j) in {1, 2, 3}^{2}, then

X(S_{1}) ∩ X(S_{2}) ∩ X(S_{3}) ≠ ∅.
Lemma 117. The axioms UN^{+} and HA imply ST^{−}.
Proof. Fix S ∈ S(G), a ∈ Φ(G, S) and u ∈ U_{a} \ {1} and first note that
{x ∈ X(S) : ux ∈ X(S)} = {x ∈ X(S) : ux = x} .

Indeed if x and ux belong to X(S), pick F ∈ F(S) with a(F) > 0. Then U_{a} ⊂ U_{F}, thus ux + tF = u(x + tF) = x + tF for t ≫ 0 by UN^{+} and ux = x since X(S) is an affine F(S)-space. For every t ∈ ℝ, we define

X(S, u, t) = X(S, u) + {F ∈ F(S) : a(F) = t}
X^{+}(S, u, t) = X(S, u, t) + {F ∈ F(S) : a(F) ≥ 0}
X^{−}(S, u, t) = X(S, u, t) + {F ∈ F(S) : a(F) ≤ 0}

If u fixes some x ∈ X(S, u, t), then also u ≡ Id on X^{+}(S, u, t) since

∀F ∈ F(S) : a(F) ≥ 0 ⟺ U_{a} ⊂ P_{F}.

By UN^{+}, u fixes some point in X(S), thus u ≡ Id on X^{+}(S, u, t) for t ≫ 0. Let us now write u_{1}uu_{2} = m(u) with u_{1}, u_{2} ∈ U_{−a} \ {1}. Then similarly for i ∈ {1, 2},

{x ∈ X(S) : u_{i}x ∈ X(S)} = {x ∈ X(S) : u_{i}x = x}

and u_{i} fixes X^{+}(S, u_{i}, t) = X^{−}(S, u, −t) for t ≫ 0. Choose T > 0 such that

u ≡ Id on X^{+}(S, u, T) and u_{1} ≡ u_{2} ≡ Id on X^{−}(S, u, −T).

Then: X(S) and uX(S) contain the half-subspace X^{+}(S, u, T), X(S) and u_{1}^{−1}X(S) contain X^{−}(S, u, −T), while uX(S) and u_{1}^{−1}X(S) contain

uX^{−}(S, u, −T) = uu_{2}X^{−}(S, u, −T) = u_{1}^{−1}m(u)X^{−}(S, u, −T) = u_{1}^{−1}X^{+}(S, u, T).

Thus by HA, there is a point x ∈ X(S)∩uX(S)∩u_{1}^{−1}X(S). Any such point is fixed by u^{−1} and u_{1}, thus also by m(u)u_{2}^{−1} = u_{1}u. In particular u_{2}^{−1}(x) = m(u)^{−1}(x) also belongs to X(S), so that again x is fixed by u_{2}, as well as m(u) = u_{1}uu_{2}. But then x belongs to X(S, u) = {x ∈ X(S) : m(u)(x) = x} and it is fixed by u, which proves ST_{2}^{−}, from which ST follows by the previous lemma. □

5.4.8. An affine F(G)-building is tight if it satisfies ST. It then also satisfies the conclusion of lemma 116, and it is determined by its type. More precisely:

Lemma 118. Suppose that X(G) is a tight affine F(G)-building and Y(G) is an affine F(G)-building which satisfies ST^{−}. Then ν_{X} = ν_{Y} ⟺ X(G) ≃ Y(G).

Proof. We have to show that ν_{X} = ν_{Y} implies X^{e}(G) ≃ Y^{e}(G). Suppose therefore that ν_{X} = ν_{Y}. Pick S ∈ S(G). By [39, 2.1.9], there is a finite subgroup of N_{G}(S) which maps surjectively onto W_{G}(S), and which thus has unique fixed points x_{S} in X^{r}(S) and y_{S} in Y^{r}(S). Let θ_{S} : X^{e}(G) → Y^{e}(G) be the unique isomorphism of affine F(S)-spaces mapping (x_{S}, 0) to (y_{S}, 0). Then θ_{S} is N_{G}(S)-equivariant, and it is the unique N_{G}(S)-equivariant isomorphism of affine F(S)-spaces from X^{e}(S) to Y^{e}(S) mapping X^{r}(S) to Y^{r}(S). If Int(g)(S) = S′, then g ∘ θ_{S} = θ_{S′} ∘ g. For x ∈ X^{e}(S) ∩ X^{e}(S′), there is such a g in G_{x} by R(i) for X(G). Thus g belongs to G_{S,x} by ST for X(G), which equals G_{S,θ_{S}(x)} by definition. Then g ∈ G_{θ_{S}(x)} by our assumption on Y(G), thus θ_{S′}(x) = θ_{S′}(gx) = gθ_{S}(x) = θ_{S}(x). Our isomorphisms θ_{S} therefore glue to θ : X^{e}(G) → Y^{e}(G), which is the desired isomorphism. □

Remark 119. A tight affine F(G)-building X(G) can be retrieved from any apartment X(S) together with its N_{G}(S)-action. It is the quotient of G × X(S) for the equivalence relation ∼ induced by (g, x) ↦ gx, which indeed only depends upon the apartment: (g, x) ∼ (g′, x′) if and only if g′ = gkn and x′ = n^{−1}x for some k ∈ G_{S,x} and n ∈ N_{G}(S).

5.5. Metric properties

Let X(G) be an affine F(G)-building. We shall here relate our mostly algebraic formalism to various notions pertaining to the non-canonical metric d = d_{τ}: rays, tangent spaces and Busemann functions. For simplicity, we furthermore assume that (X(G), d) is a CAT (0)-space.

5.5.1. There is a G-equivariant commutative diagram

\xymatrix{X(G) × F(G) \ar@{^{(}->}[r]^{α} \ar[rr]^{Id×ι} \ar[dr]_{+} & RX(G) \ar@{^{(}->}[r]^{β} \ar[d]^{_{ev_{1}}} & X(G) × C(∂X(G)) \\  & X(G)}

where the various new sets and maps are defined as follows:

• RX(G) is the set of all functions f : ℝ^{+} → X(G) such that

∃c_{f} ≥ 0 s.t. ∀t, u ∈ ℝ^{+} : d(f(t), f(u)) = c_{f} |u − t| .

• ∂X(G) is the visual boundary {f ∈ RX(G) : c_{f} = 1} / ∼ of X(G), where the equivalence relation ∼ is defined on the whole of RX(G) by

f ∼ g ⟺ t ↦ d (f(t), g(t)) is bounded.

• C(∂X(G)) is the cone (ℝ^{+} × ∂X(G)) / ≈ where the equivalence relation ≈ just collapses {0} × ∂X(G) to a single point 0 ∈ C(∂X(G)), so that

f ↦ [f] = \begin{cases}(c_{f}, class of f(c_{f}^{−1}−)) & if c_{f} ≠ 0, \\ 0 & if c_{f} = 0, \end{cases}

identifies the quotient RX(G)/ ∼ with the cone C(∂X(G)).

• α(x, F)(t) = x + tF, β(f) = (f(0), [f]) and ev_{1}(f) = f(1). By the axiom NE for X(G), β ∘ α = Id_{X(G)} × ι for some G-equivariant map

ι : F(G) ↪ C(∂X(G)).

The latter is injective: suppose that x + tF ∼ y + tG and pick z ∈ X(S) for some S ∈ S(G) such that F, G ∈ F(S). Then z + tF ∼ x + tF ∼ y + tG ∼ z + tG, thus F = G since z + tF ∼ z + tG in the affine F(S)-space X(S). It follows that α is also injective. Finally β is injective by convexity of the CAT(0)-distance d, and it is also surjective when (X(G), d) is complete [8, II.8.2] (for instance in the discrete case, by lemma 114). By the axiom L(s), α is bijective precisely when every geodesic ray in X(G) is standard, in which case ι and β are also bijective.

Remark 120. The injectivity of ι implies that the apartment map S ↦ X(S) is injective: X(S) determines C(∂X(S)) = ι(F(S)), thus also F(S) and S ∈ S(G).

5.5.2. Fix x ∈ X(G) and 0 ≠ F, G ∈ F(G), set y = x + F, z = x + G. We may then define the following five different types of angles

0 ≤ ∡_{x}(F, G) ≤ ∡_{x}(\overrightarrow{xy}, G) ≤ ∡_{x}^{c}(y, z) ≤ ∡(\overrightarrow{xy}, G) ≤ ∡^{x}(F, G) ≤ π.

First, ∡_{x}^{c}(y, z) is the angle at x in a comparison triangle for (x, y, z), so that

d(y, z) = (d(x, y)^{2} + d(x, z)^{2} − 2d(x, y)d(x, z) cos ∡_{x}^{c}(y, z))^{1/2} .

More generally for every (t, u) ∈ ℝ_{+}, the distance d(x + tF, x + uG) equals

(t^{2} ∥F∥^{2} + u^{2} ∥G∥^{2} − 2tu ∥F∥ ∥G∥ cos ∡_{x}^{c}(x + tF, x + uG))^{1/2} .

By [8, II.3.1], the comparison angle function

(t, u) ∈ ℝ_{>}^{2} ↦ ∡_{x}^{c}(x + tF, x + uG) ∈ [0, π]

is non-decreasing in both variables. We define

∡_{x}(F, G) = inf {∡_{x}^{c}(x + tF, x + uG) : t, u > 0} = lim_{t,u→0} ∡_{x}^{c}(x + tF, x + uG)
∡_{x}(\overrightarrow{xy}, G) = inf \{∡_{x}^{c}(y, x + uG) : u > 0\}  = lim_{u→0} ∡_{x}^{c}(y, x + uG)
∡(\overrightarrow{xy}, G) = sup \{∡_{x}^{c}(y, x + uG) : u > 0\}  = lim_{u→∞} ∡_{x}^{c}(y, x + uG)
∡^{x}(F, G) = sup {∡_{x}^{c}(x + tF, x + uG) : t, u > 0} = lim_{t,u→∞} ∡_{x}^{c}(x + tF, x + uG)

We will also use the notations ∡_{x}(y, z) = ∡_{x}(F, G) = ∡(F_{x}, G_{x}).

5.5.3. Let us immediately observe that:

Lemma 121. If G belongs to G(Z) ⊂ F(G), then

∡_{x}(F, G) = ∡_{x}(\overrightarrow{xy}, G) = ∡_{x}^{c}(y, z) = ∡(\overrightarrow{xy}, G) = ∡^{x}(F, G).

Proof. Pick S ∈ S(G) with x ∈ X(S), F ∈ F(S) using the axiom L(s) for X(G). Then also G ∈ F(S), thus everything stays in the flat Euclidean affine F(S)-space X(S) on which all of our angles plainly agree. □

5.5.4. The smallest of these angles, also denoted by ∡(F_{x}, G_{x}), is the Alexandrov angle at x between the rays x + tF and x + tG [8, I.12]. It satisfies a triangle inequality: if H ∈ F(G) is yet another nonzero filtration, then by [8, I.14],

∡_{x}(F, H) ≤ ∡_{x}(F, G) + ∡_{x}(G, H).

The tangent cone at x is the quotient T_{x}X(G) = F(G)/ ∼_{x}, where F ∼_{x} G if and only if ∥F∥ = ∥G∥ and ∡_{x}(F, G) = 0. This definition agrees with [8, II.3.18] by the axiom R(s) for X(G). We denote by loc_{x}(F) = F_{x} the class of F in T_{x}X(G). The norm ∥−∥ and Alexandrov angle ∡_{x}(−, −) on F(G) descend to a norm and angle on T_{x}X(G), thereby justifying our notation ∡(F_{x}, G_{x}) = ∡_{x}(F, G). We also define a scalar product and a distance function on T_{x}X(G) by the usual formulas:

⟨F_{x}, G_{x}⟩ = ∥F_{x}∥ ∥G_{x}∥ cos ∡(F_{x}, G_{x})
d(F_{x}, G_{x}) = \sqrt{∥F_{x}∥^{2} + ∥G_{x}∥^{2} − 2 ⟨F_{x}, G_{x}⟩}.

By definition of the Alexandrov angle,

d(F_{x}, G_{x}) = lim_{t→0} \frac{1}{t}d(x + tF, x + tG).

These formulas for d respectively show that d(F_{x}, G_{x}) = 0 if and only if F_{x} = G_{x}, and that d(F_{x}, H_{x}) ≤ d(F_{x}, G_{x})+d(G_{x}, H_{x}). Thus d is indeed a distance on T_{x}X(G) and F ∼_{x} G if and only if lim_{t→0} \frac{1}{t}d(x + tF, x + tG) = 0.

5.5.5. By the very definition of T_{x}X(G), there is a commutative diagram

\xymatrix{F(G) \ar@{->>}[rr]^{F↦x+F} \ar@{->>}[dr]_{_{loc_{x}}} &  & X(G) \ar@{->>}[dl]^{_{loc{a \atop x}}} \\  & T_{x}X(G)}

We may thus also define

∡_{x}(y, z) = ∡ (loc_{x}^{a}(y), loc_{x}^{a}(z)) ,
⟨y, z⟩_{x} = ⟨loc_{x}^{a}(y), loc_{x}^{a}(z)⟩ ,
d_{x}(y, z) = d (loc_{x}^{a}(y), loc_{x}^{a}(z)) .

5.5.6. Our second smallest angle actually equals the first one by [8, I.1.16]. Thus for y = x + F, we have ∡_{x}(\overrightarrow{xy}, G) = ∡(F_{x}, G_{x}). Since

lim_{t→0} \frac{1}{t} (d(y, x) − d(y, x + tG)) = ∥F∥ ∥G∥ cos ∡_{x}(\overrightarrow{xy}, G)

by definition of ∡_{x}(\overrightarrow{xy}, G), it follows that

lim_{t→0} \frac{1}{t} (d(y, x) − d(y, x + tG)) = ⟨loc_{x}^{a}(y), loc_{x}G⟩ .

5.5.7. By definition of our largest angle ∡^{x}(F, G), we have

lim_{t→∞} \frac{1}{t}d(x + tF, x + tG) = \sqrt{∥F∥^{2} + ∥G∥^{2} − 2 ∥F∥ ∥G∥ cos ∡^{x}(F, G)}.

For z_{1}, z_{2} ∈ X(G), x + tF ∼ z_{1} + tF and x + tG ∼ z_{2} + tG, thus also

lim_{t→∞} \frac{1}{t}d(z_{1} + tF, z_{2} + tG) = \sqrt{∥F∥^{2} + ∥G∥^{2} − 2 ∥F∥ ∥G∥ cos ∡^{x}(F, G)}.

In particular, ∡^{x}(F, G) is independent of x. Taking x ∈ X(S) for some S ∈ S(G) with F, G ∈ F(S), we find that ∡^{x}(F, G) = ∡(F, G) = ∡_{x}(F, G). Thus

d(F, G) = lim_{t→∞} \frac{1}{t}d(z_{1} + tF, z_{2} + tG)

for every z_{1}, z_{2} ∈ X(G) and

∡(F, G) = max{∡(F_{x}, G_{x}) : x ∈ X(G)},
⟨F, G⟩ = min{⟨F_{x}, G_{x}⟩ : x ∈ X(G)},
d(F, G) = max{d(F_{x}, G_{x}) : x ∈ X(G)}.

5.5.8. Recall from [8, II.8.18-20] that for any y, z ∈ X(G), the function

t ↦ d(y, z + tG) − t ∥G∥

is non-increasing and bounded, the functions

y ↦ d(y, z + tG) − t ∥G∥

converge uniformly on bounded subsets of X(G) as t → ∞ to

y ↦ b_{z,G}(y) = lim_{t→∞} (d(y, z + tG) − t ∥G∥) ,

and (for G ≠ 0) the Busemann function in two variables

(x, y) ↦ b_{G}(x, y) = b_{z,G}(y) − b_{z,G}(x)

does not depend upon z. Note that the proof of this last statement in loc. cit., which only uses the “if” part of [8, II.8.19], does indeed not require the ambient CAT(0)-space to be complete. For any G ∈ F(G) and x, y ∈ X(G), we set

⟨\overrightarrow{xy}, G⟩ = ∥G∥ · lim_{t→∞} (d(x, z + tG) − d(y, z + tG))

which is thus well-defined, independent of z, and equal to ∥G∥·b_{G}(y, x) if G ≠ 0. For x, y, z ∈ X(G), we have ⟨\overrightarrow{xz}, G⟩ = ⟨\overrightarrow{xy}, G⟩ + ⟨\overrightarrow{yz}, G⟩, ⟨\overrightarrow{yx}, G⟩ + ⟨\overrightarrow{xy}, G⟩ = 0. Taking z = x in the formula defining ⟨\overrightarrow{xy}, G⟩, we find that

⟨\overrightarrow{xy}, G⟩ = d(x, y) · ∥G∥ · cos ∡(\overrightarrow{xy}, G)

by definition of our second largest angle ∡(\overrightarrow{xy}, G). The function

y ↦ ⟨\overrightarrow{xy}, G⟩ = − ⟨\overrightarrow{yx}, G⟩

is ∥G∥-Lipschitzian and concave (by convexity of d).

5.5.9. Returning to y = x + F and z = x + G, we obtain

⟨F, G⟩ ≤ ⟨\overrightarrow{xy}, G⟩ ≤ \frac{1}{2} (d(x, y)^{2} + d(x, z)^{2} − d(y, z)^{2}) ≤ ⟨F_{x}, G_{x}⟩

with absolute values bounded by ∥F∥ ∥G∥, as well as

d(F_{x}, G_{x}) ≤ d(y, z) ≤ (d(x, y)^{2} + ∥G∥^{2} − 2 ⟨\overrightarrow{xy}, G⟩)^{1/2} ≤ d(F, G).

In particular, the localization functions

\begin{matrix}(F(G), d) & → & (X(G), d) & → & (T_{x}X(G), d) \\ F & ↦ & x + F = y & ↦ & y_{x} = F_{x} \end{matrix}

are non-expanding. For S ∈ S(G) with x ∈ X(S), they restrict to isometries

(F(S), d) ≃ (X(S), d) ≃ (T_{x}X(S), d)

where T_{x}X(S) = loc_{x}^{a}X(S) = loc_{x}F(S). We refer to T_{x}X(S) as the apartment of S in T_{x}X(G). It is a (complete thus) closed subset of T_{x}X(G).

5.5.10. Suppose that any two germs of geodesic segments in X(G) issuing from the same point are contained in some apartment of X(G). This is for instance the case when the strengthening L(s)^{+} of L(s) holds for X(G). Then:

(1) The axiom R(s) holds for T_{x}X(G), i.e. any two elements of T_{x}X(G) belong to T_{x}X(S) for some S in S(x) = {S ∈ S(G) : x ∈ X(S)};

(2) F_{x} = G_{x} if and only if x + tF = x + tG for all sufficiently small t ≥ 0;

(3) (T_{x}X(G), d) is a CAT(0)-space; and

5.6. THE AFFINE F(P/U)-SPACE T_{P}^{∞}X(G) 112

(4) v_{2} ↦ ⟨v_{1}, v_{2}⟩ is homogeneous, concave and ∥v_{1}∥-Lipschitzian on T_{x}X(G). The first two properties are easy. To establish (3), first note that (T_{x}X(G), d) is a geodesic space by (1), so it remains to establish the CAT (0)-inequality. Let thus v, v_{0}, v_{1} be three points of T_{x}X(G) and choose S ∈ S(x) such that v_{0}, v_{1} belong to T_{x}X(S). Lift v_{i} to F_{i} ∈ F(S) and lift v to some F ∈ F(G). For u ∈ [0, 1], let F_{u} = (1 − u)F_{0} + uF_{1} be the point at distance ud(F_{0}, F_{1}) from F_{0} on the segment [F_{0}, F_{1}] of F(S). Then x_{u,t} = x + tF_{u} is the point at distance ud(x_{0,t}, x_{1,t}) from x_{0,t} on the segment [x_{0,t}, x_{1,t}] of X(S) while v_{u} = loc_{x}F_{u} is the point at distance ud(v_{0}, v_{1}) from v_{0} on the segment [v_{0}, v_{1}] of T_{x}X(S). Set x_{t} = x + tF. By the CAT (0)-inequality in X(G) applied to the triangle (x_{t}, x_{0,t}, x_{1,t}),

d(x_{t}, x_{u,t})^{2} ≤ (1 − u) · d(x_{t}, x_{0,t})^{2} + u · d(x_{t}, x_{1,t})^{2} − u(1 − u) · d(x_{0,t}, x_{1,t})^{2}.

Dividing by t^{2} and taking the limit as t → 0 gives

d(v, v_{u})^{2} ≤ (1 − u) · d(v, v_{0})^{2} + u · d(v, v_{1})^{2} − u(1 − u) · d(v_{0}, v_{1})^{2}

which is the CAT (0)-inequality for T_{x}X(G). Given (3), the proof of (4) is entirely similar to that of corollary 91.

Remark 122. By (2), the quotient T_{x}X(G) of F(G) does not depend upon the chosen metric (i.e. chosen τ) for buildings satisfying the above condition on germs. Assuming instead that (X(G), d) is complete, [8, II.3.19] shows that the completion of (T_{x}X(G), d) is always a CAT(0)-space.

5.5.11. If the axiom L(s)^{+} holds for X(G), then G ↦ ⟨\overrightarrow{xy}, G⟩ is d(x, y)- Lipschitzian. Indeed, for G_{1}, G_{2} ∈ F(G), there is a subdivision x = x_{0}, ⋯, x_{n} = y of the segment [x, y] of X(G) and for each i ∈ {1, ⋯, n}, tori S_{i,1}, S_{i,2} ∈ S(G) such that [x_{i−1}, x_{i}] ⊂ X(S_{i,j}) and G_{j} ∈ F(S_{i,j}) for j ∈ {1, 2} by lemma 110. Then

⟨\overrightarrow{xy}, G_{1}⟩ − ⟨\overrightarrow{xy}, G_{2}⟩ = ∑_{i=0}^{n−1} ⟨\overrightarrow{x_{i}x_{i+1}}, G_{1}⟩ − ⟨\overrightarrow{x_{i}x_{i+1}}, G_{2}⟩

with ⟨\overrightarrow{x_{i}x_{i+1}}, G_{j}⟩ = ⟨loc_{x_{i}}^{a}(x_{i+1}), loc_{x_{i}}(G_{j})⟩. Thus

|⟨\overrightarrow{xy}, G_{1}⟩ − ⟨\overrightarrow{xy}, G_{2}⟩| ≤ ∑_{i=0}^{n−1} ∥loc_{x_{i}}^{a}(x_{i+1})∥ · d (loc_{x_{i}}(G_{1}), loc_{x_{i}}(G_{2}))

because v_{2} ↦ ⟨v_{1}, v_{2}⟩ is ∥v_{1}∥-Lipschitzian on T_{x_{i}}X(G). Since

∥loc_{x_{i}}^{a}(x_{i+1})∥ = d(x_{i}, x_{i+1}) and d (loc_{x_{i}}(G_{1}), loc_{x_{i}}(G_{2})) ≤ d(G_{1}, G_{2})

we obtain the desired inequality:

|⟨\overrightarrow{xy}, G_{1}⟩ − ⟨\overrightarrow{xy}, G_{2}⟩| ≤ ∑_{i=0}^{n−1}d(x_{i}, x_{i+1}) · d(G_{1}, G_{2}) = d(x, y) · d(G_{1}, G_{2}).

5.5.12. Convex projections. Let C be a closed convex subset of X(G) which is complete in the induced topology. Then for every x ∈ X(G), there is a unique point p(x) in C such that d(x, p(x)) = d(x, C) = inf {d(x, y) : y ∈ C}. We call

p : X(G) ↠ C

the convex projection onto C. It is non-expanding, constant on the segment [x, p(x)], the map H : X(G) × [0, 1] → X(G) associating to (x, t) the unique point at distance td(x, p(x)) from x on the segment [x, p(x)] is a continuous homotopy from Id_{X(G)} to p, and ∡_{p(x)}(x, y) ≥ \frac{π}{2} for every y ∈ C by [8, II.2.4], thus also

⟨x, y⟩_{p(x)} ≤ 0.

For any F ∈ F(G) such that p(x) + tF belongs to C for all sufficiently small t > 0,

⟨\overrightarrow{p(x)x}, F⟩ ≤ \frac{1}{2} (d(x, C)^{2} + ∥F∥^{2} − d(x, p(x) + F)^{2}) ≤ ⟨loc_{p(x)}^{a}(x), loc_{p(x)}(F)⟩ ≤ 0.

5.6. The affine F(P/U)-space T_{P}^{∞}X(G)

Let X(G) be an affine F(G)-building. Fix a parabolic subgroup P of G and let U be the unipotent radical of P.

5.6.1. We have already seen that a Levi subgroup L of P determines:

(1) a parabolic subgroup P_{L}^{ι} of G opposed to P with P ∩ P_{L}^{ι} = L,

(2) a splitting map F^{−1}(P) ↪ G(L),

(3) an opposition map F^{−1}(P) ∋ F ↦ F_{L}^{ι} ∈ F^{−1}(P_{L}),

(4) a section F(P/U) ∋ H ↦ H_{L} ∈ F(L) of Gr_{P} : F(G) ↠ F(P/U),

(5) a fundamental domain X(L) = ∪_{S∈S(L)}X(S) for the U-action on X(G),

(6) an L-equivariant, U-invariant retraction X(G) ∋ x ↦ x_{L} ∈ X(L). The splitting map takes F to its unique splitting G with Z_{G}(G) = L. We have F_{L}^{ι} = Fil(ιG) and P_{F_{L}^{ι}} = P_{L}^{ι}. The Gr_{P} -map and its section are discussed in 4.1.15, and Gr_{P} is defined everywhere by theorem 85. Finally (5) and (6) come from proposition 101, which also says that for any F in the facet F^{−1}(P),

x_{L} = lim_{t→∞}(x + tF) + tF_{L}^{ι} in X(G).

5.6.2. For any x, y ∈ X(G), the following conditions are equivalent:

(1) U · x = U · y,

(2) lim_{t→∞} d(x + tF, y + tF) = 0 for some (or every) F ∈ F^{−1}(P),

(3) lim_{t→∞} d(x + tF, y + tF) = 0 for some (or every) F ∈ F^{−1}(P),

(4) x_{L} = y_{L} for some (or every) Levi subgroup L of P Indeed (1) ⇒ (2) by the axiom UN, (2) ⇒ (3) is trivial, (3) ⇒ (4) because

(3) ⟹^{NE} lim_{t→∞} d ((x + tF) + tF_{L}^{ι}, (y + tF) + tF_{L}^{ι}) = 0 ⟹ (4)

and (4) ⇒ (1) is obvious. If X(G) satisfies UN^{+}, they are also equivalent to:

(5) x + tF = y + tF for t ≫ 0. Indeed (1) ⇒ (5) by UN^{+} and plainly (5) ⇒ (3).

5.6.3. For any \overline{S} ∈ S(P/U), there is a U-equivariant bijection between the set S(P, \overline{S}) of all S ∈ S(G) with Z_{G}(S) ⊂ P such that P → P/U induces an isomorphism from S to \overline{S}, and the set of all Levi subgroups L of P. It maps S to the unique Levi subgroup L_{S} containing Z_{G}(S) and L to the unique lift S_{L} of \overline{S} in L ≃ P/U, see lemma 77. In particular, S(P, \overline{S}) is a U-torsor.

5.6.4. There is a structure of affine F(P/U)-space on U\X(G),

T_{P}^{∞}X(G) = (U\X(G), +, T_{P}^{∞}X(−)) .

The P/U-equivariant apartment map is defined by

T_{P}^{∞}X(\overline{S}) = U\ ∪_{S∈S(P,\overline{S})} X(S)
= image of X(S) in U\X(G) for any S ∈ S(P, \overline{S}).

The P/U-equivariant pull map takes x ∈ U\X(G) and H ∈ F(P/U) to

x + H = image of x_{L} + H_{L} in U\X(G)

where L is any Levi subgroup of P: if L′ is another one, there is a unique u ∈ U such that Int(u)(L) = L′. Then x_{L′} = ux_{L}, H_{L′} = uH_{L}, thus x_{L′} + H_{L′} = u(x_{L} + H_{L}) and x_{L}+H_{L} have the same image in U\X(G). This defines an affine F(P/U)-space: for any \overline{S} ∈ S(P/U), x ↦ x_{L} and H ↦ H_{L} yield bijections T_{P}^{∞}X(\overline{S}) → X(S_{L}) and F(\overline{S}) → F(S_{L}), thus + indeed induces a structure of affine F(\overline{S})-space on T_{P}^{∞}X(\overline{S}).

5.6.5. There is a P/U-invariant distance on T_{P}^{∞}X(G), given by the formulas:

d(x, y) = d(x_{L}, y_{L})
= lim_{t→∞} d(x_{0} + tF, y_{0} + tF)
= inf {d(x′, y′) : (x′, y′) ∈ X(G)^{2}, (x′, y′) ↦ (x, y)}

In the first formula, L is any Levi subgroup of P. In the second formula, F ∈ F(G) belongs to the facet F^{−1}(P) and (x_{0}, y_{0}) ∈ X(G)^{2} lifts (x, y). The three formulas agree: writing d_{i} for the function defined by the i-th formula, first note that d_{2} is well-defined (by NE), independent of the chosen lift (by UN), and not greater than d(x_{0}, y_{0}) (by NE). Therefore d_{2}(x, y) ≤ d_{3}(x, y) ≤ d_{1}(x, y). But

d_{1}(x, y) = d ( lim_{t→∞} ((x_{0} + tF) + tF_{L}^{ι}) , lim_{t→∞} ((y_{0} + tF) + tF_{L}^{ι}))
= lim_{t→∞} d (((x_{0} + tF) + tF_{L}^{ι}) , ((y_{0} + tF) + tF_{L}^{ι}))
≤ lim_{t→∞} d (x_{0} + tF, y_{0} + tF) = d_{2}(x, y)

by NE, so that indeed d = d_{1} = d_{2} = d_{3}. It is obviously a P/U-invariant distance, and it restricts to a Euclidean norm on any apartment T_{P}^{∞}X(\overline{S}). The projection

Gr_{P}^{∞} : X(G) ↠ T_{P}^{∞}X(G)

is non-expanding and restricts to an isometry on X(L).

5.6.6. If T_{P}^{∞}X(G) satisfies R(s) and (X(G), d) is a CAT (0)-metric space, then so is (T_{P}^{∞}X(G), d). Indeed, it is a geodesic space by R(s), so it remains to establish the CAT (0)-inequality. Let thus v, v_{0}, v_{1} be three points of T_{P}^{∞}X(G) and choose \overline{S} ∈ S(P/U) such that v_{0}, v_{1} belong to T_{P}^{∞}X(\overline{S}). Lift \overline{S} to S ∈ S(P, \overline{S}) and v_{i} to x_{i} ∈ X(S), and lift v to x ∈ X(G). For u ∈ [0, 1], let x_{u} = (1−u)x_{0}+ux_{1} be the point at distance ud(x_{0}, x_{1}) from x_{0} on the segment [x_{0}, x_{1}] of X(S). Fix F ∈ F(S) with P_{F} = P. Then for every t ≥ 0, x_{u,t} = x_{u} + tF is the point at distance ud(x_{0,t}, x_{1,t}) from x_{0,t} on the segment [x_{0,t}, x_{1,t}] of X(S) and v_{u} = Gr_{P}^{∞}(x_{u}) is the point at distance ud(v_{0}, v_{1}) from v_{0} on the segment [v_{0}, v_{1}] of T_{P}^{∞}X(\overline{S}). Set x_{t} = x + tF. By the CAT (0)-inequality in X(G),

d(x_{t}, x_{u,t})^{2} ≤ (1 − u) · d(x_{t}, x_{0,t})^{2} + u · d(x_{t}, x_{1,t})^{2} − u(1 − u) · d(x_{0,t}, x_{1,t})^{2}.

Taking the limit as t → ∞ gives

d(v, v_{u})^{2} ≤ (1 − u) · d(v, v_{0})^{2} + u · d(v, v_{1})^{2} − u(1 − u) · d(v_{0}, v_{1})^{2}

which is the CAT (0)-inequality for T_{P}^{∞}X(G).

Remark 123. Suppose that for any x, y ∈ X(G) and F ∈ F(G) there is an S ∈ S(G) such that x + tF and y + tF belong to X(S) for t ≫ 0. Then also F ∈ F(S) and the axiom R(s) holds for T_{P}^{∞}X(G). If X(G) satisfies UN^{+}, this condition on X(G) is actually equivalent to the axiom R(s) for T_{P}^{∞}X(G).

5.6.7. We shall always equip F(P/U) with the scalar product, distance, norm… which are induced by the representation Gr_{F}^{•}(τ) of P/U. Here F is any filtration in the facet F^{−1}(P), and we view Gr_{F}^{•}(τ) = ⊕_{γ}Gr_{F}^{γ}(τ) as a representation of P/U. If L is a Levi subgroup of P and G is the corresponding splitting of F, the restriction of τ to L splits as τ|L = ⊕τ_{γ} with V (τ_{γ}) = G_{γ}(τ) and the isomorphism L ≃ P/U maps τ_{γ} to the representation Gr_{F}^{γ}(τ) of P/U, thus τ|L to Gr_{F}^{•}(τ). In particular, Gr_{F}^{•}(τ) is indeed a faithful representation of P/U and its isomorphism class Gr_{P}^{•} (τ) does not depend upon F. It follows from this conventions that the isomorphism F(L) ≃ F(P/U) is compatible with the scalar products, distances, norms… which are induced on F(L) and F(P/U) by the chosen faithful representation τ of G.

5.6.8. If T_{P}^{∞}X(G) satisfies L(s), then d(x, x + H) = ∥H∥ for x ∈ T_{P}^{∞}X(G) and H ∈ F(P/U). Indeed, choose \overline{S} ∈ S(P/U) with x ∈ T_{P}^{∞}X(\overline{S}) and H ∈ F(\overline{S}). Then x_{L} ∈ X(S_{L}) and H_{L} ∈ F(S_{L}), thus also x_{L} + H_{L} ∈ X(S_{L}). In particular, (x + H)_{L} = x_{L} + H_{L}, thus d(x, x + H) = d(x_{L}, x_{L} + H_{L}) = ∥H_{L}∥ = ∥H∥: if the affine F(P/U)-space T_{P}^{∞}X(G) actually is an affine F(P/U)-building, its “quotient” distance defined above agrees with its “building” distance defined in section 5.2.9.

5.6.9. Suppose again that (X(G), d) is a CAT (0)-space. The Busemann scalar product of section 5.5.8 on X(G)^{2} × F(G), namely

⟨\overrightarrow{xy}, F⟩ = ∥F∥ · lim_{t→∞} (d(x, z + tF) − d(y, z + tF))

induces a function on T_{P}^{∞}X(G)^{2} × F^{−1}(P). Indeed for u, v ∈ U and F ∈ F^{−1}(P),

lim_{t→∞} (d(ux, z + tF) − d(vy, z + tF)) = lim_{t→∞} (d(x, u^{−1}z + tF) − d(y, v^{−1}z + tF))
= lim_{t→∞} (d(x, z + tF) − d(y, z + tF))

by the triangle inequality and the axiom UN for X(G). If T_{P}^{∞}X(G) moreover satisfies R(s), the resulting function depends only on the image \overline{F} = Gr_{P} F of F in

G(\overline{R}(P)) = G(Z(P/U)) ⊂ F(P/U).

In fact, it is simply the corresponding Busemann scalar product

⟨\overrightarrow{xy}, \overline{F}⟩ = ∥\overline{F}∥ · lim_{t→∞} (d(x, z + t\overline{F}) − d(y, z + t\overline{F}))

on the CAT(0)-space (T_{P}^{∞}X(G), d). Indeed, pick \overline{S} ∈ S(P/U) with x, z ∈ T_{P}^{∞}X(\overline{S}). Then x_{L}, z_{L} ∈ X(S_{L}), F equals (\overline{F})_{L} and belongs to F(S_{L}), z_{L} + tF belongs to X(S_{L}), therefore (z + t\overline{F})_{L} = (z_{L} + t(\overline{F})_{L})_{L} = (z_{L} + tF)_{L} = z_{L} + tF, and finally

d(x, z + t\overline{F}) = d(x_{L}, z_{L} + tF)

for all t ≥ 0, which proves our claim since also ∥F∥ = ∥(\overline{F})_{L}∥ = ∥\overline{F}∥. However, z ↦ z + t\overline{F} is now a Clifford translation of T_{P}^{∞}X(G) [8, II 6.14]: we have just seen that it corresponds to z_{L} ↦ z_{L} + tF on the isometric space X(L), and the latter map is non-expanding by NE with non-expanding inverse z_{L} ↦ z_{L} + tF_{L}^{ι}. It is therefore an isometry, and a Clifford translation since d(z_{L}, z_{L} + tF) = t ∥F∥ for all z_{L} ∈ X(L). If F ≠ 0, it now follows from [8, II 6.15] that the comparison angle

t ↦ ∡_{x}^{c}(y, x + t\overline{F})

is constant. Taking z = x in the defining formula for ⟨\overrightarrow{xy}, \overline{F}⟩, we thus find that

⟨\overrightarrow{xy}, \overline{F}⟩ = d(x, y) · ∥F∥ · cos ( lim_{t→∞} ∡_{x}^{c}(y, x + t\overline{F}))

equals

⟨\overrightarrow{xy}, \overline{F}_{x}⟩ = d(x, y) · ∥F∥ · cos (lim_{t→0} ∡_{x}^{c}(y, x + t\overline{F})) .

Note that using [8, I.1.16] again, this is also equal to

⟨loc_{x}^{a}(y), \overline{F}_{x}⟩ = d(x, y) · ∥F∥ · cos (lim_{t→0} ∡_{x}^{c}(y_{t}, x + t\overline{F}))

5.7. EXAMPLE: F(G) AS A TIGHT AFFINE F(G)-BUILDING 116

where y_{t} = ty + (1 − t)x is the point at distance td(x, y) from x on the segment [x, y] of the CAT (0)-space T_{P}^{∞}X(G). In any case, we obtain yet another series of formulas for the relevant Busemann function on T_{P}^{∞}X(G) or X(G).

Remark 124. If the affine F(P/U)-space T_{P}^{∞}X(G) is an affine F(P/U)-building, we may also directly apply lemma 121 to G = \overline{F} = Gr_{P} (F), thereby obtaining

⟨\overrightarrow{xy}, F⟩ = ⟨loc_{Gr_{P}^{∞}(x)}^{a} (Gr_{P}^{∞}(y)) , loc_{Gr_{P}^{∞}(x)} (Gr_{P} (F))⟩

where the second scalar product is in the tangent space T_{Gr_{P} (x)} (T_{P}^{∞}X(G)).

5.6.10. If X(L) + F(L) ⊂ X(L), then X(L) inherits from X(G) a structure of affine F(L)-space, and the restriction of Gr_{P}^{∞} : X(G) → T_{P}^{∞}X(G) to X(L) is an isomorphism of affine F(L)-spaces – viewing T_{P}^{∞}X(G) as an affine F(L)-space through F(L) ≃ F(P/U). Most of the above discussion then becomes much easier.

5.7. Example: F(G) as a tight affine F(G)-building

5.7.1. Recall from example 5.2.4 that (F(G), +, F(−)) is a discrete affine F(G)-space with trivial type. Under the identification F(G) = F(ω_{K}^{∘}), the pull map may be computed as follows: for F_{1}, F_{2} ∈ F(G), ρ ∈ Rep^{∘}(G)(K) and γ ∈ ℝ,

(F_{1} + F_{2})^{γ}(ρ) = ∑_{γ_{1}+γ_{2}=γ} F_{1}^{γ_{1}}(ρ) ∩ F_{2}^{γ_{2}}(ρ).

We have already mentioned that F(G) satisfies L(s) = R(s) by theorem 85 and L(i) = R(i) by corollary 86. Actually for F, G ∈ F(G), choosing S in S(G) with F, G ∈ F(S), we find using 4.1.10 that F + ηG belongs to a fixed closed chamber of F(S) for all sufficiently small η ≥ 0, from which the stronger axiom L(s)^{+} easily follows. We have also seen in example 5.4.5 that F(G) satisfies the axiom ST. It thus satisfies UN^{+} and UN by lemma 116, and it is therefore a (tight) affine F(G)- building by proposition 106. It also trivially satisfies the axiom HA, because every apartment contains the origin 0 ∈ F(G). The latter is fixed by G, and it follows from lemma 118 that F(G) is, up to isomorphism, the unique affine F(G)-building with a point fixed by G. Indeed, any such building has trivial type and satisfies ST_{2}^{−}, thus also ST^{−} by lemma 116.

5.7.2. The retractions of corollary 88 and proposition 101 agree, and so do the decompositions of sections 2.2.13 and 5.2.19 (with base point 0 ∈ F(G)).

5.7.3. The distance d = d_{τ} of section 4.2.10 is equal to the corresponding distance on the affine F(G)-building F(G) defined in section 5.2.9. For F, G ∈ F(G) and t ∈ [0, 1], the unique point at distance td(F, G) from F on the segment [F, G] in the CAT(0)-space (F(G), d) is equal to the sum tG + (1 − t)F, as defined above, of the rescaled filtrations tG and (1 − t)F of F(G): this is obvious in any apartment.

5.7.4. For a parabolic subgroup P = U ⋊ L of G with unipotent radical U and Levi subgroup L, there is a commutative diagram

\xymatrix{ & F(L) \ar@{^{(}->}[d]^{ι_{L,G}} \ar[ddr]^{≃} \ar[ddl]_{≃}_{ι_{L}} \\  & F(G) \ar@{->>}[u]^{r_{P,L}} \ar@{->>}[dl]_{_{Gr_{P}}} \ar@{->>}[dr]^{_{Gr{∞ \atop P}}} \\ F(P/U) \ar[rr]^{≃} &  & T_{P}^{∞}F(G)}

where ι_{L} : F(L) ≃ F(P/U) and ι_{L,G} : F(L) ↪ F(G) are the L-equivariant maps functorially induced by the isomorphism L ≃ P/U and the embedding L ↪ G, r_{P,L} : F(G) ↠ F(L) is the U-invariant L-equivariant retraction of corollary 88, Gr_{P} is the P-equivariant morphism of section 2.3.4 (which is defined on the whole of F(G) by theorem 85) and Gr_{P}^{∞} : F(G) ↠ T_{P}^{∞}F(G) is the P-equivariant quotient map onto T_{P}^{∞}F(G) = U\F(G). Since Gr_{P} is P-equivariant (thus U-invariant) and Gr_{P} ∘ ι_{L,G} = ι_{L}, also Gr_{P} = ι_{L} ∘ r_{P,L}. The right hand side triangles are plainly commutative, and this implies the existence of the bottom map bijection. One checks easily that it is an isomorphism of affine F(P/U)-spaces. In particular: T_{P}^{∞}F(G) is an affine F(P/U)-building, its “quotient” and “building” metric agree by 5.6.8, thus F(P/U) → T_{P}^{∞}F(G) is an isometry while Gr_{P} : F(G) ↠ F(P/U) is non-expanding. This gives the following formula: for any F, G_{1}, G_{2} ∈ F(G),

lim_{t→∞} d (G_{1} + tF, G_{2} + tF) = d (Gr_{F}(G_{1}), Gr_{F}(G_{2})) ≤ d(G_{1}, G_{2})

where Gr_{F} = Gr_{P_{F}} : F(G) ↠ F(P_{F}/U_{F}). Also,

⟨Gr_{P} (G_{1}), Gr_{P} (G_{2})⟩ ≥ ⟨G_{1}, G_{2}⟩
∡ (Gr_{P} (G_{1}), Gr_{P} (G_{2})) ≤ ∡ (G_{1}, G_{2})

since Gr_{P} contracts the distances and preserves the norms.

Remark 125. Here is a more direct proof of the fact that Gr_{P} is non-expanding: starting with G_{1}, G_{2} ∈ F(G), cut the segment [G_{1}, G_{2}] along its facet decomposition, going from H_{0} = G_{1} to H_{n} = G_{2} with F constant on ]H_{i−1}, H_{i}[. Then observe that Gr_{P} restrict to an isometry on any facet F^{−1}(Q), Q ∈ P(G): it restricts to an isometry on any apartment containing F^{−1}(P), and there is at least one such apartment which also contains F^{−1}(Q) (along with its closure). Thus

d (G_{1}, G_{2}) = ∑_{i=1}^{n}d (H_{i−1}, H_{i})
= ∑_{i=1}^{n}d (Gr_{P} (H_{i−1}), Gr_{P} (H_{i}))
≥ d (Gr_{P} (G_{1}), Gr_{P} (G_{2}))

by the triangle inequality in F(P/U). One can also probably establish the inequalities using the explicit formulas for the scalar products, but this involves playing around with three filtrations. In any case, these approaches do not yield an exact formula relating the distances on F(P/U) and F(G).

5.7.5. For F, G ∈ F(G), choose S ∈ S(G) with F, G ∈ F(S). Then by 4.1.10, there is a facet F ⊂ F(S) of F(G) such that F +tG belongs to F for every sufficiently small t > 0. If P ⊃ Z_{G}(S) is the corresponding parabolic subgroup, then P ⊂ P_{F} since F belongs to the closure of F, thus also U_{F} ⊂ U where U is the unipotent radical of P. In particular, F + tuG = u(F + tG) = F + tG for every u ∈ U_{F}, thus

loc_{F} : F(G) ↠ T_{F}F(G) G ↦ germ of (t ↦ F + tG)

is U_{F}-invariant. Since Gr_{F} induces a bijection U_{F}\F(G) ≃ F(P_{F}/U_{F}), it follows that there is a canonical P_{F}-equivariant commutative diagram

\xymatrix{ & F(G) \ar@{->>}[dl]_{_{Gr_{F}}} \ar@{->>}[dr]^{_{loc_{F}}} \\ F(P_{F}/U_{F}) \ar[rr]^{φ} &  & T_{F}F(G)}

For S ∈ S(G) with F ∈ F(S), it restrict to a commutative diagram of isometries

\xymatrix{ & (F(S), d) \ar[dl]_{≃} \ar[dr]^{≃} \\ (F(\overline{S}), d) \ar[rr]^{≃} &  & (T_{F}F(S), d)}

where \overline{S} is the image of S in P_{F}/U_{F}. Since any two elements x, y ∈ F(P_{F}/U_{F}) are contained in one such F(\overline{S}), it follows that φ : F(P_{F}/U_{F}) → T_{F}F(G) is an isometry. It is therefore also compatible with the relevant norms, angles and scalar products. This gives the following explicit formulas:

d (Gr_{F}(G_{1}), Gr_{F}(G_{2})) = lim_{t→0} \frac{1}{t}d(G_{1} + tF, G_{2} + tF) ≤ d(G_{1}, G_{2})

and

lim_{t→0} \frac{1}{t} (d(F + G_{1}, F) − d(F + G_{1}, F + tG_{2})) = ⟨Gr_{F}(G_{1}), Gr_{F}(G_{2})⟩

for every G_{1}, G_{2} ∈ F(G), with ∥Gr_{F}(G)∥ = ∥G∥ and

⟨Gr_{F}(G_{1}), Gr_{F}(G_{2})⟩ = _{∑γ} ⟨Gr_{F}^{γ}(G_{1}, τ), Gr_{F}^{γ}(G_{2}, τ)⟩

Also: loc_{F}(G_{1}) = loc_{F}(G_{2}) if and only if U_{F} · G_{1} = U_{F} · G_{2}.

Remark 126. The previous results yield a P_{F}-equivariant isometry between the tangent space T_{F}F(G) at F viewed as a point in the affine building and the “tangent space” T_{F}^{∞}F(G) at F viewed as a boundary point. This reflects the homogeneity of the vectorial Tits building F(G): our isometry is induced by

(t small) F + tG ↦ t^{−1}(F + tG) = G + t^{−1}F (t^{−1} large)

5.7.6. There is also the localization map loc_{F}^{a} : F(G) → T_{F}F(G), which sends G to loc_{F}^{a}(G) = loc_{F}(H) if G = F + H. Define Gr_{F}^{a} = φ^{−1} ∘ loc_{F}^{a}, so that

\xymatrix{ & F(G) \ar@{->>}[dl]_{_{Gr{a \atop F}}} \ar@{->>}[dr]^{_{loc{a \atop F}}} \\ F(P_{F}/U_{F}) \ar[rr]^{φ} &  & T_{F}F(G)}

One checks easily in F(S) ∋ F, G that Gr_{F}^{a}(G) + \overline{F} = Gr_{F}(G), where

\overline{F} = Gr_{F}(F) ∈ G(Z(P_{F}/U_{F})) = Aut (F(P_{F}/U_{F}))

is the automorphism determined by F (see 2.2.6, 5.1 and 5.2.18). Thus

⟨Gr_{F}^{a}(G), Gr_{F}(H)⟩ = ⟨Gr_{F}(G), Gr_{F}(H)⟩ − ⟨\overline{F}, Gr_{F}(H)⟩
= _{∑γ} ⟨Gr_{F}^{γ}(G, τ), Gr_{F}^{γ}(H, τ)⟩ − γ deg(Gr_{F}^{γ}(H, τ))

This gives an explicit formula for

lim_{t→0} \frac{1}{t} (d(G, F) − d(G, F + tH)) = ⟨Gr_{F}^{a}(G), Gr_{F}(H)⟩ .

5.7.7. It is finally very easy to compute the Busemann functions:

b_{0,G}(F) = lim_{t→∞} (d(F, tG) − t ∥G∥) = − ∥F∥ cos ∡(F, G).

It follows that for any F, G, H ∈ F(G),

⟨\overrightarrow{FG}, H⟩ = ⟨G, H⟩ − ⟨F, H⟩ .

Thus if C is a closed convex subset of F(G), F ∈ C is the convex projection of some G ∈ F(G) and H ∈ F(G) satisfies F + tH ∈ C for all sufficiently t > 0, then

⟨G, H⟩ ≤ ⟨F, H⟩ .

5.8. Example: a symmetric space

Let K = ℝ and G = GL(V), where V is an ℝ-vector space of dimension n ∈ ℕ.

5.8.1. By corollary 65, the tautological representation V of G identifies G(G) and F(G) with the sets G(V) and F(V) of all ℝ-graduations and ℝ-filtrations on V. Similarly, the action of G on the set ℙ^{1}(V) of ℝ-lines in V identifies S(G) with

S(V) = {S ⊂ ℙ^{1}(V) : V = ⊕_{L∈S}L} .

We denote by F(S) the apartment of F(V) corresponding to S ∈ S(V). An ℝfiltration F on V thus belongs to F(S) if and only if

∀γ ∈ ℝ : F^{γ} = ⊕_{L∈S, F^{♯}(L)≥γ}L where F^{♯}(L) = sup{λ : L ⊂ F^{λ}}.

We also identify C(G) with ℝ_{≤}^{n} = {γ_{1} ≤ ⋯ ≤ γ_{n} : γ_{i} ∈ ℝ} by the map which sends t(F) to \underline{t}(F) = (t_{i}(F))_{i=1}^{n}, with ♯\{i : t_{i}(F) = γ\} = dim_{ℝ} Gr_{F}^{γ}(V) for γ ∈ ℝ. The dominance order on C(G) defined in section 5.1 corresponds to

(γ_{i})_{i=1}^{n} ≤ (γ′_{i})_{i=1}^{n} ⟺ \begin{cases}∑_{j=1}^{n} γ_{j} = ∑_{j=1}^{n} γ′_{j} & and \\ ∑_{j=i}^{n} γ_{j} ≤ ∑_{j=i}^{n} γ′_{j} & for 2 ≤ i ≤ n. \end{cases}

The length ∥−∥ : C(G) → ℝ_{+} attached to the tautological faithful representation V of G in 4.2.2 corresponds to the function ∥−∥ : ℝ_{≤}^{n} → ℝ_{+} given by

∥γ_{1} ≤ ⋯ ≤ γ_{n}∥ = \sqrt{γ_{1}^{2} + ⋯ + γ_{n}^{2}}.

5.8.2. The exponential exp : ℝ → ℝ^{×} defines an ℝ-valued section exp of the multiplicative group 𝔻(ℝ), whose evaluation at the character γ ∈ ℝ is given by

γ(exp) = exp(γ) ∈ ℝ^{×}.

For G ∈ G(V), we denote by G^{♭} the endomorphism of V which acts by γ ∈ ℝ on the direct summand G_{γ} of V. Viewing G as an ℝ-morphism 𝔻(ℝ) → G, we thus have exp(G^{♭}) = G(exp) in G = G(ℝ). The maps G ↦ G^{♭} ↦ exp(G^{♭}) yield G-equivariant bijections between G(V), the set of diagonalizable endomorphisms of V and the set of diagonalizable elements of G with positive eigenvalues.

5.8.3. Let B(V) be the space of all Euclidean norms α on V, i.e. functions

α : V → ℝ_{+}

whose square α^{2} is a positive definite quadratic form on V – thus B(V) may also be viewed as the space of all scalar products on V, or as the space of all ellipsoids in V. The group G acts transitively on B(V) by (g · α)(v) = α(g^{−1}v) and the stabilizer of α is the orthogonal group O(α) ⊂ G, thus B(V) ≃ G/O(α) is a smooth variety. We denote by G(V, α) ⊂ G(V) the set of all α-orthogonal ℝ-graduations on V, by Sym(V, α) = G(V, α)^{♭} the set of all α-symmetric endomorphisms of V, and by G(α) = exp(Sym(V, α)) the set of all α-symmetric automorphisms of V with positive eigenvalues. Thus Sym(V, α) is the tangent space of B(V) at α and the polar decomposition in G yields G = G(α) · O(α). For F ∈ F(V) and γ ∈ ℝ, we denote by G_{α}(F)_{γ} the α-orthogonal complement of F_{+}^{γ} in F^{γ}. Then G_{α}(F) is the unique splitting of F in G(V, α) and G_{α} : F(V) → G(V, α) is an O(α)-equivariant section of Fil : G(V) ↠ F(V). We obtain a sequence of O(α)-equivariant bijections

\xymatrix{F(V) \ar[r]^{G_{α}} & G(V, α) \ar[r]^{♭} & Sym(V, α) \ar[r]^{exp} & G(α) \ar[r]^{−·α} & B(V).}

We set g_{α}(F) = exp(G_{α}(F)^{♭}) = G_{α}(F)(exp) ∈ G(α) and define

α + F = g_{α}(F) · α in B(V).

Thus for any α ∈ B(V), F ∈ F(G) and v ∈ V,

(α + F)(v) = α (∑_{γ}e^{−γ}v_{γ}) : v = _{∑γ}v_{γ}, v_{γ} ∈ G_{α}(F)_{γ}.

For S ∈ S(V), we denote by B(S) the set of α’s in B(V) for which V = ⊕_{L∈S}L is an orthogonal decomposition. Thus for α ∈ B(S) and F ∈ F(S), we find that

(α + F)^{2}(v) = _{∑L∈S}(α + F)^{2}(v_{L}) = _{∑L∈S}(e^{−F^{♯}(L)}α)^{2}(v_{L})

where v = ∑_{L∈S} v_{L} with v_{L} ∈ L, therefore also α + F ∈ B(S).

5.8.4. The above formulas show that B(V) = (B(V), B(−), +) is an affine F(V)-space. It is well-known that it satisfies R(s), and L(s) follows from the existence of the α-orthogonal splittings. Moreover for any α ∈ B(V), the pull map

F(V) → B(V), F ↦ α + F

is an O(α)-equivariant bijection. The Fischer-Courant theory tells us that the orbits of the diagonal action of G on B(V) × B(V) are classified by a G-equivariant map

\underline{d} : B(V) × B(V) → ℝ_{≤}^{n}

whose i-th component d_{i} : B(V) × B(V) → ℝ is given by

d_{i}(α, β) = − log (max \{min \{\frac{β(x)}{α(x)} : x ∈ W \ \{0\}\} : W ⊂ V, dim_{ℝ} W = i\}) .

Suppose that α, β ∈ B(S) ∩ B(S′) and choose ℝ-basis e = (e_{i})_{i=1}^{n} and e′ = (e′_{i})_{i=1}^{n} of V such that S = {ℝe_{i} : i = 1, ⋯, n}, S′ = {ℝe′_{i} : i = 1, ⋯, n}, e and e′ are orthonormal for α, and β(e_{1}) ≥ ⋯ ≥ β(e_{n}), β(e′_{1}) ≥ ⋯ ≥ β(e′_{n}). Then necessarily

∀i ∈ {1, …, n} : β(e_{i}) = exp (−d_{i}(α, β)) = β(e′_{i})

The element g ∈ G mapping e to e′ satisfies gS = S′, gα = α and gβ = β, which proves R(i). The resulting vectorial distance d equals \underline{d} under the identification C(G) ≃ ℝ_{≤}^{n}, i.e. for every α ∈ B(V) and F ∈ F(V), \underline{d}(α, α + F) = \underline{t}(F). Indeed for any X ∈ Sym(V, α) and β = exp(X) · α, we have

\underline{d}(α, β) = (γ_{1}, ⋯, γ_{n}) in ℝ_{≤}^{n}

where γ_{1} ≤ ⋯ ≤ γ_{n} are the eigenvalues of X counted with multiplicities.

5.8.5. Define d^{i}(α, β) = ∑_{j=0}^{i−1} d_{n−j}(α, β), so that
d^{i}(α, β) = max {d^{i}(α|W, β|W) : W ⊂ V, dim_{ℝ} W = i}
= log max \{\frac{Λ^{i}(α)(v)}{Λ^{i}(β)(v)} : v ∈ Λ^{i}(V) \ \{0\}\}

where Λ^{i}(α) is the Euclidean norm on Λ^{i}(V) induced by α. We have

d^{n}(α, β) = log (\frac{∫_{β(v)≤1} dv}{∫_{α(v)≤1} dv} )

for any Borel measure dv on V, therefore

d^{n}(α, γ) = d^{n}(α, β) + d^{n}(β, γ),
d^{n}(α, gα) = log |det(g)| ,
d^{n}(α, α + F) = ∑_{γ}γ dim_{ℝ} Gr_{F}^{γ}.

In particular, if d^{i}(α, γ) = d^{i}(α|W, γ|W) for some W ⊂ V, dim_{ℝ} W = i, then

d^{i}(α, γ) = d^{i}(α|W, β|W) + d^{i}(β|W, γ|W) ≤ d^{i}(α, β) + d^{i}(β, γ)

i.e. \underline{d} satisfies the triangle inequality T R.

5.8.6. We next show that for any α ∈ B(V) and F, G ∈ F(V),

2 · \underline{d}(α + F, α + G) ≤ \underline{d}(α + 2F, α + 2G) in ℝ_{≤}^{n}.

Put f = g_{α}(F), g = g_{α}(G). Then f^{2} = g_{α}(2F), g^{2} = g_{α}(2G) and we have to show

2 · \underline{d}(fα, gα) ≤ \underline{d}(f^{2}α, g^{2}α).

Let h ↦ h^{∗} be the involution of G defined by α, so that f^{∗} = f, g^{∗} = g and f^{−2}g^{2} is conjugated to gf^{−2}g = (gf^{−1})(gf^{−1})^{∗}. For 1 ≤ i ≤ n and h ∈ G, write λ_{i}(h) for the largest real eigenvalue of h acting on Λ^{i}(V) and denote by ⟨−, −⟩_{α,i} the scalar product on Λ^{i}(V) attached to its Euclidean norm Λ^{i}(α). Then

exp (d^{i}(f^{2}α, g^{2}α)) = max \{\frac{Λ^{i}(α)(f^{−2}v)}{Λ^{i}(α)(g^{−2}v)} : v ∈ Λ^{i}(V) \ \{0\}\}
= max \{\frac{Λ^{i}(α)(f^{−2}g^{2}v)}{Λ^{i}(α)(v)} : v ∈ Λ^{i}(V) \ \{0\}\}
≥ λ_{i}(f^{−2}g^{2}) = λ_{i}(gf^{−2}g)
= max \begin{cases}⟨gf^{−2}gx, x⟩ & α,i \\ ⟨x, x⟩ & α,i \end{cases} : x ∈ Λ^{i}(V) \ \{0\}\}

= max \begin{cases}⟨f^{−1}x, f^{−1}x⟩ & α,i \\ ⟨g^{−1}x, g^{−1}x⟩ & α,i \end{cases} : x ∈ Λ^{i}(V) \ \{0\}\}
= exp (2d^{i}(fα, gα))

with equality for i = n, which proves our claim. Thus \underline{d} satisfies CO′′ and CO.

5.8.7. For α ∈ B(V) and F ∈ F(V), we denote by Gr_{F}(α) the Euclidean norm on Gr_{F}^{•}(V) induced by α through the isomorphism V ≃ Gr_{F}(V) provided by the α-orthogonal splitting G_{α}(F) of F. We claim that for every α, β ∈ B(V),

lim_{t→∞} \underline{d}(α + tF, β + tF) = \underline{d}(Gr_{F}(α), Gr_{F}(β)) in ℝ_{≤}^{n}.

Indeed, choosing an isomorphism (G_{α}(F)_{γ}, α|G_{α}(F)_{γ}) ≃ (G_{β}(F)_{γ}, β|G_{β}(F)_{γ}) for every γ ∈ ℝ, we obtain an element g ∈ G which fixes F and maps α to β. It then also maps α + tF = g_{α}(tF) · α to β + tF = gg_{α}(tF) · α, so that

\underline{d}(α + tF, β + tF) = \underline{d} (α, g_{α}^{−1}(tF)gg_{α}(tF) · α) .

Let L_{α}(F) be the centralizer of G_{α}(F), so that P_{F} = U_{F} ⋊ L_{α}(F). Write g = u · ℓ with u ∈ U_{F} and ℓ ∈ L_{α}(F), so that g_{α}^{−1}(tF)gg_{α}(tF) = g_{α}^{−1}(tF)ug_{α}(tF) · ℓ. Let then 𝔲_{F} = ⊕_{γ>0}𝔲_{γ} be the weight decomposition of 𝔲_{F} = Lie(U_{F})(ℝ) induced by

ad ∘ G_{α}(F) : 𝔻(ℝ) → G → GL(𝔤)

where 𝔤 = Lie(G)(ℝ). Then g_{α}(tF) acts on 𝔲_{γ} by exp(tγ), from which easily follows that g_{α}^{−1}(tF)ug_{α}(tF) converges to 1 in U_{F} (for the real topology). It follows that

lim_{t→∞} \underline{d}(α + tF, β + tF) = \underline{d} (α, ℓα) = \underline{d} (Gr_{F}(α), Gr_{F}(β)) .

Taking β = uα with u ∈ U_{F}, we obtain UN. On the other hand for any β, since

ℝ_{+} ∋ t ↦ \underline{d}(α + tF, β + tF) ∈ ℝ_{≤}^{n}

is convex and bounded, it is non-increasing, which proves NE.

5.8.8. Let d = ∥\underline{d}∥ be the G-invariant distance on B(V) attached to the faithful representation V of G, as in 5.2.9. We claim that the metric space (B(V), d) is CAT (0). In particular, it is uniquely geodesic, thus B(V) also satisfies UG. To establish our claim, fix α ∈ B(V), choose an α-orthonormal basis (e_{i})_{i=1}^{n} of V and use it to identify G with GL(n, ℝ), Sym(V, α) with the vector space S(n, ℝ) of symmetric matrices in M(n, ℝ) and G(α) with the open cone P (n, ℝ) ⊂ S(n, ℝ) of positive definite matrices. Let ⟨−, −⟩ be the scalar product on V attached to α and g ↦ g^{∗} the corresponding involution of G. For p ∈ G(α), g ∈ G and v ∈ V, set g · p = gpg^{∗} and α_{p}(v) = ⟨pv, v⟩^{1/2}. This defines an action of G on G(α) and p ↦ α_{p} is an isomorphism of differentiable manifold G(α) → B(V) such that

α_{g·p} = (g^{∗})^{−1} · α_{p} and α + F = α_{g_{α}(F)^{−2}} in B(V)

for any g ∈ G, p ∈ G(α) and F ∈ F(V). In [8, II.10.31], G(α) is equipped with a G-invariant Riemannian structure. Let d_{α} be the corresponding G-invariant Riemannian metric on G(α) or B(V). For X ∈ Sym(V, α) and p = exp(X) ∈ G(α),

d_{α}^{2}(α, α_{p}) = Tr(X^{2})

by [8, II.10.42.(2)]. Thus for any F ∈ F(V),

d_{α}^{2}(α, α + F) = 4Tr ((G_{α}(F)^{♭})^{2}) = 4 ∥F∥^{2} = 4d^{2}(α, α + F)

since α + F = α_{p} with p = exp(−2G_{α}(F)^{♭}) in G(α). Therefore d_{α}(α, β) = 2d(α, β) for any β ∈ B(V) by R(s) and d_{α} = 2d on B(V) since G acts transitively on B(V). Since the metric space (B(V), d_{α}) is CAT (0) by [8, II.10.39], so is (B(V), d).

5.8.9. We have thus established that (B(V), B(−), +) is an affine F(V)- building. If S ∈ S(G) corresponds to S ∈ S(V), the type map ν_{B,S} : S → G(S) maps s ∈ S to the unique morphism 𝔻_{ℝ}(ℝ) → S whose composite with the character χ_{L} through which S acts on L ∈ S is the character log |χ_{L}(s)| ∈ ℝ of 𝔻(ℝ).

5.8.10. The computations of section 5.8.7 show that Gr_{F} induces an isometry

Gr_{F} : T_{P_{F}}^{∞} B(V) ≃ _{∏γ}B (Gr_{F}^{γ}(V)) .

On the other hand, the tangent space T_{α}B(V) as defined in section 5.5.4 is equal to the corresponding tangent space Sym(V, α) of the differential manifold B(V), its scalar product is given by ⟨X, Y ⟩ = Tr(XY) and the localization map

loc_{α} : F(V) ↠ T_{α}B(V)

maps F ∈ F(V) to the α-symmetric endomorphism

loc_{α}(F) = (\frac{d}{dt}g_{α}(tF))_{t=0} = G_{α}(F)^{♭} in Sym(V, α).

CHAPTER 6 Bruhat-Tits buildings

We first keep the assumptions and notations of the previous chapter. Thus G will be a reductive group over a field K, G = G(K) and Γ = (ℝ, +, ≤). In addition, we assume that K is equipped with a non-trivial, non-archimedean absolute value

|−| : K → ℝ^{+}.

However, we will eventually return to the setting of chapter 4, with G a reductive group over the valuation ring O_{K} = {x ∈ K : |x| ≤ 1} of K and Γ = ℝ. Note that then P(G) = P(G_{K}), F(G) = F(G_{K}) but S(G) ⊊ S(G_{K}) and G(G) ⊊ G(G_{K}).

6.1. The Bruhat-Tits building of GL(V)

6.1.1. Let G = GL(V), where V ≠ 0 is a K-vector space of dimension n ∈ ℕ. As in section 5.8, we thus have G-equivariant bijections

S(G) ≃ S(V) = {S ⊂ ℙ^{1}(V)(K) : V = ⊕_{L∈S}L} ,
F(G) ≃ F(V) = {ℝ − filtrations on V } ,
G(G) ≃ G(V) = {ℝ − graduations on V } .

6.1.2. A K-norm (or simply: norm) on V is a function α : V → ℝ^{+} such that (1) α(v) = 0 if and only if v = 0,

(2) α(λv) = |λ| α(v) for every λ ∈ K and v ∈ V, and

(3) α(u + v) ≤ max {α(u), α(v)} for every u, v ∈ V. The K-norm α is split by S ∈ S(V) if and only if

∀v ∈ V : α(v) = max {α(v_{L}) : L ∈ S} where v = ∑_{L∈S}v_{L}, v_{L} ∈ L.

It is splittable if it is split by S for some S ∈ S(V). If K is locally compact, every K-norm on V is splittable by [22, Proposition 1.1].

6.1.3. We denote by B(V) the set of all splittable K-norms on V, by B(S) the subset of all K-norms split by S. We let G act on B(V) by (g ·α)(v) = α(g^{−1}v), and define the pull map + : B(V) × F(V) → B(V) by

(α + F)(v) = min {max {e^{−γ}α(v_{γ}) : γ ∈ ℝ} : v = _{∑γ∈ℝ}v_{γ}, v_{γ} ∈ F^{γ}}

where the sums ∑_{γ∈ℝ} v_{γ} have finite support. We have to verify that this operation is well-defined. Note first that the axiom L(s) follows from the second proof of [11, 1.5.ii]: for any α ∈ B(V) and F ∈ F(V), there is an S ∈ S(V) with α ∈ B(S) and F ∈ F(S). Let us then identify F(S) with ℝ^{S} by F ↦ F^{♯} where

F^{♯}(L) = max {γ ∈ ℝ : L ⊂ F^{γ}} , F^{γ} = ⊕_{L:F^{♯}(L)≥γ}L.

Then for v = ∑_{L∈S} v_{L} in V = ⊕_{L∈S}L, we find that

inf \{max \{e^{−γ}α(v_{γ}) : γ ∈ ℝ\} | \begin{matrix}v=∑_{γ} v_{γ} \\ v_{γ}∈F^{γ} \end{matrix} \} = max \{e^{−F^{♯}(L)}α(v_{L}) : L ∈ S\} .

6.1. THE BRUHAT-TITS BUILDING OF GL(V) 125

Indeed for v = ∑_{γ} v_{γ} with v_{γ} = ∑_{L} v_{γ,L}, v_{γ,L} ∈ L and v_{γ,L} = 0 if γ > F^{♯}(L),

max {e^{−γ}α(v_{γ}) : γ ∈ ℝ} = max {e^{−γ}α(v_{γ,L}) : γ ∈ ℝ, L ∈ S}
≥ max {e^{−F^{♯}(L)}α(v_{γ,L}) : γ ∈ ℝ, L ∈ S}
≥ max {e^{−F^{♯}(L)}α(v_{L}) : L ∈ S}

since α ∈ B(S) (for the first equality) and v_{L} = ∑_{γ} v_{γ,L} (for the last inequality), which provides the non-trivial required inequality in the displayed formula. Thus

(6.1.1)  (α + F)(v) = max {e^{−F^{♯}(L)}α(v_{L}) : L ∈ S}

from which follows that α + F is well-defined and again belongs to B(S).

6.1.4. The apartment and pull maps are plainly G-equivariant, and the above formula shows that the latter turns B(S) into an affine F(S)-space, thus

B(V) = (B(V), +, B(−))

is an affine F(G)-space. If S ∈ S(G) corresponds to S ∈ S(V), the type map

ν_{B,S} : S → G(S)

maps s to the unique F ∈ F(S) with γ_{L}(F) = log |χ_{L}(s)| for all L ∈ S, where χ_{L} : S → 𝔾_{m,k} is the character through which S acts on L.

6.1.5. In [37, §3], Parreau shows that the closely related set Δ = ℝ_{+}^{×}\B(V) is an affine building in the sense of [37, 1.1] (see also [11, 22]). The axioms R(s), R(i)^{+}, HA and L(s)^{+} for B(V) respectively follow from the axioms A3, A2, A5 and proposition 1.8 for Δ in [37]. For α ∈ B(V), F ∈ F(V) and u ∈ U_{F}, pick S ∈ S(V) such that α ∈ B(S) and F ∈ F(S) using L(s). Write S = {Kv_{1}, ⋯, Kv_{n}} with i ↦ γ_{i} = F^{♯}(Kv_{i}) non-increasing and identify B(S) with ℝ^{n} by α ↦ (α_{1}, ⋯, α_{n}) where α_{i} = − log(α(v_{i})). Then for t ≥ 0, tF ∈ F(S) acts on B(S) ≃ ℝ^{n} by

(α_{1}, ⋯, α_{n}) ↦ (α_{1} + tγ_{1}, ⋯, α_{n} + tγ_{n})

and the matrix (u_{i,j}) of u ∈ U_{F} in the basis (v_{1}, ⋯, v_{n}) of V satisfies u_{i,i} = 1 and u_{i,j} ≠ 0 if and only if γ_{i} > γ_{j} for i ≠ j. Moreover, u fixes α if and only if α_{j} − α_{i} ≤ − log |u_{i,j}| for all 1 ≤ i, j ≤ n by [37, 3.5]. Therefore u fixes α + tF for all t ≫ 0, i.e. B(V) satisfies UN^{+}. Thus by proposition 106, B(V) is an affine F(G)-building whose underlying metric space is CAT (0).

6.1.6. Let Z ≃ 𝔾_{m,K} be the center of G, so that G(Z) ≃ ℝ by the isomorphism which maps G : 𝔻_{K}(ℝ) → Z to the unique weight G^{♯} ∈ ℝ of the corresponding representation of 𝔻_{K}(ℝ) on V. For S ∈ S(G) corresponding to S ∈ S(V), the projection from F(S) = G(S) to G(Z) then maps F ∈ F(S) to the unique G with

G^{♯} = \frac{1}{n} ∑_{L∈S}F^{♯}(L).

It follows that the projection

d^{c} : B(V) × B(V) → G(Z)

of the distance d : B(V) × B(V) → C(G) maps (α, β) to the unique G with

G^{♯} = \frac{1}{n} ∑_{i=1}^{n} log α(v_{i}) − log β(v_{i})

6.2. THE BRUHAT-TITS BUILDING OF G 126

for any K-basis (v_{1}, ⋯, v_{n}) of V such that α, β ∈ B(S) with S = {Kv_{1}, ⋯, Kv_{n}}. From [37, 3.2], we then deduce that the morphism

ν_{B}^{c} : G → G(Z)

maps g to the unique G with G^{♯} = \frac{1}{n} log |det g|. In particular, |det G_{α}| = 1 for every α ∈ B(V), and then [37, 3.5] implies ST: G_{α} = G_{S,α} for all α ∈ B(S), S ∈ S(G). Therefore B(V) is a tight affine F(G)-building.

6.1.7. If the valuation of K is discrete, the map

α ↦ L = {x ∈ V : α(x) ≤ 1}

identifies the subset B^{∘}(V) ⊂ B(V) of K-norms α on V such that

α(V \ {0}) = |K^{×}|

with the set L(V) of O_{K}-lattices in V. Then B^{∘}(V) is stable under the action of G and of the subset F^{log|K^{×}|}(V) ⊂ F(V) of log |K^{×}|-filtrations on V. It is then convenient to either normalize the valuation by requiring that |K^{×}| = e^{ℤ}, or to rescale the pull map as in 5.2.20. Then F^{ℤ}(V) acts on B^{∘}(V) ≃ L(V) by

L + F = _{∑i∈ℤ}π^{−i}L ∩ F^{i}

for L ∈ L(V) and F ∈ F^{ℤ}(V), where π ∈ O_{K} is any uniformizer.

6.1.8. The space B(V) is known to be a realization of the Bruhat-Tits building of G; for a more general case, see [11].

6.2. The Bruhat-Tits building of G

6.2.1. For a general reductive group G over K, we have to make some assumption on the triple (G, K, |−|): the existence of a valuation on the root datum (Z_{G}(S), (U_{a})_{a∈Φ(G,S)}) of G = G(K), in the sense of [9, 6.2.1]. Here S is a fixed element of S(G) and the notations are taken from section 5.4.1.

6.2.2. Let then B^{r}(G) and B^{e}(G) = B^{r}(G) × G(Z) be respectively the reduced and extended Bruhat-Tits buildings of G, as defined in [9, §7] and [10, 4.2.16 & 5.1.29]. These two sets have compatible actions of G, they are covered by apartments B^{r}(S) and B^{e}(S) = B^{r}(S) × G(Z) which are G-equivariantly parametrized by S(G), B^{e}(S) is an affine F(S)-space on which N_{G}(S) acts by affine transformations with linear part ν_{S}^{v} : N_{G}(S) ↠ W_{G}(S) and the resulting action of Z_{G}(S) is given by a morphism ν_{B,S} : Z_{G}(S) → G(S) which is uniquely characterized by the following property: for every morphism χ : Z_{G}(S) → 𝔾_{m,K}, the induced morphism

G(χ|_{S}) ∘ ν_{B,S} : Z_{G}(S) ⟶ G(𝔾_{m,K})

maps z in Z_{G}(S) to log |χ(z)| in ℝ = G(𝔾_{m,K}). Similarly, the action of G on G(Z) is given by a morphism ν_{B}^{c} : G → G(Z) which is uniquely characterized by the following property: for every morphism χ : G → 𝔾_{m,K}, the induced morphism

G(χ|Z) ∘ ν_{B}^{c} : G → G(𝔾_{m,K})

maps g in G to log |χ(g)| in ℝ = G(𝔾_{m,K}). There is a G-invariant distance

d : B^{e}(G) × B^{e}(G) → ℝ^{+}

inducing a Euclidean distance on each apartment, which turns B^{e}(G) into a CAT(0)- space. Finally, B^{e}(G) already satisfies our axiom R(s) by [9, 7.4.18.i] as well as the following strengthening of ST and R(i)^{+}:

For every subset Ω ≠ ∅ of B^{e}(S), the pointwise stabilizer G_{Ω} ⊂ G of Ω equals G_{S,Ω} by [9, 7.4.4], and it acts transitively on the set of apartments containing Ω by [9, 7.4.9].

We denote by +_{S} : B^{e}(S) × F(S) → B^{e}(S) the given structure of affine F(S)-space on B^{e}(S). These maps are compatible in the following sense:

g · (x +_{S} F) = (g · x +_{g·S} g · F).

6.2.3. Let us first prove L(s): starting with x ∈ B^{e}(G) and F ∈ F(G), choose a minimal parabolic subgroup B′ ⊂ P_{F} with Levi Z_{G}(S′), pick c ∈ B^{e}(S′) and form the sector C′ = c′+_{S′} F^{−1}(B′) in B^{e}(S′). By [9, 7.4.18.ii], there is another apartment B^{e}(S) containing x and a subsector C of C′, which a priori is of the form C = c +_{S} F^{−1}(B) for some minimal parabolic B with Levi Z_{G}(S). Since C ⊂ B^{e}(S) ∩ B^{e}(S′), there is a g ∈ G fixing C and mapping S to S′. Then g ∈ B since G_{C} = G_{S,C} ⊂ B ∩ G_{c}, thus Int(g)(B) = B and Z_{G}(S′) ⊂ B. Moreover

C = gC = g(c +_{S} F^{−1}(B)) = c +_{S′} F^{−1}(B)

thus actually B = B′ because

C = c +_{S′} F^{−1}(B) ⊂ c′ +_{S′} F^{−1}(B′) = C′

in the affine F(S′)-space B^{e}(S′). Now x ∈ B^{e}(S) and also F ∈ F(S) since

Z_{G}(S) ⊂ B = B′ ⊂ P_{F}.

This proves L(s). Note also that for any G ∈ F(S)∩F(S′) in the closure of F^{−1}(B),

c +_{S} G = g(c +_{S} G) = c +_{S′} G

since g fixes c, c +_{S} G and G. In particular, c +_{S} tF = c +_{S′} tF for all t ∈ ℝ^{+}.

6.2.4. Suppose now that x ∈ B^{e}(S_{1}) ∩ B^{e}(S_{2}) and F ∈ F(S_{1}) ∩ F(S_{2}) with S_{1}, S_{2} ∈ S(G). We now show that x +_{S_{1}} F = x +_{S_{2}} F in B^{e}(G). For t ≥ 0, put

x_{i}(t) = x +_{S_{i}} tF in B^{e}(S_{i}).

The CAT (0)-property of d implies that t ↦ d(x_{1}(t), x_{2}(t)) is a convex function, and it is therefore sufficient to show that it is also bounded. Let us choose minimal parabolic subgroups Z_{G}(S_{i}) ⊂ B_{i} ⊂ P_{F}, and form the corresponding sectors

C_{i} = x +_{S_{i}} F^{−1}(B_{i}) in B^{e}(S_{i}).

By [9, 7.4.18.iii], there is another apartment B^{e}(S) which contains subsectors

C′_{1} ⊂ C_{1} and C′_{2} ⊂ C_{2}.

We have just seen that then Z_{G}(S) ⊂ B_{i} (thus F ∈ F(S)) and C′_{i} = y_{i} +_{S} F^{−1}(B_{i}) in B^{e}(S) for some y_{i}’s in B^{e}(S), with moreover

y_{i}(t) = y_{i} +_{S} tF = y_{i} +_{S_{i}} tF

for t ≥ 0. Then t ↦ d(y_{1}(t), y_{2}(t)) and t ↦ d(x_{i}(t), y_{i}(t)) are constant by elementary computations in B^{e}(S) and B^{e}(S_{i}) respectively, thus t ↦ d(x_{1}(t), x_{2}(t)) is indeed bounded by the triangle inequality in (B^{e}(G), d).

6.2.5. We may at last define our pull map: for x ∈ B^{e}(G) and F ∈ F(G), choose S ∈ S(G) with x ∈ B^{e}(S) and F ∈ F(S) and set x + F = x +_{S} F in B^{e}(G): this does not depend upon the chosen S. Our pull map is plainly G-equivariant, and induces the given structure of affine F(S)-space on B^{e}(S). Therefore

B^{e}(G) = (B^{e}(G), +, B^{e}(−))

is an affine F(S)-space.

6.2.6. For x ∈ B^{e}(G) and F, G ∈ F(G), choose S ∈ S(G) with x ∈ B^{e}(S), F ∈ F(S), let F be the “facet” in B^{e}(S) denoted by γ(x, E) in [5, 7.2.4] with E = {tF : t > 0}, let C be a “chamber” of B^{e}(S) containing F in its closure. Using [9, 7.4.18.ii] as above, we find that there is an apartment B^{e}(S′) containing C with G ∈ F(S′). It then also contains F by [9, 7.4.8], which means that for some ϵ > 0, it contains x + ηG for every η ∈ [0, ϵ]: this proves L(s)^{+}.

6.2.7. We already have the axioms R(s), R(i)^{+}, L(s)^{+} and ST, thus B^{e}(G) is a tight affine F(G)-building by proposition 106 and lemma 116. Note that the axiom HA also holds for B^{e}(G) by [37, 1.4] and [9, 7.4.19]. If G = GL(V), then B^{e}(G) ≃ B(V) by lemma 118 (see also [22, 11, 37]).

6.2.8. The CAT (0)-distance d used above may be chosen to be one of our d_{τ} ’s, for some faithful representation τ of G. The affine F(G)-space B^{e}(G) is discrete when (K, |−|) is discrete, in which case (B^{e}(G), d) is a complete metric space by lemma 114 or [9, 2.5.12]. If (K, |−|) is complete, then every geodesic ray or line in B^{e}(G) is contained in some apartment by [39, 2.3.8] and R(s). Thus with the notations of section 5.5.1, F(G) ≃ C(∂B^{e}(G)) if (K, |−|) is complete.

6.2.9. The Bruhat-Tits building B^{e}(G) = B^{e}(G, |−|) depends upon the choice of the valuation |−| on K, and so does its structure of affine F(G)-building. However for ν > 0 , there is a G(K)-equivariant commutative diagram

B^{e}(G, |−|) × F(G_{K}) ⟶^{+} B^{e}(G, |−|)
a ↓ b ↓ a ↓
B^{e}(G, |−|^{ν}) × F(G_{K}) ⟶^{+} B^{e}(G, |−|^{ν})

where a is a canonical G(K)-equivariant map and b(F) = νF.

6.3. Functoriality for Bruhat-Tits buildings

6.3.1. Suppose for this section that the valuation ring of (K, |−|), namely

O_{K} = {x ∈ K : |x| ≤ 1}

is Henselian. Then for every algebraic extension L of K, there is a unique absolute value |−| : L → ℝ^{+} on L which extends |−| : K → ℝ^{+}, and its valuation ring

O_{L} = {x ∈ L : |x| ≤ 1}

is the integral closure of O_{K} in L, also Henselian. We say that L/K has a geometric property P over O_{K} if the corresponding morphism Spec(O_{L}) → Spec(O_{K}) does.

Proposition 127. Let G be a reductive group over O_{K}.

(1) There is an extension L/K, finite étale and Galois over O_{K}, splitting G.

(2) The Bruhat-Tits building B^{e}(G_{K}) exists and contains a canonical point

∘_{G,K}^{e} = ∘_{G}^{e} = (∘_{G}^{r}, 0) ∈ B^{e}(G_{K}) = B^{r}(G_{K}) × G(Z(G_{K}))

with stabilizer G(O_{K}) in G(K). The projection ∘_{G}^{r} of ∘_{G}^{e} is the unique fixed point of G(O_{K}) in B^{r}(G_{K}) if the residue field of O_{K} is neither 𝔽_{2} nor 𝔽_{3}.

(3) The apartments of B^{e}(G_{K}) containing ∘_{G}^{e} are the B^{e}(S_{K})’s for S ∈ S(G).

Proof. Let S be a maximal split torus of G and let T be a maximal torus of Z_{G}(S) [1, XIV 3.20]. Then G and T are isotrivial by proposition 48, thus split by a finite étale cover of Spec(O_{K}) which we may assume to be connected and Galois, i.e. of the form Spec(O_{L}) → Spec(O_{K}) where O_{L} is the normalization of O_{K} in a finite étale Galois extension L/K over O_{K} by [28, 18.10.12]. Since O_{K} is Henselian, O_{L} is also the valuation ring of (L, |−|). Let (x_{α}) be a Chevalley system for (G_{O_{L}}, T_{O_{L}}), as defined in [21, XXIII 6.2], giving rise to a Chevalley valuation φ_{L} for G_{L}, as explained in [9, 6.2.3.b] and [10, 4.2.1], thus also to the reduced Bruhat-Tits building B^{r}(G_{L}) with its distinguished apartment B^{r}(T_{L}) and the distinguished point ∘_{G}^{r} ≡ φ_{L} in B^{r}(T_{L}), as defined in [9, §7]. For f = 0, the group schemes 𝔊_{f}^{0} ⊂ 𝔊_{f} ⊂ 𝔊̂_{f} ⊂ 𝔊_{f}^{†} constructed in [10, 4.3-6] are all equal to G_{O_{L}} [10, 4.6.22]. Thus by [10, 4.6.28], G(O_{L}) is the stabilizer of the distinguished point ∘_{G}^{e} = (∘_{G}^{r}, 0) of B^{e}(T_{L}) ⊂ B^{e}(G_{L}) in G(L), and ∘_{G}^{r} is the unique fixed point of G(O_{L}) in B^{r}(G_{L}) by [10, 5.1.39] if the residue field of O_{L} is not equal to 𝔽_{2} or 𝔽_{3}, which we can always assume.

The pair (G_{K}, K) satisfies the conditions of the pair denoted by (H, K^{♮}) in [10, 5.1.1]. The Galois group Σ = Gal(L/K) acts compatibly on G(L) and B^{e}(G_{L}). It therefore fixes ∘_{G}^{e}, which thus belongs to B^{e}(T_{L})^{Σ} = B^{e}(S_{K}). Applying this to Z_{G}(S) instead of G, we see that (G_{K}, K) also satisfies the assumption (DE) of [10, 5.1.5]. Then by [10, 5.1.20], the valuation φ_{L} descends to a valuation φ for G_{K}. The corresponding building B^{e}(G_{K}) is the fixed point set of Σ in B^{e}(G_{L}) by [10, 5.1.25]. The stabilizer of ∘_{G}^{e} ∈ B^{e}(G_{K}) in G(K) equals G(O_{K}) = G(K) ∩ G(O_{L}) and again by [10, 5.1.39], ∘_{G}^{r} is the unique fixed point of G(O_{K}) in B^{r}(G_{K}) if the residue field of O_{K} is not equal to 𝔽_{2} or 𝔽_{3}. By construction, ∘_{G}^{e} belongs to B^{e}(S_{K}). Therefore [9, 7.4.9] proves our last claim, since G(O_{K}) also acts transitively on S(G). □

6.3.2. We denote by B^{e}(G, K, |−|) the pointed affine F(G_{K})-building

B^{e}(G, K, |−|) = (B^{e}(G_{K}), ∘_{G}^{e})

attached to a reductive group G over O_{K}. It easily follows from [10, 5.1.41] that this construction is functorial in the Henselian pair (K, |−|). More precisely, let HV be the category whose objects are pairs (K, |−|) where K is a field and |−| : K → ℝ^{+} is a non-trivial, non-archimedean absolute value whose valuation ring O_{K} is Henselian. Then for every morphism f : (K, |−|) → (L, |−|) in HV and every reductive group G over O_{K}, there is a canonical morphism f : B^{e}(G_{K}) → B^{e}(G_{L}) such that

f(∘_{G}^{e}) = ∘_{G}^{e}, f(gx) = f(g)f(x) and f(x + F) = f(x) + f(F)

for every x ∈ B^{e}(G_{K}), g ∈ G(K) and F ∈ F(G_{K}). The first and last property already determine this morphism uniquely: by the axiom T (s) for B^{e}(G_{K}), any element x of B^{e}(G_{K}) equals ∘_{G}^{e} + F for some for F ∈ F(G_{K}).

Remark 128. The above functoriality amounts to saying that the mapping

B^{e}(G_{K}) ∋ ∘_{G}^{e} + F ↦ ∘_{G}^{e} + f(F) ∈ B^{e}(G_{L})

is well-defined and equivariant with respect to G(K) → G(L). This indeed implies the equivariance with respect to f : F(G_{K}) → F(G_{L}) as follows. For S ∈ S(G) mapping into S′ ∈ S(G_{O_{L}}), the above mapping restricts to a well-defined map B^{e}(S_{K}) → B^{e}(S′_{L}) which is equivariant with respect to f : F(S_{K}) → F(S′_{K}); by the axiom L(s) for B^{e}(G_{K}) and proposition 79, any pair (x, F) in B^{e}(G_{K})×F(G_{K}) is conjugated by some g ∈ G(K) to one in B^{e}(S_{K}) × F(S_{K}), thus

f (x + F) = f(g^{−1})f (gx + gF) = f(g^{−1}) (f(gx) + f(gF)) = f(x) + f(F).

Theorem 129. The pointed affine F(G)-building B^{e}(G, K, |−|) is also functorial in the reductive group G over O_{K}: for every morphism f : G → H of reductive groups over O_{K}, there is a unique morphism f : B^{e}(G_{K}) → B^{e}(H_{K}) such that

f(∘_{G}^{e}) = ∘_{H}^{e}, f(gx) = f(g)f(x) and f(x + F) = f(x) + f(F)

for every x ∈ B^{e}(G_{K}), g ∈ G(K) and F ∈ F(G_{K}).

This essentially follows from Landvogt’s work in [32], which has no assumptions on the reductive groups over K but requires (K, |−|) to be quasi-local, in particular discrete. The main difficulty there is the construction of base points with good properties, which is here trivialized by the given points ∘_{G}^{e} and ∘_{H}^{e}. Note that again, the uniqueness of f : B^{e}(G_{K}) → B^{e}(H_{K}) follows from the first and last displayed requirements, and its existence amounts to showing that the mapping

B^{e}(G_{K}) ∋ ∘_{G}^{e} + F ↦ ∘_{H}^{e} + f(F) ∈ B^{e}(H_{K})

is well-defined and equivariant with respect to f : G(K) → H(K). Given the identification B^{e}(GL(V)) ≃ B(V), this theorem is closely related to the Tannakian theorem 132 below. We will prove the former as a corollary of the latter.

6.3.3. Assuming theorem 129, we may work out an analog of the discussion of section 5.7.4 for the pointed Bruhat-Tits building B^{e}(G, K). First, recall that P(G_{K}) = P(G) since ℙ(G) is projective over O_{K}. Let thus P ∈ P(G) be a parabolic subgroup of G with unipotent radical U. For every Levi subgroup L of P, there is a canonical commutative diagram

\xymatrix{ & B^{e}(L, K) = B^{e}(L_{K}) \ar@{^{(}->}[d]^{ι_{L,G}} \ar[ddr]^{≃} \ar[ddl]_{≃} \\  & B^{e}(G, K) = B^{e}(G_{K}) \ar@{->>}[u]^{r_{P,L}} \ar@{->>}[dl]_{_{Gr_{P}}} \ar@{->>}[dr]^{_{Gr{∞ \atop P}}} \\ B^{e}(P/U, K) \ar[rr]_{≃}^{ψ} &  & T_{P}^{∞}B^{e}(G_{K})}

where ι_{L} : B^{e}(L, K) ≃ B^{e}(P/U, K) and ι_{L,G} : B^{e}(L, K) ↪ B^{e}(G, K) are the L(K)-equivariant maps functorially induced by L ≃ P/U and L ↪ G, r_{P,L} is the U(K)-invariant, L(K)-equivariant retraction of proposition 101 onto the image ∪_{S∈S(L_{K})}B^{e}(S) of ι_{L,G}, Gr_{P} = ι_{L} ∘ r_{P,L} is a P (K)-equivariant map, and the right hand side triangle comes from 5.6.4. Both Gr_{P} and Gr_{P}^{∞} identify there codomain with U(K)\B^{e}(G_{K}), which yields the existence and unicity of the P (K)-equivariant bijection ψ : B^{e}(P/U, K) ≃ T_{P}^{∞}B^{e}(G_{K}) at the bottom of our diagram. Neither ψ nor Gr_{P} depends upon the choice of L: if L′ is another Levi subgroup of P, there is a u ∈ U(O_{K}) such that L′ = uLu^{−1}. The automorphism Int(u) : G → G then induces by functoriality a commutative diagram

\xymatrix{B^{e}(G, K) \ar[d]^{Int(u)} \ar@{->>}[r]^{r_{P,L}} & ∪_{S∈S(L_{K})}B^{e}(S) \ar[d]^{Int(u)} & B^{e}(L, K) \ar[l]_{ι_{L,G}} \ar[d]^{Int(u)} \ar[r]^{ι_{L}} & B^{e}(P/U, K) \ar[d]^{Id} \\ B^{e}(G, K) \ar@{->>}[r]^{r_{P,L′}} & ∪_{S′∈S(L′_{K})}B^{e}(S′) & B^{e}(L′, K) \ar[l]_{ι_{L′,G}} \ar[r]^{ι_{L′}} & B^{e}(P/U, K)}

The first vertical map is also equal to the multiplication by u map on B^{e}(G, K):

Int(u)(∘_{G}^{e} + F) = ∘_{G}^{e} + uF = u(∘_{G}^{e} + F)

for all F ∈ F(G) since u ∈ G(O_{K}) fixes ∘_{G}^{e}. Thus Gr_{P} and ψ indeed do not depend upon the choice of L. One checks easily that ψ is an isomorphism of affine F(P_{K}/U_{K})-spaces. In particular: T_{P}^{∞}B^{e}(G_{K}) is an affine F(P_{K}/U_{K})-building, its “quotient” and “building” metric agree by 5.6.8, thus ψ : B^{e}(P/U, K) → T_{P}^{∞}B^{e}(G_{K}) is an isometry while Gr_{P} : B^{e}(G, K) ↠ B^{e}(P/U, K) is non-expanding when everyone is equipped with the metrics induced by a chosen faithful representation τ of G_{K}. This gives the following formula: for every x, y ∈ B^{e}(G_{K}) and F ∈ F(G),

lim_{t→∞} d_{τ} (x + tF, y + tF) = d_{Gr_{F}^{•} (τ)}(Gr_{F}(x), Gr_{F}(y)) ≤ d_{τ} (x, y)

where Gr_{F} = Gr_{P_{F}} : B^{e}(G, K) ↠ B^{e}(P_{F}/U_{F}, K). Also:

⟨\overrightarrow{xy}, F⟩ = ⟨\overrightarrow{Gr_{F}(x)Gr_{F}(y)}, \overline{F}⟩

for every x, y ∈ B^{e}(G, K), with \overline{F} = Gr_{F}(F) in G(Z(P_{F}/U_{F})) = G(\overline{R}(P_{F})).

6.3.4. We may also establish some partial functoriality results when no base point is given, as in Landvogt’s work. Fix a quasi-local (discrete, Henselian) pair (K, |−|). For any reductive group G over K, there is a finite Galois extension L of K splitting G such that the reduced building B^{r}(G_{L}) contains a special point ∘ fixed by Gal(L/K). Indeed, let first L_{1} be a finite Galois extension of K splitting G, and choose a facet F of B^{r}(G_{L_{1}}) fixed by Gal(L_{1}/K), for instance one which intersects B^{r}(G_{K}). Then the barycenter ∘ of F is also fixed by Gal(L_{1}/K), and just like any barycenter of a facet of the Bruhat-Tits building of a split group, it becomes special over a sufficiently ramified extension L of L_{1}, which we may assume to be Galois over K. Write ∘_{G}^{e} = (∘, 0) for the corresponding Gal(L/K)-invariant point of B^{e}(G_{L}) and let G_{∘} be the reductive group over O_{L} with generic fiber G_{L} such that G_{∘}(O_{L}) is the stabilizer of ∘_{G}^{e} in G(L). Since ∘_{G}^{e} is fixed by Gal(L/K), the Hopf O_{L}-sub-algebra A(G_{∘}) of A(G_{L}) = A(G)_{L} is fixed by the action of Gal(L/K).

Let now τ be a finite dimensional K-representation of G, corresponding to a morphism f : G → H, with H = GL(V), V = V (τ). By [44, 1.5], every finitely generated O_{L}-submodule M of V_{L} is contained in some A(G_{∘})-sub-comodule F of V_{L} which is finitely generated (hence free) over O_{L}. Since A(G_{∘}) is flat over O_{L}, there is a smallest such F, which we denote by F (M). Since A(G_{∘}) is stabilized by Gal(L/K), the map M ↦ F (M) is Gal(L/K)-equivariant. Thus starting with a Gal(L/K)-stable O_{L}-lattice M of V_{L}, for instance the base change of an O_{K}- lattice of V, we obtain an O_{L}-model f : G_{∘} → H_{∘} of f_{L} : G_{L} → H_{L}, with H_{∘} = GL(F (M)), such that the point ∘_{H}^{e} = (∘, 0) corresponding to H_{∘} in B^{e}(H_{L}) is also fixed by Gal(L/K). Applying now the previous functoriality results to this O_{L}-morphism f : G_{∘} → H_{∘}, we obtain: for every extension (L′, |−|) of (L, |−|) in HV, there is a unique morphism f : B^{e}(G_{L′}) → B^{e}(H_{L′}) such that

f(∘_{G}^{e}) = ∘_{H}^{e}, f(gx) = f(g)f(x) and f(x + F) = f(x) + f(F)

for every x ∈ B^{e}(G_{L′}), g ∈ G(L′) and F ∈ F(G_{L′}). Moreover, for every K-linear morphism σ : (L′, |−|) → (L′′, |−|) between two such extensions,

f(σx) = σf(x) in B^{e}(G_{L′′})

for every x ∈ B^{e}(G_{L′}). Indeed if x = ∘_{G}^{e} + F with F ∈ F(G_{L′}), then

f(σx) = f(σ ∘_{G}^{e} +σF) = f(∘_{G}^{e} + σF) = ∘_{H}^{e} + f(σF)
= σ ∘_{H}^{e} +σf(F) = σ(∘_{H}^{e} + f(F)) = σf(x).

6.4. A Tannakian formalism for Bruhat-Tits buildings

6.4.1. Let again (K, |−|) be a field with a non-trivial, non-archimedean absolute value |−| : K → ℝ^{+}, with valuation ring O_{K} and residue field k. We denote by Norm^{∘}(K, |−|) the category whose objects are pairs (V, α) where V is a finite dimensional K-vector space and α : V → ℝ^{+} is a splittable K-norm on V. A morphism f : (V, α) → (V′, α′) is a K-linear morphism f : V → V′ such that α′(f(x)) ≤ α(x) for every x ∈ V. This defines an O_{K}-linear rigid ⊗-category with neutral object 1_{K} = (K, |−|). The ⊗-products, inner homs and duals

(V_{1}, α_{1}) ⊗ (V_{2}, α_{2}) = (V_{1} ⊗ V_{2}, α_{1} ⊗ α_{2})
Hom ((V_{1}, α_{1}) , (V_{2}, α_{2})) = (Hom (V_{1}, V_{2}) , Hom (α_{1}, α_{2}))
(V, α)^{∗} = (V^{∗}, α^{∗})

are respectively given by : α_{1} ⊗ α_{2} = Hom(α_{1}^{∗}, α_{2}) under V_{1} ⊗ V_{2} = Hom(V_{1}^{∗}, V_{2}),

Hom (α_{1}, α_{2}) (f) = sup \{\frac{α_{2}(f(x))}{α_{1}(x)} : x ∈ V_{1} \ \{0\}\} ,
α^{∗}(f) = sup \{\frac{|f(x)|}{α(x)} : x ∈ V \ \{0\}\} .

In addition, Norm^{∘}(K, |−|) is an exact category in Quillen’s sense: a short sequence

(V_{1}, α_{1}) ⟶^{f_{1}} (V_{2}, α_{2}) ⟶^{f_{2}} (V_{3}, α_{3})

is exact precisely when the underlying sequence of K-vector spaces is exact and

α_{1}(x) = α_{2}(f_{1}(x)), α_{3}(z) = inf {α_{2}(y) : y ∈ f_{2}^{−1}(z)}

for every x ∈ V_{1} and z ∈ V_{3}. For γ ∈ ℝ and (V, α) ∈ Norm^{∘}(K, |−|), we set

B(α, γ) = {x ∈ V : α(x) < exp(−γ)}}
\overline{B}(α, γ) = \{x ∈ V : α(x) ≤ exp(−γ)\}\}

These are O_{K}-submodules of V and the functors (V, α) ↦ B(α, γ) are easily seen to be exact. However, (V, α) ↦ \overline{B}(α, γ) is also exact, because in fact every exact sequence in Norm^{∘}(K) is split by [11, 1.5.ii + Appendix]! If M is an O_{K}-lattice in V (by which we mean a finitely generated, thus free, O_{K}-submodule spanning V), we denote by α_{M} the splittable K-norm on V with \overline{B}(α_{M}, 0) = M defined by

α_{M}(x) = inf {|λ| : λ ∈ K, x ∈ λM} = min {|λ| : λ ∈ K, x ∈ λM} .

6.4. A TANNAKIAN FORMALISM FOR BRUHAT-TITS BUILDINGS 133

6.4.2. For (K, |−|) → (L, |−|), there is an exact O_{K}-linear ⊗-functor
− ⊗ L : Norm^{∘}(K, |−|) → Norm^{∘}(L, |−|)

defined by (V, α) ⊗ L = (V_{L}, α_{L}) where V_{L} = V ⊗ L and

α_{L}(v) = inf {max{|x_{k}| α(v_{k})} : v = ∑v_{k} ⊗ x_{k}, v_{k} ∈ V, x_{k} ∈ L} ,
= min {max{|x_{k}| α(v_{k})} : v = ∑v_{k} ⊗ x_{k}, v_{k} ∈ V, x_{k} ∈ L} .

For (V, α) ∈ Norm^{∘}(K, |−|), γ ∈ ℝ and x ∈ V,

B(α_{L}, γ) = B(α, γ) ⊗ O_{L}, \overline{B}(α_{L}, γ) = \overline{B}(α, γ) ⊗ O_{L} and α = α_{L}|V. If M is an O_{K}-lattice in V, then α_{M,L} = α_{M⊗O_{L}}.

6.4.3. We shall also consider the category Norm′(K) whose objects are triples (V, α, M) where (V, α) is an object of Norm^{∘}(K) and M is an O_{K}-lattice in V, with the obvious morphisms. It is again an O_{K}-linear ⊗-category. The formula

loc^{γ}(V, α, M) = image of \overline{B}(α, γ) ∩ M in M_{k} = M ⊗_{O_{K}} k

defines an O_{K}-linear ⊗-functor with values in Fil(k) = Fil^{ℝ}LF(k),

loc : Norm′(K) → Fil(k).

Indeed by the axiom R(s) for B(V), there is an O_{K}-basis (e_{1}, ⋯, e_{n}) of M adapted to α, thus α(∑ x_{i}e_{i}) = max {|x_{i}| e^{−γ_{i}}} where γ_{i} = − log α(e_{i}) and

loc^{γ}(V, α, M) = ⊕_{γ_{i}≥γ}ke_{i}

from which easily follows that loc is well-defined and compatible with ⊗-products.

6.4.4. For an extension (K, |−|) → (L, |−|) and a reductive group G over O_{K}, we denote by B′(ω_{G}^{∘}, L, |−|) or simply B′(ω_{G}^{∘}, L) the set of all factorizations

Rep^{∘}(G)(O_{K}) ⟶^{α} Norm^{∘}(L, |−|) ⟶^{forg} Vect(L)

of the fiber functor ω_{G,L}^{∘} through an O_{K}-linear ⊗-functor α. For τ ∈ Rep^{∘}(G)(O_{K}) and α ∈ B′(ω_{G}^{∘}, L), we denote by α(τ) the corresponding L-norm on V_{L}(τ).

6.4.5. For g ∈ G(L) and F ∈ F(G_{L}), the following formulas

(g · α)(τ) = τ_{L}(g) · α(τ) and (α + F)(τ) = α(τ) + F(τ)

respectively define an action of G(L) on B′(ω_{G}^{∘}, L) and a G(L)-equivariant map

+ : B′(ω_{G}^{∘}, L) × F(G_{L}) → B′(ω_{G}^{∘}, L). 6.4.6. We define the canonical L-norm α_{G,L} on ω_{G,L}^{∘} by the formula α_{G,L}(τ) = α_{V_{O_{L}} (τ)} = α_{V (τ),L}.

By propositions 45 and 48, G(O_{L}) is the stabilizer of α_{G,L} in G(L). We set

B(ω_{G}^{∘}, L) =^{def} α_{G,L} + F(G_{L}).

This is a G(O_{L})-stable subset of B′(ω_{G}^{∘}, L) equipped with a G(O_{L})-equivariant map can : F(G_{L}) ↠ B(ω_{G}^{∘}, L), can(F) = α_{G,L} + F.

6.4.7. Any L-norm α on ω_{G,L}^{∘} induces an O_{K}-linear ⊗-functor
α′ : Rep^{∘}(G)(O_{K}) → Norm′(L)

by the formula α′(τ) = (V_{L}(τ), α(τ), V_{O_{L}}(τ)), thus also an O_{K}-linear ⊗-functor

loc(α) : Rep^{∘}(G)(O_{K}) → Fil(k_{L}), loc(α) = loc ∘ α′

where k_{L} is the residue field of O_{L}. We may thus define

B^{?}(ω_{G}^{∘}, L) = {α ∈ B′(ω_{G}^{∘}, L) : loc(α) is exact} .

This is a G(O_{L})-stable subset of B′(ω_{G}^{∘}, L) equipped with a G(O_{L})-equivariant map

loc : B^{?}(ω_{G}^{∘}, L) → F(G_{k_{L}}).

6.4.8. All of the above constructions are functorial in G, (K, |−|) and (L, |−|), using pre- or post-composition with the obvious exact ⊗-functors

Rep^{∘}(G_{2})(O_{K}) ⟶ Rep^{∘}(G_{1})(O_{K})
Rep^{∘}(G)(O_{K_{1}}) ⟶ Rep^{∘}(G)(O_{K_{2}})
Norm^{∘}(L_{1}, |−|_{1}) ⟶ Norm^{∘}(L_{2}, |−|_{2})

for

G_{1} → G_{2}_{(K_{1}, |−|_{1}) → (K_{2}, |−|_{2})}_{(L_{1}, |−|_{1}) → (L_{2}, |−|_{2})}

Lemma 130. For any reductive group G over O_{K}, we have

B(ω_{G}^{∘}, L) ⊂ B^{?}(ω_{G}^{∘}, L) ⊂ B′(ω_{G}^{∘}, L)

and the composition loc ∘ can : F(G_{L}) → F(G_{k_{L}}) is the reduction map

F(G_{L}) ⟵^{≃} F(G_{O_{L}}) ⟶^{red} F(G_{k_{L}}).

For any S ∈ S(G_{O_{L}}), the functorial map B′(ω_{S}^{∘}, L) → B′(ω_{G}^{∘}, L) is injective.

Proof. By proposition 79, any F ∈ F(G_{L}) belongs to F(S_{L}) for some S in S(G_{O_{L}}). Pre-composing with Rep^{∘}(G)(O_{K}) → Rep^{∘}(S)(O_{L}) yields the vertical maps of the commutative diagram

\xymatrix{F(S_{L}) \ar@{->>}[r]^{can} \ar@{^{(}->}[d] & B(ω_{S}^{∘}, L) \ar@{^{(}->}[r] \ar[d] & B′(ω_{S}^{∘}, L) \ar[d] & B^{?}(ω_{S}^{∘}, L) \ar@{^{(}->}[l] \ar[r]^{loc} \ar[d] & F(S_{k_{L}}) \ar@{^{(}->}[d] \\ F(G_{L}) \ar@{->>}[r]^{can} & B(ω_{G}^{∘}, L) \ar@{^{(}->}[r] & B′(ω_{G}^{∘}, L) & B^{?}(ω_{G}^{∘}, L) \ar@{^{(}->}[l] \ar[r]^{loc} & F(G_{k_{L}})}

which reduces us to the case K = L, G = S treated below. □

Lemma 131. Suppose that G = S is a split torus. Then all maps in

\xymatrix{F(S_{L}) \ar@{->>}[r]^{can} & B(ω_{S}^{∘}, L) \ar@{^{(}->}[r] & B′(ω_{S}^{∘}, L) & B^{?}(ω_{S}^{∘}, L) \ar@{^{(}->}[l] \ar[r]^{loc} & F(S_{k_{L}})}

are isomorphisms of pointed affine G(S)-spaces. Moreover, S(L) acts on

B(ω_{S}^{∘}, L) = B^{?}(ω_{S}^{∘}, L) = B′(ω_{S}^{∘}, L)

by translations through the morphism

ν_{B,S} : S(L) → G(S)

which maps s ∈ S(L) to the unique morphism ν_{B,S}(s) : 𝔻_{O_{K}} (ℝ) → S whose composition with any character χ of S is the character log |χ(s)| ∈ ℝ of 𝔻_{O_{K}} (ℝ).

Proof. Put M = Hom(S, 𝔾_{m,O_{K}}) and let ρ_{m} be the representation of S on O_{K} given by the character m ∈ M. For τ ∈ Rep^{∘}(S)(O_{K}), let τ = ⊕τ_{m} be the weight decompositions of τ. Recall from section 3.10.6 that the formulas

F^{γ}(τ) = ⊕_{F^{♯}(m)≥γ}V (τ_{m}), F^{♯}(m) = sup{γ : F^{γ}(ρ_{m}) ≠ 0}

yield isomorphisms between F(S) = G(S) and Hom(M, ℝ). Similarly, the formulas

α(τ)(x) = max {e^{−α^{♯}(m)}α_{V_{O_{L}} (τ_{m})}(x_{m}) : m ∈ M} , α^{♯}(m) = − log α(ρ_{m})(1_{O_{K}})

where x = ∑ x_{m} is the decomposition of x in V_{L}(τ) = ⊕V_{L}(τ_{m}) yield isomorphisms between B′(ω_{S}^{∘}, L) and Hom(M, ℝ). One then checks easily that

α_{S,L}^{♯} = 0, (α + F)^{♯} = α^{♯} + F^{♯} and s · α = α + ν_{B,S}(s)

as well as loc^{γ}(α)(τ) = ⊕_{α^{♯}(m)≥λ}V_{k_{L}}(τ_{m}), from which the lemma follows. □

6.4.9. For S ∈ S(G_{O_{L}}), we identify B(ω_{S}^{∘}, L) with its image in B(ω_{G}^{∘}, L) and call it the apartment attached to S. The pull map on B′(ω_{G}^{∘}, L) thus induces a structure of affine F(S_{L})-space on B(ω_{S}^{∘}, L), and the action of G(L) on B′(ω_{G}^{∘}, L) restricts to an action of S(L) on B(ω_{S}^{∘}, L), by translations through the above morphism ν_{B,S} : S(L) → G(S_{L}).

6.4.10. We now restrict our attention to Henselian fields, so that B^{e}(G, L, |−|) is also well-defined, functorial in (L, |−|), and equal to ∘_{G}^{e} + F(G_{L}) by T (s). Given the functorial properties of B(ω_{G}^{∘}, L), theorem 129 immediately follows from:

Theorem 132. The formula ∘_{G}^{e} + F ↦ α_{G,L} + F defines a functorial bijection

α : B^{e}(G, L, |−|) → B(ω_{G}^{∘}, L, |−|)

such that for every x ∈ B^{e}(G_{L}), g ∈ G(L) and F ∈ F(G_{L}),

α(∘_{G}^{e}) = α_{G}, α(g · x) = g · α(x) and α(x + F) = α(x) + F.

Proof. Fix an extension (L, |−|) → (L′, |−|) such that G′ = G_{O_{L′}} splits and consider the following diagram, where F ∈ F(G_{L}) and F′ ∈ F(G_{L′}) = F(G′_{L′}):

\xymatrix{∘_{G}^{e} + F \ar@{.>}[d]_{?} & B^{e}(G, L) \ar@{.>}[d]_{α} \ar@{.>}[dr]^{β} \ar[rr] &  & B^{e}(G′, L′) \ar@{.>}[d]^{_{α_{′}}} \ar@{.>}[dl]_{_{β_{′}}} & ∘_{G′}^{e} + F′ \ar@{.>}[d]^{?} \\ α_{G,L} + F & B′(ω_{G}^{∘}, L) \ar[r]^{−⊗L′} & B′(ω_{G}^{∘}, L′) & B′(ω_{G′}^{∘}, L′) \ar[l]_{Res} & α_{G′,L′} + F′}

The bottom maps are respectively induced by post and pre-composition with

−⊗L′ : Norm^{∘}(L) → Norm^{∘}(L′) and −⊗O_{L′} : Rep^{∘}(G)(O_{K}) → Rep^{∘}(G′)(O_{L′}).

If α′ is well-defined and equivariant with respect to the operations of G(L′) and F(G_{L′}), so is β′. Then β is well-defined and equivariant with respect to the operations of G(L) and F(G_{L}). But B′(ω_{G}^{∘}, L) → B′(ω_{G}^{∘}, L′) is injective, thus α is also well-defined and equivariant with respect to the operations of G(L) and F(G_{L}). Its image equals B(ω_{G}^{∘}, L) by definition, which is thus stable under the operations of G(L) and F(G_{L}) on B′(ω_{G}^{∘}, L). Since loc(α_{G,L} + F) = F_{k_{L}} for every F ∈ F(G_{L}), the restriction of α to any apartment B^{e}(S_{L}) = ∘_{G}^{e} + F(S_{L}) for S ∈ S(G_{O_{L}}) is injective. Since any pair of points in B^{e}(G, L) is G(L)-conjugated to one in such an apartment by the axiom R(s) for B^{e}(G, L), α : B^{e}(G, L) → B(ω_{G}^{∘}, L) is a bijection. This reduces us to the case where G is split over O_{K} and K = L.

Suppose that ∘_{G}^{e} + F_{1} = ∘_{G}^{e} + F_{2} = x in B^{e}(G_{K}) for some F_{1}, F_{2} ∈ F(G_{K}), choose S_{i} ∈ S(G_{K}) such that F_{i} ∈ F(S_{i}) and ∘_{G}^{e} ∈ B^{e}(S_{i}) using L(s) for B^{e}(G_{K}), and then choose g ∈ G(K) fixing ∘_{G}^{e} and x such that Int(g)(S_{1}) = S_{2} using R(i) for B^{e}(G_{K}). Then S_{i} ∈ S(G) and g ∈ G(O_{K}) by proposition 127, moreover gF_{1} = F_{2} since B^{e}(S_{2}) is an affine F(S_{2})-space. Thus g(α_{G} + F_{1}) = α_{G} + F_{2} in B′(ω_{G}^{∘}, K), since G(O_{K}) fixes α_{G}. But g fixes the point x = ∘_{G}^{e} + F_{1} of B^{e}(S_{1}), thus g fixes α_{G} + F_{1} in B′(ω_{G}^{∘}, K) by lemma 133 below, therefore α_{G} + F_{1} = α_{G} + F_{2} and our map α : B^{e}(G, K) → B′(ω_{G}^{∘}, K) is indeed well-defined.

It is plainly G(O_{K})-equivariant. For any S ∈ S(G), the G(K)-equivariant map α_{S} of lemma 133 below coincides with α on B^{e}(S_{K}), thus α equals α_{S} everywhere since every point of B^{e}(G_{K}) is conjugated to one in B^{e}(S_{K}) by some element in G(O_{K}). Therefore α is G(K)-equivariant. Since every pair in B^{e}(G_{K}) × F(G_{K}) is conjugated to one in B^{e}(S_{K}) × F(S_{K}) by some element in G(K), our α is also compatible with the operations of F(G_{K}). □

Lemma 133. Suppose that G is split over O_{K} and let (K, |−|) → (L, |−|) be any extension in HV. Then for any S ∈ S(G), there is a unique map

α_{S} : B^{e}(S_{L}) → B′(ω_{S}^{∘}, L)

such that for all x ∈ B^{e}(S_{L}) and F ∈ F(S_{L}),

α_{S}(∘_{G}^{e}) = α_{G,L} and α_{S}(x + F) = α_{S}(x) + F

Moreover, it extends uniquely to a G(L)-equivariant map

α_{S} : B^{e}(G_{L}) → B′(ω_{G}^{∘}, L).

Proof. The uniqueness of both maps is obvious. Since B^{e}(S_{L}) and B(ω_{S}^{∘}, L) are affine G(S_{L})-spaces on which S(L) acts by translations through the same morphism ν_{B,S} : S(L) → G(S_{L}), the unique isomorphism of affine G(S_{L})-spaces

α_{S} : B^{e}(S_{L}) → B(ω_{S}^{∘}, L)

mapping ∘_{G}^{e} ∈ B^{e}(S_{L}) to α_{G,L} ∈ B(ω_{S}^{∘}, L) is S(L)-equivariant. Since G(O_{L}) fixes ∘_{G}^{e} ∈ B^{e}(G_{L}) and α_{G,L} ∈ B′(ω_{G}^{∘}, L), the induced embedding

α_{S} : B^{e}(S_{L}) → B′(ω_{G}^{∘}, L)

is also equivariant for the actions of N_{G}(S)(L) = N_{G}(S)(O_{L}) · S(L). To extend the latter map to a G(L)-equivariant morphism on the whole tight building B^{e}(G_{L}), it remains to establish the following claim – see remark 119:

For every x ∈ B^{e}(S_{L}), the G(L)-stabilizer of x ∈ B^{e}(G_{L}) is

contained in the G(L)-stabilizer of α_{S}(x) ∈ B′(ω_{G}^{∘}, L).

This is true for x = ∘_{G}^{e}, where both stabilizers equal G(O_{L}). This is therefore also true for any x in S(L) · ∘_{G}^{e} = ∘_{G}^{e} + ν_{B,S}(S(L)) since α_{S} is S(L)-equivariant. To clarify the proof, note that the base change maps from K to L identify

F = F(S_{K})  with  F(S_{L}) ⊂ F(G_{L})
A = B^{e}(S_{K}) with B^{e}(S_{L}) ⊂ B^{e}(G_{L})
B = B(ω_{S}^{∘}, K) with B(ω_{S}^{∘}, L) ⊂ B′(ω_{G}^{∘}, L)

and the isomorphism of affine F -space α_{S} : A → B also does not depend upon L. What depends upon L is the subset Λ(L) = ∘_{G}^{e} + ν_{B,S}(S(L)) of A on which we know the validity of our claim. So let us fix x and α = α_{S}(x) as above, as well as some g ∈ G(L) such that gx = x. By lemma 134 below, there is an extension (L, |−|) → (L′, |−|) in HV such that log |L^{′×}| = ℝ. Then Λ(L′) = A, thus gα = α in B′(ω_{G}^{∘}, L′) since gx = x in B^{e}(G_{L′}). But B′(ω_{G}^{∘}, L) → B′(ω_{G}^{∘}, L′) is injective and G(L)-equivariant, thus also gα = α in B′(ω_{G}^{∘}, L), which proves our claim. □

Lemma 134. Let L be a field with a non-archimedean absolute value |−|. There is an extension (L′, |−|) of (L, |−|) with L′ algebraically closed and log |L^{′×}| = ℝ.

Proof. By [6, VI, §8, Proposition 9], we may assume that L is algebraically closed. Then log |L^{×}| is a divisible subgroup of ℝ, i.e. a ℚ-vector space. Let (δ_{i})_{i∈I} be a ℚ-basis of ℝ/ log |L^{×}| and lift each δ_{i} to d_{i} ∈ ℝ. Let (t_{i})_{i∈I} be independent variables and let L′ be an algebraic closure of the purely transcendental extension M = K((t_{i})_{i∈I}) of K. By Zorn’s lemma and [6, VI, §10, Proposition 1], there is a unique extension of |−| to a non-archimedean absolute value on M such that log |t_{i}| = d_{i} for every i ∈ I. The latter again extends to L′, and then log |L^{′×}| equals ℝ, being a divisible subgroup of ℝ which contains log |L^{×}| and all d_{i}’s. □

6.4.11. The theorem implies various properties of B(ω_{G}^{∘}, K), for instance: B(ω_{G}^{∘}, K) is a tight affine F(G_{K})-building. For an extension (K, |−|) → (L, |−|), the map B(ω_{G_{O_{L}}}^{∘} , L) → B(ω_{G}^{∘}, L) is an isomorphism of affine F(G_{L})-buildings. For a closed immersion G_{1} ↪ G_{2}, the map B(ω_{G_{1}}^{∘}, K) → B(ω_{G_{2}}^{∘}, K) is injective. For a central isogeny G_{1} ↠ G_{2}, the map B(ω_{G_{1}}^{∘}, K) → B(ω_{G_{2}}^{∘}, K) is an isomorphism. Thus B(ω_{G}^{∘}, K) has canonical decompositions analogous to those of section 2.2.13. This last property also follows from 5.2.19.

6.4.12. Fix a faithful representation τ in Rep^{∘}(G)(O_{K}) and drop it from the notations for the induced distances, angles, scalar products… For x, y ∈ B^{e}(G, K),

d(x, y) = d (α(x), α(y)) = d (α(x)(τ), α(y)(τ))

where the last distance is computed in the space of K-norms on V_{K}(τ).

6.4.13. Fix F_{1}, F_{2} ∈ F(G_{K}). Suppose that for some ϵ > 0,
∀t ∈ [0, ϵ] : α_{G} + tF_{1} = α_{G} + tF_{2} in B(ω_{G}^{∘}, K).

Then the reductions F_{1,k} and F_{2,k} are equal in F(G_{k}) by lemma 130. Suppose conversely that F_{1,k} = F_{2,k}, and choose an apartment B^{e}(S) in B^{e}(G_{K}) containing the germs of t ↦ ∘_{G}^{e} + tF_{i} for i ∈ {1, 2} – in particular, S belongs to S(G) since ∘_{G}^{e} belongs to B^{e}(S). Then there are unique F_{i}^{∗} in F(S) such that, for some ϵ > 0, ∘_{G}^{e} +tF_{i} = ∘_{G}^{e} +tF_{i}^{∗} in B^{e}(G_{K}) for all t ∈ [0, ϵ]. But then also α_{G} +tF_{i} = α_{G} +tF_{i}^{∗} in B(ω_{G}^{∘}, K), thus F_{i,k} = F_{i,k}^{∗} in F(G_{k}), therefore F_{1,k}^{∗} = F_{2,k}^{∗} and F_{1}^{∗} = F_{2}^{∗} since the reduction map is injective on F(S), thus again α_{G} + tF_{1} = α_{G} + tF_{2} for all t ∈ [0, ϵ]. This yields canonical identifications

\xymatrix{ & F(G_{K}) \ar@{->>}[dl]_{loc^{∘^{e}_{G}}} \ar@{->>}[d]^{_{loc_{α_{G}}}} \ar@{->>}[dr]^{red} \\ T_{∘_{G}^{e}}B^{e}(G_{K}) \ar[r]^{≃} \ar[rr]_{κ}_{≃} & T_{α_{G}}B(ω_{G}^{∘}, K) \ar[r]^{≃} & F(G_{k})}

between the localization maps of 5.5.4 and the reduction map on F(G_{K}). By restriction to an apartment F(S_{K}) with S ∈ S(G), one checks that the isomorphism

κ : T_{∘_{G}^{e}}B^{e}(G_{K}) ⟶^{≃} F(G_{k})

is compatible with the distances, scalar products etc… attached to our chosen τ as in 5.5.4 and 4.2.12, and also that κ fits in a commutative diagram

\xymatrix{B^{e}(G, K) \ar@{->>}[d]_{_{loc{a \atop ∘^{e}_{G}}}} \ar[r]^{α} & B(ω_{G}^{∘}, K) \ar@{->>}[d]^{loc} \\ T_{∘_{G}^{e}}B^{e}(G_{K}) \ar[r]^{κ} & F(G_{k})}

Thus for every F, G ∈ F(G) and x, y ∈ B^{e}(G_{K}),

∡_{∘}(F, G) = ∡(F_{k}, G_{k}) and ∡_{∘}(x, y) = ∡(loc ∘ α(x), loc ∘ α(y))

where we have abbreviated ∘_{G}^{e} = ∘. In particular,

lim_{t→0} \frac{1}{t}d(∘ + tF, ∘ + tG) = d(F_{k}, G_{k})
lim_{t→0} \frac{1}{t} (d(x, ∘ + tF) − d(x, ∘)) = ⟨loc ∘ α(x), F_{k}⟩

As for the vector valued distance d : B^{e}(G_{K}) × B^{e}(G_{K}) → C(G_{K}), we have

d(∘, x) = t(loc ∘ α(x)) in C(G_{K}) = C(G_{k}).

6.4.14. For a parabolic subgroup P of G with unipotent radical U, the Gr_{P} - map of section 6.3.3 induces an analogous P (K)-equivariant map

Gr_{P} : B(ω_{G}^{∘}, K) → B(ω_{P/U}^{∘}, K).

For F ∈ F(G), set Gr_{F} = Gr_{P_{F}}. For ρ ∈ Rep^{∘}(G)(O_{K}), γ ∈ ℝ and α ∈ B′(ω_{G}^{∘}, K), let Gr_{F}^{γ}(α, ρ) be the K-norm on Gr_{F}^{γ}(ρ)_{K} induced by α(ρ) on V_{K}(ρ), i.e.

Gr_{F}^{γ}(α, ρ)(\overline{x}) = inf \{α(ρ)(x) : x ∈ F_{K}^{γ}(ρ), x ≡ \overline{x} mod F_{+,K}^{γ}(ρ)\}

for every \overline{x} in Gr_{F}^{γ}(ρ)_{K} = F_{K}^{γ}(ρ)/F_{+,K}^{γ}(ρ). By the axiom L(s) for the F(V_{K}(ρ))- building B(V_{K}(ρ)) of splittable K-norms on V_{K}(ρ), Gr_{F}^{γ}(α, ρ) is a splittable Knorm on Gr_{F}^{γ}(ρ)_{K}. Viewing Gr_{F}^{γ}(ρ) as a representation of P_{F}/U_{F}, we have:

∀α ∈ B(ω_{G}^{∘}, K) : Gr_{F}(α) (Gr_{F}^{γ}(ρ)) = Gr_{F}^{γ}(α, ρ).

Indeed, both sides only depend upon the U_{F}(K)-orbit of α, and we may thus assume that α belongs to the image of B(ω_{L}^{∘}, K) → B(ω_{G}^{∘}, K) for some fixed Levi subgroup L of P_{F}, i.e. α = α_{G} + H for some H ∈ F(L). Then r_{P_{F} ,L}(α) = α, thus Gr_{F}(α) = α_{P/U} + \overline{H} where \overline{H} is the image of H in F(P_{F}/U_{F}). On the other hand, the chosen L gives a splitting G ∈ G(Z(L)) of F, thus also a splitting ρ|_{L} = ⊕_{γ}ρ_{γ} with ρ_{γ} ∈ Rep^{∘}(L)(O), V (ρ_{γ}) = G_{γ}(ρ). Since α is the image of α_{L}+H in B^{e}(ω_{G}^{∘}, K), this splitting is adapted to α: α(ρ) = ⊕α_{γ}(ρ) where α_{γ}(ρ) = α_{L}(ρ_{γ})+H(ρ_{γ}). Thus Gr_{F}^{γ}(α, ρ) ≃ α_{γ}(ρ) under the K-linear isomorphism V_{K}(ρ_{γ}) ≃ Gr_{F}^{γ}(ρ)_{K} induced by the (L → P_{F}/U_{F})-equivariant isomorphism ρ_{γ} ≃ Gr_{F}^{γ}(ρ). It follows that indeed

Gr_{F}^{γ}(α, ρ) = α_{P/U}(Gr_{F}^{γ}(ρ)) + \overline{H}(Gr_{F}^{γ}(ρ)) = Gr_{F}(α)(Gr_{F}^{γ}(ρ)).

For the distances attached to our chosen τ, we thus obtain

lim_{t→∞} d_{τ} (x + tF, y + tF) = d (Gr_{F}^{•}(α(x), τ), Gr_{F}^{•}(α(y), τ))

for every x, y ∈ B^{e}(G_{K}), F ∈ F(G).

6.4.15. Combining the previous two computations, we also obtain a formula for the Busemann scalar product on B^{e}(G_{K}). Recall from section 6.3.3 (and 5.6.9) that for any x, y ∈ B^{e}(G_{K}) and F ∈ F(G), we have

⟨\overrightarrow{xy}, F⟩ = ⟨\overrightarrow{Gr_{F}(x)Gr_{F}(y)}, \overline{F}⟩ = ⟨loc_{Gr_{F} (x)}^{a} (Gr_{F}(y)) , loc_{Gr_{F} (x)} (\overline{F})⟩

where the second and third scalar product are respectively the Busemann scalar product on B^{e}(P_{K}/U_{K}) and the scalar product on its tangent space at Gr_{F}(x), with (P, U) = (P_{F}, U_{F}). For x = ∘_{G}^{e}, Gr_{F}(x) = ∘_{P/U}^{e} and we thus obtain

⟨\overrightarrow{xy}, F⟩ = ⟨loc (Gr_{F}(α(y))) , \overline{F}_{k}⟩

with the scalar product of F(P_{k}/U_{k}) attached to the faithful representation

Gr_{F}^{•}(τ) = ⊕_{γ}Gr_{F}^{γ}(τ)

of P/U. Since \overline{F}(Gr_{F}^{γ}(τ)) is the ℝ-filtration with a single jump at γ,

⟨\overrightarrow{xy}, F⟩ = _{∑γ}γ · deg (loc (Gr_{F}(α(y))) (Gr_{F}^{γ}(τ))) .

By definition of the morphism loc : B(ω_{G}^{∘}, K) → F(G_{k}),

loc (Gr_{F}(α(y))) (Gr_{F}^{γ}(τ)) = loc (Gr_{F}^{γ}(τ)_{K}, Gr_{F}^{γ}(α(y), τ), Gr_{F}^{γ}(τ)) .

The degree of this filtration is the degree of its determinant. Since the functors

loc : Norm′(K) → Fil(k) and Gr_{F}^{•} (α(y))′ : Rep^{∘}(P/U)(O_{K}) → Norm′(K)

are exact ⊗-functors, they both commute with the determinant. The degrees which occur in the last displayed formula for ⟨\overrightarrow{xy}, F⟩ are therefore given by

deg (loc (Λ_{F}^{γ}(τ)_{K}, Λ_{F}^{γ}(α(y), τ), Λ_{F}^{γ}(τ)))

where Λ_{F}^{γ}(τ) = det (Gr_{F}^{γ}(τ)) is a rank one representation of P/U and

Λ_{F}^{γ}(α(y), τ) = det (Gr_{F}^{γ}(α(y), τ)) = Gr_{F}^{•} (α(y)) (Λ_{F}^{γ}(τ))

is a K-norm on Λ_{F}^{γ}(τ)_{K}. For a rank one object (V, α, L) in Norm′(K), the degree of loc(V, α, L) is simply the largest γ ∈ ℝ such that L ⊂ \overline{B}(α, γ). Equivalently,

deg (loc(V, α, L)) = − log (sup {α(ℓ) : ℓ ∈ L}) = − log (α(ℓ_{0}))

where L = O_{K} · ℓ_{0}. Thus, still assuming that x = ∘_{G}^{e}, we finally obtain

⟨\overrightarrow{xy}, F⟩ = −_{∑γ}γ · log (sup \{Λ_{F}^{γ} (α(y), τ) |Λ_{F}^{γ}(τ)\})
= −_{∑γ}γ · log (Λ_{F}^{γ} (α(y), τ) (e_{1}^{γ} ∧ ⋯ ∧ e_{r_{γ}}^{γ}))

where (e_{1}^{γ}, ⋯, e_{r_{γ}}^{γ}) is an O_{K}-basis of Gr_{F}^{γ}(τ). For a general x in B^{e}(G_{K}), we find:

⟨\overrightarrow{xy}, F⟩ = _{∑γ}γ · log (\frac{Λ_{F}^{γ} (α(x), τ)}{Λ_{F}^{γ} (α(y), τ)} (e_{1}^{γ} ∧ ⋯ ∧ e_{r_{γ}}^{γ} )) .

Note that if we are given some G ∈ F(P/U) with Gr_{F}(y) = Gr_{F}(x)+G, then simply

⟨\overrightarrow{xy}, F⟩ = ⟨G, \overline{F}⟩ = _{∑γ}γ · deg (G (Gr_{F}^{γ}(τ))) .

6.4.16. For every ν > 0, there is a G(K)-equivariant commutative diagram

B(ω_{G}^{∘}, K, |−|) × F(G_{K}) ⟶^{+} B(ω_{G}^{∘}, K, |−|)
a ↓ b ↓ a ↓
B(ω_{G}^{∘}, K, |−|^{ν}) × F(G_{K}) ⟶^{+} B(ω_{G}^{∘}, K, |−|^{ν})

where a(α) = α^{ν} and b(F) = νF. It is compatible with the analogous diagram of section 6.2.9 via the relevant α-maps.

6.4.17. For x ∈ B^{e}(G_{K}), the K-norm α(x) ∈ B(ω_{G}^{∘}, K) is exact and extends to a K-norm on ω′_{G} as in 3.6.4. Thus by proposition 49, it yields a K-norm α(x)(ρ) on V_{K}(ρ) for any representation ρ of G on a flat O_{K}-module V (ρ). We set

α_{ad}(x) = α(x)(ρ_{ad}), α_{reg}(x) = α(x)(ρ_{reg}) and α_{adj}(x) = α(x)(ρ_{adj}).

Proposition 135. Suppose that (K, |−|) is discrete, say |K^{×}| = q^{ℤ} with q > 1. Let (𝔤_{x,r})_{r∈ℝ} be the Moy-Prasad filtration attached to x on 𝔤_{K} = Lie(G_{K}). Then

∀x ∈ ℝ : 𝔤_{x,r} = {v ∈ 𝔤_{K} : α_{ad}(x)(v) ≤ q^{−r}} .

Proof. Given the definition of 𝔤_{x,r} (by étale descent from the quasi-split case) and proposition 127, we may assume that G splits over O_{K}. Changing |−| to |−|^{ν} with ν = \frac{1}{log q}, we may also assume that q = e. Fix S ∈ S(G) with x in B^{e}(S_{K}) and write x = ∘_{G}^{e} + F for some F ∈ F(S_{K}), so that also α(x) = α_{G} + F. Let 𝔤 = 𝔤_{0} ⊕ ⊕_{β∈Φ(G,S)}𝔤_{β} be the weight decomposition of 𝔤 and F^{♯} : M → ℝ the morphism corresponding to F, where M = Hom(S, 𝔾_{m,O_{K}}). Then for every r ∈ ℝ,

\overline{B} (α_{ad}(x), r) = 𝔤_{0,r} ⊕ ⊕_{β∈Φ(G,S)}𝔤_{β,r}

where 𝔤_{β,r} = \overline{B} (α_{𝔤_{β}}, r − F^{♯}(β)) for β ∈ Φ(G, S) ∪ \{0\}. For r = 0, this is the Lie algebra 𝔤_{x} of the group scheme 𝔊_{x} over O_{K} attached to x in [10]. Comparing now this formula with the definition of 𝔤_{x,r} in [3, 2.1.3] proves our claim. □

Let G_{K}^{an} be the analytic Berkovich space attached to G_{K}. In [38, 2.2], the authors construct a canonical map ϑ : B^{e}(G_{K}) → G_{K}^{an}, thus attaching to every x ∈ B^{e}(G_{K}) a multiplicative K-semi-norm ϑ(x) on A(G_{K}).

Proposition 136. For every x ∈ B^{e}(G_{K}), α_{adj}(x) = ϑ(x). In particular, the K-norm α_{adj}(x) on A(G_{K}) is multiplicative and ϑ(x) is a norm.

Proof. Equip G_{K}^{an} with the action of G(K) induced by ρ_{adj}. Then x ↦ ϑ(x) is G(K)-equivariant and compatible with extensions (K, |−|) → (L, |−|) in the sense that ϑ(x) = ϑ(x_{L})|A(G_{K}) for every x ∈ B^{e}(G_{K}) [38, Proposition 2.8]. The map x ↦ α_{adj}(x) has the same properties. We may thus assume that G splits over O_{K}, and again choosing L with log |L^{×}| = ℝ, we merely have to show that ϑ(∘_{G}^{e}) = α_{adj}(∘_{G}^{e}) = α_{A(G)}. By definition: {ϑ(x)} is the Shilov boundary of a K-affinoid subgroup G_{x} of G_{K}^{an}. For x = ∘_{G}^{e}, G_{x} is the affinoid group G^{an} attached to G, and its Shilov boundary is the gauge norm attached to A(G), i.e. α_{A(G)}. □

Since the multiplication on A(G) is a morphism ρ_{reg} ⊗ ρ_{reg} → ρ_{reg} in Rep′(G)(O_{K}), the K-norm α_{reg}(x) on A(G_{K}) is sub-multiplicative. Since for τ ∈ Rep^{∘}(G)(O_{K}), the co-module map V (τ) → V (τ)⊗A(G) is a pure monomorphism τ ↪ τ_{0} ⊗ρ_{reg} in Rep′(G)(O_{K}), α(x)(τ) is the restriction of α_{V (τ_{0})} ⊗ α_{reg}(x) to V_{K}(τ), thus α_{reg}(x) determines α(x) and α_{reg} is a G(K)-equivariant embedding of B^{e}(G_{K}) into the space of sub-multiplicative K-norms on A(G) (equipped with the regular action).

6.4.18. Some final remarks:

(1) We have not given an intrinsic characterization of the subset B(ω_{G}^{∘}, K) of B′(ω_{G}^{∘}, K). We expect that B(ω_{G}^{∘}, K) = B^{?}(ω_{G}^{∘}, K), or perhaps even that B(ω_{G}^{∘}, K) is equal to the G(K)-stable subset of exact norms in B′(ω_{G}^{∘}, K).

(2) Suppose that O is a valuation ring of height > 1 with fraction field K. Then Γ = K^{×}/O^{×} is a totally ordered commutative group which can not be embedded into ℝ. Let G be a reductive group over O. Replacing ℝ with Γ in the above constructions, it might be possible to define a “Bruhat-Tits” building B(ω_{G}^{∘}, K) with compatible actions of G(K) and F^{Γ}(G_{K}), made of factorizations of the fiber functor ω_{G,K}^{∘} : Rep^{∘}(G)(O) → Vect(K) through a suitable category of “Γ-norms”. The type maps should be the tautological morphisms ν : S(K) → G^{Γ}(S) mapping s ∈ S(K) to the unique morphism ν(s) : 𝔻_{K}(Γ) → S whose composite with a character χ of S is the image of χ(s) in Γ = K^{×}/O^{×}.

(3) There might also be a similar Tannakian formalism for the symmetric spaces of reductive groups over ℝ, with factorizations of fiber functors through a category of Euclidean spaces, using compact forms of the adjoint groups as base point.

Nomenclature

∥−∥_{τ} Length on F^{Γ}(G) or C^{Γ}(G) defined page 83. ∡_{τ} (−, −) Angle on Std^{Γ}(G) defined page 83. ∡_{τ}^{os}(−, −) Osculatory angle on C^{Γ}(G) defined page 84. ∡_{τ}^{tr}(−, −) Transverse angle on C^{Γ}(G) defined page 84. ∡_{x}(F, G) Alexandrov angle at x between x + tF and x + tG, page 109. ∡_{x}^{c}(y, z) Angle at x in a comparison Euclidean triangle for (x, y, z). ∡(\overrightarrow{xy}, G) Busemann angle at x between y and x + tG, page 109. ⟨−, −⟩_{τ} Scalar product on Std^{Γ}(G) defined page 83. ⟨−, −⟩_{τ}^{os} Osculatory scalar product on C^{Γ}(G) defined page 84. ⟨−, −⟩_{τ}^{tr} Transverse scalar product on C^{Γ}(G) defined page 84. ≤ Weak dominance partial order on ℂ^{Γ}(G), page 20. ⪯ Strong dominance partial order on ℂ^{Γ}(G), page 20. ⟨\overrightarrow{xy}, G⟩ Busemann scalar product between [x, y] and x+tG, defined page 111. 1_{G}^{♮} Counit 1_{G}^{♮} : A(G) → O_{S} of A(G). 1_{G} Unit section 1_{G} : S → G of a group scheme G over S. 1_{S} Trivial representation of G on O_{S}. A(G) Hopf algebra of G. ACF^{Γ}(G) Set of all triples (S, B, F) with S ∈ S(G) and F ∈ F^{−1}(B) ⊂ F^{Γ}(S),

page 78.

ad Morphism ad : G → G^{ad}. AF^{Γ}(G) Set of all pairs (S, F) with S ∈ S(G) and F ∈ F^{Γ}(S), page 78. α Functorial isomorphism B^{e}(G, L) → B(ω_{G}^{∘}, L) defined page 135. α_{G,L} Canonical L-norm on ω_{G,L}^{∘} defined page 133. Aut^{⊗}(F) Sheaf of tensor automorphisms of a fiber functor X preserving a Γ-

filtration F on X, page 38.

Aut^{⊗}(G) Sheaf of tensor automorphisms of a fiber functor X preserving a Γ-

graduation G on X, page 38.

Aut^{⊗}(ω) Sheaf of tensor automorphisms of ω, page 38. Aut^{⊗}(ω^{∘}) Sheaf of tensor automorphisms of ω^{∘}, page 42. Aut^{⊗}(V) Sheaf of tensor automorphisms of V, page 38. Aut^{⊗}(V^{∘}) Sheaf of tensor automorphisms of V^{∘}, page 42. B(α, γ) Open ball of radius exp(−γ) for α, page 132. \overline{B}(α, γ) Closed ball of radius exp(−γ) for α, page 132. B(G) Set of all minimal parabolic subgroups of G, page 76. B(ω_{G}^{∘}, L) Space of all good L-norms on ω_{G,L}^{∘}, defined page 133. B′(ω_{G}^{∘}, L) Space of all L-norms on ω_{G,L}^{∘}, defined page 133. B^{?}(ω_{G}^{∘}, L) Space of all nice L-norms on ω_{G,L}^{∘}, defined page 134. B^{e}(G) Extended Bruhat-Tits building of G, page 126.

B^{e}(G, K) Pointed extended Bruhat-Tits building for G over O_{K}, page 129. B^{r}(G) Reduced Bruhat-Tits building of G, page 126. C(∂X(G)) Cone on the visual boundary of X(G), page 108. ℂ^{Γ}(G) Scheme of types of Γ-graduations of Γ-filtrations on G, page 16. C^{Γ}(G) Cone of types of Γ-filtrations or Γ-graduations on G, page 78. ℂ^{Γ}(G)^{c} Central part of ℂ^{Γ}(G), page 21. ℂ^{Γ}(G)^{r} Reduced part of ℂ^{Γ}(G), page 21. c_{ρ} Comodule structure morphism V (ρ) → V (ρ) ⊗ A(G) of ρ. can Map F(G_{L}) ↠ B(ω_{G}^{∘}, L), defined page 133. C^{∘} Category opposed to C. d Vectorial distance on an affine F(G)-building, defined page 94. d_{τ} Distance on Std^{Γ}(G) defined page 83. 𝔻_{S}(M) Diagonalizable group scheme over S with character group M. ∂X(G) Visual boundary of X(G), page 108. 𝔻𝕐ℕ(G) Dynkin scheme of G, defined page 13. e 2.71828182846... F Facet morphism on 𝔾^{Γ}(G), 𝔽^{Γ}(G)) or ℂ^{Γ}(G), page 16. 𝔽^{Γ}(G) Scheme of Γ filtrations on G, page 16. F^{Γ}(G) Set of all Γ-filtrations on G, page 78. 𝔽^{Γ}(G)^{c} Central part of 𝔽^{Γ}(G), page 21. 𝔽^{Γ}(G)^{r} Reduced part of 𝔽^{Γ}(G), page 21. 𝔽^{Γ}(ω) Sheaf of Γ-filtrations on ω, page 38. 𝔽^{Γ}(ω^{∘}) Sheaf of Γ-filtrations on ω^{∘}, page 42. 𝔽^{Γ}(V) Sheaf of Γ-filtrations on V, page 38. 𝔽^{Γ}(V^{∘}) Sheaf of Γ-filtrations on V^{∘}, page 42. F_{L}^{ι} Filtration opposed to F with respect to a Levi L of P_{F}, page 95. Fil Functor Fil : Gr^{Γ}LF → Fil^{Γ}LF, page 39. Fil Functor Fil : Gr^{Γ}QCoh → Fil^{Γ}QCoh, page 35. Fil Morphism Fil : 𝔾^{Γ}(G) → 𝔽^{Γ}(G) defined on page 16. Fil Morphism Fil : 𝔾^{Γ}(X) → 𝔽^{Γ}(X) for X = V or ω, page 38. Fil Morphism Fil : 𝔾^{Γ}(X) → 𝔽^{Γ}(X) for X = V^{∘} or ω^{∘}, page 42. Fil(G) Γ-filtration induced by a Γ-graduation G, page 35. Fil^{Γ}LF Category of Γ-filtered finite locally free sheaves on schemes, page 39. Fil^{Γ}LF(X) Category of Γ-filtered finite locally free sheaves on X, page 39. Fil^{Γ}QCoh Category of Γ-filtered quasi-coherent sheaves on schemes, page 35. Fil^{Γ}QCoh(X) Category of Γ-filtered quasi-coherent sheaves on X, page 35. \overline{F} Morphism 𝔻(Γ) → \overline{R}(P_{F}) attached to a Γ-filtration F, page 17. G G = G(O) for a group scheme G over a local ring O. G^{ab} Abelianization G^{ab} = G/G^{der} of G. G^{ad} Adjoint group G^{ad} = G/Z(G) of G. G^{der} Derived group of G. 𝔾^{Γ}(G) Scheme of Γ-graduations on G, defined page 12. G^{Γ}(G) Set of all Γ-graduations on G, page 78. 𝔾^{Γ}(G)^{c} Central part of 𝔾^{Γ}(G), page 21. 𝔾^{Γ}(G)^{r} Reduced part of 𝔾^{Γ}(G), page 21. 𝔾^{Γ}(ω) Sheaf of Γ-graduations on ω, page 38. 𝔾^{Γ}(ω^{∘}) Sheaf of Γ-graduations on ω^{∘}, page 42. 𝔾^{Γ}(V) Sheaf of Γ-graduations on V, page 38.

𝔾^{Γ}(V^{∘}) Sheaf of Γ-graduations on V^{∘}, page 42. G^{ss} Semi-simplification G^{ss} = G/R(G) of G. 𝔾_{a} Additive group over Spec(ℤ), 𝔾_{a}(R) = R. 𝔾_{m} Multiplicative group over Spec(ℤ), 𝔾_{m}(R) = R^{×}. G_{Ω} Pointwise stabilizer of Ω in G, page 105. G_{S,Ω} Apartment based avatar of G_{Ω}, page 105. ΓR^{∗} Γ-subgroup spanned by the coroots, page 26. (ΓR^{∗})_{sat} Saturation of ΓR^{∗}, page 26. Γ_{+}R_{+}^{∗} Γ_{+}-cone spanned by the positive coroots, page 26. (Γ_{+}R_{+}^{∗})_{sat} Saturation of Γ_{+}R_{+}^{∗}, page 26. Γ(X, F) Sections of a sheaf F over X. Γ(X/S) Sections of a morphism X → S. Γ_{+} Γ_{+} = {γ ∈ Γ : γ ≥ 0}. 𝔾𝔼ℕ(G) Scheme of pairs of parabolic subgroups of G in generic relative posi-

tion, page 22.

𝔾𝔼ℕ^{Γ}(G) Scheme of pairs of Γ-filtrations on G in generic relative position, page

22.

Gr Functor Gr : Fil^{Γ}LF → Gr^{Γ}LF, page 39. Gr Functor Gr : Fil^{Γ}QCoh → Gr^{Γ}QCoh, page 35. Gr^{•} Morphism Gr^{•} : Aut^{⊗}(F) → Aut^{⊗}(Gr_{F}^{•}), page 38. Grr Growling sound indicative of frustration with useless generalities. Gr^{Γ}LF Category of Γ-graded finite locally free sheaves on schemes, page 39. Gr^{Γ}LF(X) Category of Γ-graded finite locally free sheaves on X, page 39. Gr^{Γ}QCoh Category of Γ-graded quasi-coherent sheaves on schemes, page 35. Gr^{Γ}QCoh(X) Category of Γ-graded quasi-coherent sheaves on X, page 35. Gr_{F}(M) Γ-graded quasi-coherent sheaf associated with a Γ-filtration F on a

quasi-coherent sheaf M, page 35.

Gr_{F}^{γ}(τ) Graded piece of the Γ-filtration F(τ) on V (τ). Gr_{P} P (K)-equivariant map B^{e}(G, K) ↠ B^{e}(P/U, K), page 130. Gr_{P} P-equivariant map F(G) ↠ F(P/U), page 117. Gr_{P} (α) K-norm on ω_{P/U}^{∘} induced by a K-norm α on ω_{G}^{∘}, page 138. Gr_{P} (F) Γ-filtration on P/U induced by a Γ-filtration F on G in standard

relative position with a parabolic subgroup P of G, page 25.

Gr_{P}^{∞} Projection X(G) ↠ T_{P}^{∞}X(G), page 114. Group Category of groups. S − Group Category of group schemes over S. H^{∘} Neutral component of a group scheme H. \underline{Hom} Sheafified version of Hom. Hom^{+}(M, Γ) Dominant morphisms in Hom(M, Γ). HV Category of Henselian valued fields, page 129. I(G) Augmentation ideal of A(G). Int(g) Inner automorphism h ↦ ghg^{−1}. ι Opposition involution of 𝔾^{Γ}(G) or ℂ^{Γ}(G), defined page 17. ι Opposition involution of 𝕆ℙℙ(G) or 𝕆(G), defined page 14. ι_{S} Opposition involution on the apartment of S in F^{Γ}(G), page 78. ι_{S} Opposition involution on the apartment of S in P(G), page 77. K_{0}(G) Grothendieck ring of Rep^{∘}(G)(S), page 55. LF Category of finite locally free sheaves on schemes, page 39.

LF(X) Category of finite locally free sheaves on X, page 39. Lie(G) Lie algebra of G. loc Functor Norm′(K) → Fil(k), defined page 133. loc Map B^{?}(ω_{G}^{∘}, L) → F(G_{k_{L}}), defined page 134. loc_{x} Projection F(G) ↠ T_{x}X(G), page 109. loc_{x}^{a} Localization X(G) ↠ T_{x}X(G), page 110. M_{d} Dominant weights. μ_{G}^{♮} Comultiplication μ_{G}^{♮} : A(G) → A(G) ⊗ A(G) of A(G). N_{G}(x) Normalizer of x in G. Norm^{∘}(K) Category of splittable normed finite K-vector spaces, page 132. Norm′(K) Category of normed K-spaces with a lattice, defined page 133. ν_{X} Type morphism of an F(G)-building X(G) defined page 92. ν_{X,S} Morphism Z_{G}(S) → G(S) defined page 92. ∘ Smallest element of O(G), page 78. 𝕆(G) Scheme of types of parabolic subgroups of G, defined page 13. O(G) Set of all types of parabolic subgroups of G, page 76. ∘_{G}^{e} Canonical point of B^{e}(G_{K}) attached to G over O_{K}, page 129. ∘_{G}^{r} Canonical point of B^{r}(G_{K}) attached to G over O_{K}, page 129. ω_{X} Fiber functor ω_{X} : Rep(G)(S) → QCoh(X), page 36. ω_{X}^{∘} Fiber functor ω_{X}^{∘} : Rep^{∘}(G)(S) → LF(X), page 41. 𝕆ℙℙ(G) Scheme of pairs of opposed parabolic subgroups of G, page 14. OPP(G) Set of all pairs of opposed parabolic subgroups of G, page 76. os Osculatory section os : 𝕆(G) → 𝕋𝕊𝕋𝔻(G), defined page 22. ℙ(G) Scheme of parabolic subgroups of G, defined page 13. P(G) Set of all parabolic subgroups of G, page 76. P_{F} Parabolic subgroup of G fixing F, page 17. P_{u} Universal parabolic subgroup of G_{ℙ(G)}. Φ(S, G) Roots of S in Lie(G). π 3.14159265359... QCoh Category of quasi-coherent sheaves on schemes, page 35. QCoh(X) Category of quasi-coherent sheaves on X, page 35. R(G) Radical of G. R^{u}(P) Unipotent radical of P. R_{𝕆(G)} A torus over 𝕆(G) defined on page 13. R_{𝕆ℙℙ(G)} Radical of the universal Levi subgroup of G_{𝕆ℙℙ(G)}. R_{ℙ(G)} Radical \overline{R}(P_{u}) of P_{u}/U_{u}, a torus over ℙ(G). r_{P,L} Retraction r_{P,L} : F^{Γ}(G) ↠ F^{Γ}(L), defined page 86. r_{P,L} Retraction r_{P,L} : X(G) ↠ X(L), defined page 96. Rep(G) Fibered category of algebraic representations of G on quasi-coherent

sheaves, page 36.

Rep(G)(X) Category of algebraic representations of G on quasi-coherent sheaves

over X, page 36.

Rep^{∘}(G) Fibered category of algebraic representations of G on finite locally

free sheaves, page 41.

Rep^{∘}(G)(X) Category of algebraic representations of G on finite locally free sheaves

over X, page 41.

Rep′(G)(S) Full sub-category of Rep(G)(S) defined on page 47. ρ^{n} Adjoint representation of G on I(G)/I(G)^{n+1}, page 52. ρ^{∨} Dual of ρ. ρ_{0} Trivial representation of G on V (ρ). ρ_{ad} Adjoint representation of G on Lie(G). ρ_{adj} Adjoint representation of G on A(G), page 52. ρ_{adj}^{∘} Adjoint representation of G on I(G), page 52. ρ_{n} Dual of ρ^{n}, page 52. ρ_{reg} Regular representation of G on A(G), page 45. Ring Category of commutative rings. \overline{R}(P) Radical of P/U, where U is the unipotent radical of P. RX(G) Rays in X(G), page 108. S(G) Set of all maximal split tori of G, page 76. SBP(G) Set of triples (S, B, P) in S(G)×B(G)×P(G) with Z_{G}(S) ⊂ B ⊂ P,

page 76.

Sch Category of schemes. Sch/S Category of schemes over S. Set Category of sets. ∼_{Par} Par-equivalence on 𝔾^{Γ}(G), defined page 16. SP(G) Set of all pairs (S, P) in S(G) × P(G) with Z_{G}(S) ⊂ P, page 76. 𝕊𝕋𝔻(G) Scheme of pairs of parabolic subgroups of G in standard relative

position, defined page 22.

Std(G) Set of all pairs of parabolic subgroups of G in standard relative po-

sition, page 80.

𝕊𝕋𝔻(Z) Pull-back of 𝕊𝕋𝔻(G) ↪ ℙ(G)^{2} through Z → ℙ(G)^{2}. 𝕊𝕋𝔻^{Γ}(G) Scheme of pairs of Γ-filtrations on G in standard relative position,

defined page 22.

Std^{Γ}(G) Set of all pairs of Γ-filtrations on G in standard relative position,

page 80.

t Type morphism t : ℙ(G) → 𝕆(G), defined page 13. t Type morphism t : 𝔽^{Γ}(G) → ℂ^{Γ}(G), defined page 16. t_{2} Type morphism t_{2} : 𝕊𝕋𝔻(G) → 𝕋𝕊𝕋𝔻(G), defined page 22. t_{2} Type morphism t_{2} : 𝕊𝕋𝔻^{Γ}(G) → 𝕋𝕊𝕋𝔻^{Γ}(G), defined page 22. T_{P}^{∞}X(G) Quotient of X(G) by the unipotent radical U of P, page 112. T_{x}X(G) Tangent space at x in X(G), defined page 109. tr Transverse section tr : 𝕆(G) → 𝕋𝕊𝕋𝔻(G), defined page 22. 𝕋𝕊𝔻𝕋(G) Scheme of types of pairs of parabolic subgroups of G in standard

relative position, defined page 22.

𝕋𝕊𝔻𝕋^{Γ}(G) Scheme of types of pairs of Γ-filtrations on G in standard relative

position, defined page 22.

U_{a} Root subgroup of G for a ∈ Φ(G, S), page 104. U_{F} Unipotent radical of P_{F}. U_{u} Unipotent radical of P_{u}. V Fiber functor V : Rep(G) → QCoh/S, page 36. V^{∘} Fiber functor V^{∘} : Rep^{∘}(G) → LF/S, page 41. W_{G}(S) Weyl group of S in G, W_{G}(S) = N_{G}(S)/Z_{G}(S). X(ρ) Filtered set of subrepresentations of ρ on finite locally free subsheaves

of V (ρ), page 47.

X_{T} Pull-back or base change of some X over S through T → S. Z(G) Center of G.

Z_{G}(x) Centralizer of x in G.

Bibliography

1. Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 152, Springer-Verlag, Berlin, 1970. MR 0274459 (43 #223b)

2. Revêtements étales et groupe fondamental (SGA 1), Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 3, Société Mathématique de France, Paris, 2003, Séminaire de géométrie algébrique du Bois Marie 1960–61. [Algebraic Geometry Seminar of Bois Marie 1960-61], Directed by A. Grothendieck, With two papers by M. Raynaud, Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin; MR0354651 (50 #7129)]. MR 2017446 (2004g:14017)

3. Jeffrey D. Adler and Stephen DeBacker, Some applications of Bruhat-Tits theory to harmonic analysis on the Lie algebra of a reductive p-adic group, Michigan Math. J. 50 (2002), no. 2, 263–286. MR 1914065 (2003g:22016)

4. M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615. MR 702806 (85k:14006)

5. Armand Borel and Jacques Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. (1965), no. 27, 55–150. MR 0207712 (34 #7527)

6. N. Bourbaki, Éléments de mathématique. Fasc. XXX. Algèbre commutative. Chapitre 5: Entiers. Chapitre 6: Valuations, Actualités Scientifiques et Industrielles, No. 1308, Hermann, Paris, 1964. MR 0194450 (33 #2660)

7. , Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968. MR 0240238 (39 #1590)

8. M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486 (2000k:53038)

9. F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. (1972), no. 41, 5–251. MR 0327923 (48 #6265)

10. , Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math. (1984), no. 60, 197–376. MR 756316 (86c:20042)

11. , Schémas en groupes et immeubles des groupes classiques sur un corps local, Bull. Soc. Math. France 112 (1984), no. 2, 259–301. MR 788969 (86i:20064)

12. , Schémas en groupes et immeubles des groupes classiques sur un corps local. II. Groupes unitaires, Bull. Soc. Math. France 115 (1987), no. 2, 141–195. MR 919421

13. Brian Conrad, Ofer Gabber, and Gopal Prasad, Pseudo-reductive groups, New Mathematical Monographs, vol. 17, Cambridge University Press, Cambridge, 2010. MR 2723571 (2011k:20093)

14. Christophe Cornut, Mazur’s Inequality and Laffaille’s theorem, Preprint, (2014).

15. , A fixed point theorem in Euclidean buildings, Adv. Geom. 16 (2016), no. 4, 487–496. MR 3595184

16. Christophe Cornut and Marc-Hubert Nicole, Cristaux et immeubles, Bull. Soc. Math. France 144 (2016), no. 1, 125–143. MR 3481264

17. J.-F. Dat, S. Orlik, and M. Rapoport, Period domains over finite and p-adic fields, Cambridge Tracts in Mathematics, vol. 183, Cambridge University Press, Cambridge, 2010. MR 2676072 (2012a:22026)

18. P. Deligne, Catégories tannakiennes, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 111–195. MR 1106898 (92d:14002)

19. Jean-Marc Fontaine and Michael Rapoport, Existence de filtrations admissibles sur des isocristaux, Bull. Soc. Math. France 133 (2005), no. 1, 73–86. MR 2145020 (2005m:14032)

20. Philippe Gille and Patrick Polo (eds.), Schémas en groupes (SGA 3). Tome I. Propriétés générales des schémas en groupes, Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 7, Société Mathématique de France, Paris, 2011, Séminaire de Géométrie Algébrique du Bois Marie 1962–64. [Algebraic Geometry Seminar of Bois Marie 1962–64], A seminar directed by M. Demazure and A. Grothendieck with the collaboration of M. Artin, J.-E. Bertin, P. Gabriel, M. Raynaud and J-P. Serre, Revised and annotated edition of the 1970 French original. MR 2867621

21. Philippe Gille and Patrick Polo (eds.), Schémas en groupes (SGA 3). Tome III. Structure des schémas en groupes réductifs, Documents Mathématiques (Paris) [Mathematical Documents (Paris)], vol. 8, Société Mathématique de France, Paris, 2011, Séminaire de Géométrie Algébrique du Bois Marie 1962–64. [Algebraic Geometry Seminar of Bois Marie 1962–64], A seminar directed by M. Demazure and A. Grothendieck with the collaboration of M. Artin, J.-E. Bertin, P. Gabriel, M. Raynaud and J-P. Serre, Revised and annotated edition of the 1970 French original. MR 2867622

22. O. Goldman and N. Iwahori, The space of p-adic norms, Acta Math. 109 (1963), 137–177. MR 0144889 (26 #2430)

23. A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. (1960), no. 4, 228. MR MR0163908 (29 #1207)

24. , Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. (1961), no. 8, 222. MR MR0163909 (29 #1208)

25. , Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Inst. Hautes Études Sci. Publ. Math. (1964), no. 20, 259. MR MR0173675 (30 #3885)

26. , Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. (1965), no. 24, 231. MR MR0199181 (33 #7330)

27. , Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. (1966), no. 28, 255. MR MR0217086 (36 #178)

28. , Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. (1967), no. 32, 361. MR MR0238860 (39 #220)

29. Bruce Kleiner and Bernhard Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Inst. Hautes Études Sci. Publ. Math. (1997), no. 86, 115–197 (1998). MR 1608566 (98m:53068)

30. R. E. Kottwitz, Isocrystals with additional structure, Compositio Math. 56 (1985), no. 2, 201–220. MR 809866 (87i:14040)

31. Guy Laffaille, Groupes p-divisibles et modules filtrés: le cas peu ramifié, Bull. Soc. Math. France 108 (1980), no. 2, 187–206. MR 606088 (82i:14028)

32. E. Landvogt, Some functorial properties of the Bruhat-Tits building, J. Reine Angew. Math. 518 (2000), 213–241. MR 1739403 (2001g:20029)

33. F. W. Levi, Ordered groups, Proc. Indian Acad. Sci., Sect. A. 16 (1942), 256–263. MR 0007779 (4,192b)

34. Hideyuki Matsumura, Commutative ring theory, second ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989, Translated from the Japanese by M. Reid. MR 1011461 (90i:13001)

35. Allen Moy and Gopal Prasad, Unrefined minimal K-types for p-adic groups, Invent. Math. 116 (1994), no. 1-3, 393–408. MR 1253198 (95f:22023)

36. Anne Parreau, La distance vectorielle dans les immeubles affines et les espaces symétriques.

37. , Immeubles affines : construction par les normes et étude des isométries., In Crystallographic groups and their generalizations (Kortrijk, 1999), Contemp. Math., vol. 262, Amer. Math. Soc., Providence, RI, 2000, pp. 263–302.

38. B. Rémy, A. Thuillier, and A. Werner, Bruhat-Tits theory from Berkovich’s point of view. I. Realizations and compactifications of buildings, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), no. 3, 461–554. MR 2667022 (2011j:20075)

39. Guy Rousseau, Immeubles des groupes réductifs sur les corps locaux, U.E.R. Mathématique, Université Paris XI, Orsay, 1977, Thèse de doctorat, Publications Mathématiques d’Orsay,

No. 221-77.68, http://www.iecl.univ-lorraine.fr/~Guy.Rousseau/Textes/. MR 0491992

(58 #11158)

40. , Euclidean buildings, Géométries à courbure négative ou nulle, groupes discrets et rigidités, Sémin. Congr., vol. 18, Soc. Math. France, Paris, 2009, pp. 77–116. MR 2655310 (2011m:20072)

41. Neantro Saavedra Rivano, Catégories Tannakiennes, Lecture Notes in Mathematics, Vol. 265, Springer-Verlag, Berlin, 1972. MR 0338002 (49 #2769)

42. Daniel Schäppi, A characterization of categories of coherent sheaves of certain algebraic stacks, Preprint, (2012).

43. , The formal theory of Tannaka duality, Astérisque (2013), no. 357, viii+140. MR 3185459

44. Jean-Pierre Serre, Groupes de Grothendieck des schémas en groupes réductifs déployés, Inst. Hautes Études Sci. Publ. Math. (1968), no. 34, 37–52. MR 0231831 (38 #159)

45. John R. Stembridge, The partial order of dominant weights, Adv. Math. 136 (1998), no. 2, 340–364. MR 1626860 (2000a:06038)

46. Jacques Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, Berlin-New York, 1974. MR 0470099

47. , Reductive groups over local fields, Automorphic forms, representations and Lfunctions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 29–69. MR 546588 (80h:20064)

48. Angelo Vistoli, Grothendieck topologies, fibered categories and descent theory, Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Amer. Math. Soc., Providence, RI, 2005, pp. 1–104. MR 2223406

49. Torsten Wedhorn, On Tannakian duality over valuation rings, J. Algebra 282 (2004), no. 2, 575–609. MR 2101076 (2005j:18007)

50. Jr Kevin Michael Wilson, A Tannakian description for parahoric Bruhat-Tits group schemes, ProQuest LLC, Ann Arbor, MI, 2010, Thesis (Ph.D.)–University of Maryland, College Park. MR 2941465

51. Paul Ziegler, Graded and filtered fiber functors on Tannakian categories, J. Inst. Math. Jussieu 14 (2015), no. 1, 87–130. MR 3284480