MAZUR’S INEQUALITY AND LAFFAILLE’S THEOREM

CHRISTOPHE CORNUT

Abstract. We look at various questions related to filtrations in p-adic Hodge theory, using a blend of building and Tannakian tools. Specifically, Fontaine and Rapoport used a theorem of Laffaille on filtered isocrystals to establish a converse of Mazur’s inequality for isocrystals. We generalize both results to the setting of (filtered) G-isocrystals and also establish an analog of Totaro’s ⊗-product theorem for the Harder-Narasimhan filtration of Fargues.

1. Introduction

Many spaces that show up in p-adic Hodge theory are closely related to buildings, if only because they involve filtrations and lattices (or norms), which are respectively classified by Tits and Bruhat-Tits buildings. Quite naturally, building theoretical tools have thus occasionally been used to study these spaces, as in [22], [25] or [3]. In continuation with [7], this paper tries to promote the idea that a more systematic use of buildings sometimes leads to streamlined proofs and new results in p-adic Hodge theory, and certainly already provides an enlightening new point of view on various classical notions, simply by recasting them in a metric geometry framework.

For instance in [7], we showed that the Newton decomposition of an isocrystal (V, φ) over an algebraically closed residue field k of characteristic p > 0 can be identified with the translation vector of φ, viewing φ as a semi-simple isometry of the extended Bruhat-Tits building of G = GL(V) – this still holds when k is merely perfect. Filtrations F on V come into the picture in the metric guise of nonexpanding maps on the building, and the weak admissibility of a filtered isocrystal (V, φ, F) is related to the joint dynamic of the (φ, F)-actions on the building: this was somehow first observed by Laffaille [19], who showed that (V, φ, F) is weakly admissible if and only if V contains a strongly divisible lattice, giving rise to a special point of the building fixed by the composition of φ and F. This criterion for weak admissibility implies that V contains a lattice L such that L and φL are in a given relative position μ if and only if there is a weakly admissible filtration F of type μ on V. Assuming that the residue field k is algebraically closed, Fontaine and Rapoport [12] gave a criterion for the existence of such an F, thereby establishing a converse to the already known necessary condition for the existence of L – Mazur’s inequality. Our first target is the generalization of these results to more general reductive groups G (Theorem 7 and 10). Note that the converse of Mazur’s inequality for unramified groups was already established by [13], following an entirely different strategy suggested by [16]. Our method yields a somewhat weaker result: we say nothing about the existence of hyperspecial (strongly divisible) points.

2000 Mathematics Subject Classification. 14F 30, 20G25.

Key words and phrases. Filtered G-isocrystals, Mazur’s inequality, Harder-Narasimhan filtrations and tensor products.

Here is a rough dictionary that may help the reader digest the previous paragraph and warm-up for the sequel. Norms on V correspond to points on the (extended) Bruhat-Tits building X of G, and lattices correspond to a G-orbit X^{∘} of hyperspecial points in X. There is a canonical G-equivariant distance d on X with nice convexity properties: (X, d) is a complete CAT(0)-space, of which X^{∘} is a closed discrete subset. The ℝ-graduations G and (non-increasing) ℝ-filtrations F on V correspond to asymptotic classes of, respectively, (constant speed) geodesic lines ℓ : ℝ → X, and geodesic rays c : ℝ_{+} → X. In particular, ℝ-filtrations act on X by non-expanding maps as follows: x + F = c(1), where c is the unique geodesic ray in F emanating from x ∈ X. If x ∈ X^{∘} corresponds to a lattice L in V and F is a ℤ-filtration, then x + F also belongs to X^{∘} and corresponds to the lattice

L + F = ∑_{i∈ℤ} p^{−i}L ∩ F^{i}.

The G-orbits of ℝ-filtrations on V are classified by their type t(F) ∈ ℝ_{≥}^{r}, where

ℝ_{≥}^{r} = {(x_{1}, ⋯, x_{r}) : x_{1} ≥ ⋯ ≥ x_{r}}

is usually identified with the set of concave polygons over [0, r] whose brake points have integral x-coordinate. We equip ℝ_{≥}^{r} with its usual partial order, given by

(x_{1}, ⋯, x_{r}) ≤ (x′_{1}, ⋯, x′_{r}) ⟺ \begin{cases}∑_{i=1}^{j} x_{i} ≤ ∑_{i=1}^{j} x′_{i} & ∀j ∈ \{1, ⋯, r − 1\} \\ ∑_{i=1}^{r} x_{i} = ∑_{i=1}^{r} x′_{i}. \end{cases}

We set ℚ_{≥}^{r} = ℝ_{≥}^{r} ∩ ℚ^{r} and ℤ_{≥}^{r} = ℝ_{≥}^{r} ∩ ℤ^{r}. The formula d(x, x + F) = t(F) defines a G-invariant vector-valued distance d : X × X → ℝ_{≥}^{r}, whose composition with the standard euclidean norm on ℝ_{≥}^{r} ⊂ ℝ^{r} is equal to the canonical distance d of X. The restriction of d to X^{∘} yields a bijection G\ (X^{∘} × X^{∘}) ≃ ℤ_{≥}^{r}, which is the usual invariant describing the relative position of two lattices in V.

The Frobenius φ of V induces an isometry φ of (X, d) which preserves X^{∘}. It is a semi-simple isometry, which means that if min(φ) = inf {d(φx, x) : x ∈ X}, then

Min(φ) = {x ∈ X : d(φx, x) = min(φ)}

is non-empty. This is a closed, convex, φ-stable subset of X, equal to the disjoint union of the φ-stable geodesic lines of X. The common asymptotic class of these lines corresponds to the Newton ℚ-graduation G_{N} of (V, φ). The induced pair of opposed Newton ℚ-filtrations (F_{N}, F_{N}^{ι}) on V act on x ∈ Min(φ) as follows:

x + F_{N} = φ^{−1}(x) and x + F_{N}^{ι} = φ(x).

In particular, the Newton type t_{N} = t(F_{N}) ∈ ℚ_{≥}^{r} is equal to d(φ(x), x) for every x ∈ Min(φ), and min(φ) = ∥t_{N}∥. Moreover, we have

Min(φ) = X_{φ}(t_{N}) = X_{φ}(F_{N})

where for any μ ∈ ℝ_{≥}^{r} and any ℝ-filtration F on V of type t(F) = μ,

X_{φ}(μ) = {x ∈ X : d(φx, x) = μ} ,
X_{φ}(F) = {x ∈ X : x = φx + F} .

By definition, X_{φ}(μ) is a closed Aut(V, φ)-stable subset of X, X_{φ}(F) is a closed convex Aut(V, φ, F)-stable subset of X, and we have a covering

(1.1)  X_{φ}(μ) = ∪_{t(F)=μ}X_{φ}(F).

For μ ∈ ℤ_{≥}^{r} and any ℤ-filtration F on V of type t(F) = μ, we set

X_{φ}^{∘}(μ) = X_{φ}(μ) ∩ X^{∘} and X_{φ}^{∘}(F) = X_{φ}(F) ∩ X^{∘}

so that X_{φ}^{∘}(μ) is stable under Aut(V, φ), X_{φ}^{∘}(F) is stable under Aut(V, φ, F), and

(1.2)  X_{φ}^{∘}(μ) = ∪_{t(F)=μ}X_{φ}^{∘}(F).

The subsets X_{φ}^{∘}(μ) are usually called affine Deligne-Lusztig “varieties”, and there is an extensive literature about them (and various generalizations): they show up in the description of special fibers of Shimura varieties, and indeed can be equipped with some sort of algebraic structure over the relevant residue field. The covering subsets X_{φ}^{∘}(F) are made of strongly divisible lattices in the sense of [18, 11], and they are related to stable lattices in crystalline Galois representations, but the covering (1.2) itself has not attracted much attention, even though it was crucially used in [12] to obtain a converse to Mazur’s inequality using Laffaille’s criterion for weak admissibility, as we shall now explain.

In this simple setting where G = GL(V), Mazur’s inequality and Laffaille’s criterion can be stated as follows: for any μ ∈ ℤ_{≥}^{r} and any ℤ-filtration F on V,

(1.3)  X_{φ}^{∘}(μ) ≠ ∅ ⟹ μ ≥ t_{N} in ℝ_{≥}^{r},
(1.4)  X_{φ}^{∘}(F) ≠ ∅ ⟺ (V, φ, F) is (weakly) admissible.

Let Adm_{φ}(μ) be the set of all (weakly) admissible ℤ-filtrations F of type t(F) = μ on (V, φ). Using (1.2) and (1.4), the converse of Mazur’s inequality (1.3) becomes

(1.5)  μ ≥ t_{N} in ℝ_{≥}^{r} ⟹ Adm_{φ}(μ) ≠ ∅.

This existence result is established by Fontaine and Rapoport in [12]: they show that any sufficiently generic ℤ-filtration of type μ with μ ≥ t_{N} is weakly admissible.

Forgetting hyperspecial points, let us now explain the results that we will eventually generalize in the first part of this paper – from GL(V) to an arbitrary but unramified reductive group G. First, we have the following variants of Mazur’s inequality and Laffaille’s criterion: for any μ ∈ ℝ_{≥}^{r} and any ℝ-filtration F on V,

(1.6)  X_{φ}(μ) ≠ ∅ ⟹ μ ≥ t_{N} in ℝ_{≥}^{r},
(1.7)  X_{φ}(F) ≠ ∅ ⟺ (V, φ, F) is weakly admissible.

Note that weak admissibility makes perfect sense for ℝ-filtrations. Let Adm_{φ}(μ) be the set of all weakly admissible ℝ-filtrations F of type t(F) = μ on (V, φ). Using (1.1) and (1.7) as above, the converse of Mazur’s inequality (1.6) becomes

(1.8)  μ ≥ t_{N} in ℝ_{≥}^{r} ⟹ Adm_{φ}(μ) ≠ ∅.

As in [12], we establish the latter existence result by showing that any sufficiently generic ℝ-filtration of type μ with μ ≥ t_{N} is weakly admissible, with a proof now based on a mixture of algebraic geometry and building theoretical tools.

Remark 1. The Fontaine-Rapoport method sketched above amounts to reduce a relatively hard existence problem in a CAT(0)-space (the converse of (1.3) or (1.6)) to an easier existence problem on the boundary of that space ((1.5) or (1.8)), using a suitable fixed point theorem ((1.4) or (1.7)). This line of thought has been used in other problems unrelated to p-adic Hodge theory, see [14] or the appendix of [17].

Various criteria for weak admissibility of unramified filtered G-isocrystals are listed in section 2, including our generalization of Laffaille’s criterion (the equivalence (1) ⟺ (3) of Theorem 7). Mazur’s inequality is addressed in section 3 (Theorem 10). In the ramified case, we establish another list of criteria for weak admissibility in section 4, using a Harder-Narasimhan filtration and its compatibility with tensor products. There are now many proofs of this compatibility, which essentially says that the tensor product of weakly admissible filtered isocrystals is weakly admissible. In the unramified case, this was an immediate consequence of Laffaille’s criterion. Faltings [9] generalized Laffaille’s proof to the unramified case, and Totaro [24] found yet another proof related to Geometric Invariant Theory. It turns out that Totaro’s proof has a very clean building theoretical translation. Instead of just repeating his arguments in this smoother framework, we chose to treat the slightly more difficult case of semi-stable weakly admissible filtered isocrystals in section 5 (Theorem 15), where semi-stability here refers to the notion introduced by Fargues in connection with his own Harder-Narasimhan filtrations [10]. In particular, we prove that Fargues’s filtration on weakly admissible filtered isocrystals is compatible with tensor products (Theorem 15), and moreover show that it is given by a convex projection (Lemma 13 and Proposition 16): the algebraically defined Fargues ℚ-filtration F_{F} of a weakly admissible filtered isocrystal (V, φ, F) is the best approximation (in a metric sense) of the Newton ℚ-filtration F_{N}^{ι} of (V, φ) by a filtration whose steps are weakly admissible subspaces of (V, φ, F).

In sections 2 and 3, our filtered isocrystals are unramified (i.e. K = K_{0} in standard notations). In the remaining sections, they are defined with respect to an extension L of the fraction field K of the Witt vectors W (k), which we do not require to be finite or unramified. The residue field k is a perfect field of characteristic p > 0, which is only required to be algebraically closed in Theorem 10. Throughout the paper, we work with Γ-filtrations (defined in 2.1), where Γ is a non-trivial subgroup of ℝ. For applications to p-adic Hodge theory, one should take Γ = ℤ for Hodge filtrations and Γ = ℚ for Newton, Harder-Narasimhan and Fargues filtrations.

The need to properly identify ℝ-filtrations with non-expanding operators on Bruhat-Tits buildings led us to write [5], which was initially meant to be the first part of this paper, but eventually grew way out of proportion. This will be our general background reference for everything pertaining to Γ-graduations, Γ-filtrations and their types. We similarly refer to [15, 20] for G-isocrystals, [21, 8] for Γ-filtered G-isocrystals, [1] for CAT(0)-spaces and [23, 5] for Bruhat-Tits buildings.

2. Unramified Filtered Isocrystals

In this section, we first review various basic notions related to graduations and filtrations (2.1), isocrystals (2.2), and unramified filtered isocrystals (2.3). We then add G-structures to them using the Tannakian framework (2.4), and gently shift from the latter to the building framework (2.5-2.9) using [5] for the translation. Our main result is Theorem 7, in particular the equivalence (1) ⟺ (3) relating weak admissibility of an unramified filtered G-isocrystal (φ, F) to the joint dynamic of the induced operators φ and F on the extended Bruhat-Tits building of G.

2.1. Let K be a field, Γ ≠ 0 a subgroup of ℝ, V a finite dimensional K-vector space. A Γ-graduation on V is a collection of K-subspaces G = (G_{γ})_{γ∈Γ} such that V = ⊕_{γ∈Γ}G_{γ}. A Γ-filtration on V is a collection of K-subspaces F = (F^{γ})_{γ∈Γ} for which there exists a Γ-graduation G such that F = Fil(G), i.e. F^{γ} = ⊕_{η≥γ}G_{η} for every γ ∈ Γ. We call any such G a splitting of F. If W is a K-subspace of V, then F|W = (F^{γ} ∩ W)_{γ∈Γ} is again a Γ-graduation on W. The degree of F is given by

deg(F) = ∑_{γ} dim_{K} (Gr_{F}^{γ}) · γ

where Gr_{F}^{γ} = F^{γ}/F_{+}^{γ} with F_{+}^{γ} = ∪_{η>γ}F^{η} for every γ ∈ Γ. The scalar product of two Γ-filtrations F_{1} and F_{2} on V is defined by the analogous formula

⟨F_{1}, F_{2}⟩ = ∑_{γ_{1},γ_{2}} dim_{K} (\frac{F_{1}^{γ_{1}} ∩ F_{2}^{γ_{2}}}{F_{1,+}^{γ_{1}} ∩ F_{2}^{γ_{2}} + F_{1}^{γ_{1}} ∩ F_{2,+}^{γ_{2}}} ) · γ_{1}γ_{2}
= ∑_{γ} γ · deg Gr_{F_{1}}^{γ}(F_{2}) = ∑_{γ} γ · deg Gr_{F_{2}}^{γ}(F_{1})

where Gr_{F_{i}}^{γ}(F_{j}) is the Γ-filtration induced by F_{j} on Gr_{F_{i}}^{γ}. We denote by Gr^{Γ}(K) and Fil^{Γ}(K) the categories of Γ-graded and Γ-filtered finite dimensional K-vector spaces, and equip them with their usual structure of exact K-linear ⊗-categories. The Fil and Gr-constructions yield exact K-linear ⊗-functors

Fil : Gr^{Γ}(K) ↔ Fil^{Γ}(K) : Gr.

The formula (ιG)_{γ} = G_{−γ} defines an involutive exact ⊗-endofunctor of Gr^{Γ}(K).

2.2. Let k be a perfect field of characteristic p > 0, W (k) the ring of Witt vectors, K = W (k)[\frac{1}{p}] its fraction field, σ the Frobenius of k, W (k) or K. An isocrystal over k is a finite dimensional K-vector space V equipped with a σ-linear isomorphism φ. Since k is perfect, there is a canonical slope decomposition (V, φ) = ⊕_{λ∈ℚ}(V_{λ}, φ|V_{λ}). The φ-stable K-subspace V_{λ} is the union of all finitely generated W (k)-submodules L of V such that φ^{(h)}(L) = p^{d}L, where λ = \frac{d}{h} with (d, h) ∈ ℤ × ℕ. We denote by Iso(k) the category of isocrystals over k. It is a ℚ_{p}-linear (non-neutral) Tannakian category and the above slope decomposition yields an exact ℚ_{p}-linear ⊗-functor

ν : Iso(k) → Gr^{ℚ}(K).

2.3. A Γ-filtered isocrystal over k is an isocrystal (V, φ) together with a Γ-filtration F on V. The K-vector space V thus carries three filtrations: the Hodge Γ-filtration F_{H} = F and the pair of opposed φ-stable Newton ℚ-filtrations F_{N} = Fil(G_{N}) and F_{N}^{ι} = Fil(ιG_{N}) attached to the slope decomposition G_{N} = ν(V, φ), given by

F_{N}^{λ} = ⊕_{η≥λ}V_{η} and F_{N}^{ιλ} = ⊕_{η≥λ}V_{−η}.

For Γ = ℤ, these objects are usually called unramified filtered isocrystals over k.

Lemma 2. The following conditions are equivalent:

(1) (V, φ, F) is weakly admissible, i.e.: (a) deg(F_{H}) = deg(F_{N}) and

(b) deg(F_{H}|W) ≤ deg(F_{N}|W) for every φ-stable K-subspace W of V,

(2) For every φ-stable Δ-filtration Ξ on V, ⟨F_{H}, Ξ⟩ ≤ ⟨F_{N}, Ξ⟩.

(3) For every φ-stable Δ-filtration Ξ on V, ⟨F_{H}, Ξ⟩ + ⟨F_{N}^{ι}, Ξ⟩ ≤ 0.

In (2) and (3), Δ is any non-trivial subgroup of ℝ.

Proof. For a φ-stable K-subspace W of V and a, b, δ in Δ with a ≤ b, set

Ξ_{W,a,b}^{δ} = \begin{cases}V & if δ ≤ a, \\ W & if a < δ ≤ b, \\ 0 & if b < δ. \end{cases}

This defines a φ-stable Δ-filtration Ξ_{W,a,b} = (Ξ_{W,a,b}^{δ})_{δ∈Δ} on V with

⟨F_{H}, Ξ_{W,a,b}⟩ = a · deg(F_{H}) + (b − a) · deg(F_{H}|W)
and ⟨F_{N}, Ξ_{W,a,b}⟩ = a · deg(F_{N}) + (b − a) · deg(F_{N}|W)

Now (2) implies that ⟨F_{H}, Ξ_{W,a,b}⟩ ≤ ⟨F_{N}, Ξ_{W,a,b}⟩ for every choice of W and a ≤ b, from which (1) easily follows. Conversely, let Ξ be any φ-stable Δ-filtration on V. Write {δ ∈ Δ : Gr_{Ξ}^{δ} ≠ 0} = {δ_{1} < ⋯ < δ_{n}}. Then Ξ^{δ_{i}} is a φ-stable K-subspace of V. Put d_{H}(i) = deg(F_{H}|Ξ^{δ_{i}}) and d_{N}(i) = deg(F_{N}|Ξ^{δ_{i}}). Then

⟨F_{H}, Ξ⟩ = ∑_{i=1}^{n} δ_{i} · Δ_{H}(i) and ⟨F_{N}, Ξ⟩ = ∑_{i=1}^{n} δ_{i} · Δ_{N}(i)

where Δ_{⋆}(i) = d_{⋆}(i) − d_{⋆}(i + 1) for 1 ≤ i < n and Δ_{⋆}(n) = d_{⋆}(n), so that also

⟨F_{H}, Ξ⟩ = ∑_{i=1}^{n} Δ_{i} · d_{H}(i) and ⟨F_{N}, Ξ⟩ = ∑_{i=1}^{n} Δ_{i} · d_{N}(i)

where Δ_{i} = δ_{i}−δ_{i−1} for 1 < i ≤ n and Δ_{1} = δ_{1}. Now (1) implies that d_{H}(i) ≤ d_{N}(i) for all i and d_{H}(1) = d_{N}(1) from which (2) follows since Δ_{i} > 0 for i > 1. Note that with these notations, we also find that

⟨F_{N}^{ι}, Ξ⟩ = ∑_{i=1}^{n} Δ_{i} · d_{N}^{ι}(i) with d_{N}^{ι}(i) = deg(F_{N}^{ι}|Ξ^{δ_{i}}).

But deg(F_{N}) + deg(F_{N}^{ι}) = 0 and more generally deg(F_{N}|W) + deg(F_{N}^{ι}|W) = 0 for every φ-stable K-subspace W of V by functoriality of the slope ℚ-graduation, thus d_{N}(i) + d_{N}^{ι}(i) = 0 for all i. Therefore ⟨F_{N}, Ξ⟩ + ⟨F_{N}^{ι}, Ξ⟩ = 0 for every φ-stable ℝ-graduation Ξ on V, which proves that (2) ⇔ (3). □

We denote by Fil^{Γ}Iso(k) the category of Γ-filtered isocrystals over k and denote by Fil^{Γ}Iso(k)^{wa} its full sub-category of weakly admissible objects, as defined in the previous lemma. These are both exact ℚ_{p}-linear ⊗-categories, the smaller one is also abelian, and it is even a neutral Tannakian category when Γ = ℤ [4].

2.4. Fix a reductive group G over ℚ_{p}, let Rep(G) be the Tannakian category of algebraic representations of G on finite dimensional ℚ_{p}-vector spaces, and let

V : Rep(G) → Vect(ℚ_{p})

be the natural fiber functor. A Γ-graduation on V_{K} = V ⊗ K (resp. Γ-filtration on V_{K}, G-isocrystal over k or Γ-filtered G-isocrystal over k) is a factorization of

V_{K} : Rep(G) → Vect(K)

through the natural fiber functor of the relevant category Gr^{Γ}(K) (resp. Fil^{Γ}(K), Iso(k), Fil^{Γ}Iso(k)). We require these factorizations to be exact and compatible with the ⊗-products and their neutral objects. The set of all such factorizations is the set of K-valued points of a smooth and separated ℚ_{p}-scheme 𝔾^{Γ}(G) (resp. 𝔽^{Γ}(G), G, G × 𝔽^{Γ}(G)). The G-isocrystal attached to b ∈ G(K) maps ρ ∈ Rep(G) to (V_{K}(ρ), ρ(b) ∘ Id ⊗ σ) (this would be (V, ρ) ↦ (V_{K}, ρ(b)σ) in standard notations). We identify b with the corresponding Frobenius element φ = (b, σ) in G(K) ⋊ ⟨σ⟩. For ρ ∈ Rep(G), we denote by G(ρ), F(ρ) or φ(ρ) the Γ-graduation, Γ-filtration or Frobenius on V_{K}(ρ) = V (ρ) ⊗ K attached to a Γ-graduation G, Γ-filtration F or Frobenius φ on V_{K}. We say that a Γ-filtered G-isocrystal

(φ, F) : Rep(G) → Fil^{Γ}Iso(k)

is weakly admissible if it factors through the subcategory Fil^{Γ}Iso(k)^{wa} of Fil^{Γ}Iso(k).

Remark 3. The G-isocrystals or Γ-filtered G-isocrystals considered in this paper are trivial in the sense that the underlying fiber functor is required to be the trivial fiber functor V_{K}. We caution our reader not to apply our results carelessly on more general G-isocrystals or Γ-filtered G-isocrystals, unless he or she has checked that the underlying fiber functors are at least isomorphic (over K) to the trivial one.

2.5. There is a G-equivariant sequence of morphisms of ℚ_{p}-schemes

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

where ℂ^{Γ}(G) = G\𝔾^{Γ}(G) = G\𝔽^{Γ}(G) and the Fil-morphism is induced by the previous Fil-functors. We set G^{Γ}(G_{K}) = 𝔾^{Γ}(G)(K), F^{Γ}(G_{K}) = 𝔽^{Γ}(G)(K) and denote by C^{Γ}(G_{K}) the image of F^{Γ}(G_{K}) in ℂ^{Γ}(G)(K) under the type morphism t, giving rise to a G(K)-equivariant sequence of surjective maps [5, 4.1]

\xymatrix{G^{Γ}(G_{K}) \ar@{->>}[r]^{Fil} & F^{Γ}(G_{K}) \ar@{->>}[r]^{t} & C^{Γ}(G_{K}).}

Then C^{Γ}(G_{K}) = G(K)\G^{Γ}(G_{K}) = G(K)\F^{Γ}(G_{K}), and it is a monoid. When Γ = ℝ, it is the usual (relative) Weyl cone attached to G over K, and it comes equipped with a partial order (the dominance order). If S ⊂ G_{K} is a maximal K-split torus and B ⊂ G_{K} is a minimal parabolic subgroup of G_{K} containing the centralizer of S, then C^{Γ}(G_{K}) ≃ Hom^{+}(X^{∗}(S), Γ), where Hom^{+}(X^{∗}(S), Γ) is the monoid of all morphisms f : X^{∗}(S) → Γ which are non-negative on the roots of S in the Lie algebra of B [5, 4.1.10]; when Γ = ℝ, f_{1} ≤ f_{2} if and only if f_{2} − f_{1} is a non-negative linear combination of B-positive (relative) coroots [5, 2.4 and 5.1.2].

The Newton slope decomposition of section 2.2 yields a morphism

ν : G(K) → G^{ℚ}(G_{K})

such that ν(g ⋄ b) = g · ν(b) for g, b ∈ G(K) with g ⋄ b = gbσ(g)^{−1}, i.e. a G(K)- equivariant morphism φ ↦ ν(φ) from the G(K)-coset G(K) · σ of Frobenius elements in G(K) ⋊ ⟨σ⟩ (with the action by conjugation) to G^{ℚ}(G_{K}).

All of these constructions are covariantly functorial in G, Γ and k.

2.6. A Γ-filtered G-isocrystal (φ, F) ∈ G(K) · σ × F^{Γ}(G_{K}) thus again defines three filtrations on V_{K}: the Hodge Γ-filtration F_{H} = F in F^{Γ}(G_{K}) and the pair of opposed φ-stable Newton ℚ-filtrations F_{N} = Fil(G_{N}) and F_{N}^{ι} = Fil(ιG_{N}) in F^{ℚ}(G_{K}) attached to the slope ℚ-graduation G_{N} = ν(φ) in G^{ℚ}(G_{K}).

2.7. On the other hand, a Γ-filtered G-isocrystal (φ, F) also “acts” on the extended Bruhat-Tits building B^{e}(G_{K}) of G over K, through the classical action of the group G(K) ⋊ ⟨σ⟩ on B^{e}(G_{K}) and the canonical G(K) ⋊ ⟨σ⟩-equivariant +-operation

+ : B^{e}(G_{K}) × F^{ℝ}(G_{K}) → B^{e}(G_{K})

which is defined in [5, 6.2.5]. We recall that the extended Bruhat-Tits building is the direct product of the usual (or reduced) isogeny-invariant Bruhat-Tits building B(G_{K}) by X_{∗}(A)⊗ℝ, where A is the maximal K-split torus in the center of G_{K} and X_{∗}(A) is the group of cocaracters of A. The extended Bruhat-Tits building B^{e}(G_{K}) and vectorial Tits building F^{ℝ}(G_{K}) are covered by apartments, both indexed by the maximal K-split subtori of G_{K}; for any such torus, the corresponding apartment in the Bruhat-Tits building is canonically equipped with the structure of an affine space whose underlying vector space is the matching apartment in the vectorial Tits building, and the above +-map is obtained by gluing the resulting faithfully transitive actions on all apartments. To simplify our notations, we set

X = B^{e}(G_{K}) = B(G_{K}) × X_{∗}(A) ⊗ ℝ.

For ρ ∈ Rep(G), φ(ρ) and F(ρ) similarly act on

X(ρ) = B^{e}(GL(V (ρ))_{K}).

Definition 4. For a Γ-filtered G-isocrystal (φ, F), we set

X_{φ}(F) = {x ∈ X : φ(x) + F = x} = {x ∈ X : α(x) = x}

where α(x) = φ(x) + F for all x in X. For ρ ∈ Rep(G), we set

X_{φ}(F)(ρ) = {x ∈ X(ρ) : φ(ρ)(x) + F(ρ) = x} = {x ∈ X(ρ) : α(ρ)(x) = x}

where α(ρ)(x) = φ(ρ)(x) + F(ρ) for all x in X(ρ).

2.8. As explained in [5, §5], the choice of a faithful τ ∈ Rep(G) induces a bunch of numerical functions on our buildings. The G(K) ⋊ ⟨σ⟩-invariant “scalar product”

⟨−, −⟩_{τ} : F^{ℝ}(G_{K}) × F^{ℝ}(G_{K}) → ℝ, ⟨F_{1}, F_{2}⟩_{τ} = ⟨F_{1}(τ), F_{2}(τ)⟩

yields a ⟨σ⟩-invariant length function

∥−∥_{τ} : C^{ℝ}(G_{K}) → ℝ_{+}, ∥t(F)∥_{τ} = ∥F∥_{τ} = ⟨F, F⟩τ^{\frac{1}{2}}

whose composition with the G(K) ⋊ ⟨σ⟩-equivariant “vector valued distance”

d : B^{e}(G_{K}) × B^{e}(G_{K}) → C^{ℝ}(G_{K}), d(x, x + F) = t(F)

yields a genuine G(K) ⋊ ⟨σ⟩-invariant distance

d_{τ} : B^{e}(G_{K}) × B^{e}(G_{K}) → ℝ_{+}

which turns B^{e}(G_{K}) into a complete CAT(0)-space. The above +-map defines a G(K) ⋊ ⟨σ⟩-equivariant “action” of F^{ℝ}(G_{K}) on X = B^{e}(G_{K}) by non-expanding maps, which yields a G(K) ⋊ ⟨σ⟩-equivariant identification of F^{ℝ}(G_{K}) with the cone 𝒞 (∂X) on the visual boundary ∂X of X (acting on X as in [6]). Under this identification, ∂X = {ξ ∈ F^{ℝ}(G_{K}) : ∥ξ∥_{τ} = 1}. On the other hand, the Frobenius φ is an isometry of (X, d_{τ}). Thus α is a non-expanding map of (X, d_{τ}). In particular,

X_{φ}(F) ≠ ∅ ⟺ α has bounded orbits.

Lemma 5. The isometry φ of (X, d_{τ}) is semi-simple, i.e. Min(φ) ≠ ∅ where

Min(φ) = {x ∈ X : d_{τ} (x, φ(x)) = min(φ)}
with min(φ) = inf {d_{τ} (x, φ(x)) : x ∈ X} .

Moreover, φ(x) = x + F_{N}^{ι} for every x ∈ Min(φ) and 𝒞 (∂Min(φ)) is the fixed point set of φ acting on 𝒞 (∂X) = F^{ℝ}(G_{K}), i.e. the set of φ-stable ℝ-filtrations on V_{K}.

Proof. The first two assertions are established in [7] when k is algebraically closed and the general case follows from [1, II.6.2 (4)]. The final claim holds true for any semi-simple isometry φ of a CAT(0)-space (X, d). Indeed, ∂Min(φ) is contained in (∂X)^{φ=Id} by [1, II.6.8 (4)]. Conversely, suppose that ξ ∈ ∂X is fixed by φ and choose some x in Min(φ) ≠ ∅. Then for every t ≥ 0,

φ(x + tξ) = φ(x) + tφ(ξ) = φ(x) + tξ,

thus x + tξ also belongs to Min(φ) since

min(φ) ≤ d(x + tξ, φ(x + tξ)) = d(x + tξ, φ(x) + tξ) ≤ d(x, φ(x)) = min(φ)

by convexity of the CAT(0)-distance d, therefore ξ ∈ ∂Min(φ). □

Remark 6. We should rather write min_{τ} (φ) to reflect the dependency of min(φ) on the choice of τ, but the subset Min(φ) of X really does not depend upon that choice: it is precisely the set of all x ∈ X such that φ(x) = x + F_{N}^{ι}.

2.9. For any ρ ∈ Rep(G), applying these constructions to GL(V (ρ)) and its tautological faithful representation on V (ρ), we obtain a canonical distance d_{ρ} on X(ρ) for which φ(ρ) is an isometry while α(ρ) is a non-expanding map. Moreover:

X_{φ}(F)(ρ) ≠ ∅ ⟺ α(ρ) has bounded orbits.

Here the cones 𝒞 (∂X(ρ)) and 𝒞 (∂Min(φ(ρ))) are respectively identified with the set of all ℝ-filtrations on V_{K}(ρ) and its subset of φ(ρ)-stable ℝ-filtrations.

2.10. Fix a faithful τ ∈ Rep(G) and some subgroup Δ ≠ 0 of ℝ. Then:

Theorem 7. The following conditions are equivalent:

(1) (φ, F) is weakly admissible. (1_{ρ}) For every ρ ∈ Rep(G), (V_{K}(ρ), φ(ρ), F(ρ)) is weakly admissible. (1_{τ}) (V_{K}(τ), φ(τ), F(τ)) is weakly admissible. (2^{Δ}) For every φ-stable Δ-filtration Ξ on V_{K},

⟨F_{H}, Ξ⟩_{τ} ≤ ⟨F_{N}, Ξ⟩_{τ} .

(2_{ρ}^{Δ}) For every ρ ∈ Rep(G) and every φ-stable Δ-filtration Ξ on V_{K}(ρ),

⟨F_{H}(ρ), Ξ⟩ ≤ ⟨F_{N}(ρ), Ξ⟩ .

(2_{τ}^{Δ}) For every φ-stable Δ-filtration Ξ on V_{K}(τ),

⟨F_{H}(τ), Ξ⟩ ≤ ⟨F_{N}(τ), Ξ⟩ .

(3) X_{φ}(F) ≠ ∅. (3_{ρ}) For every ρ ∈ Rep(G), X_{φ}(F)(ρ) ≠ ∅. (3_{τ}) X_{φ}(F)(τ) ≠ ∅.

(4) α has bounded orbits in (X, d_{τ}). (4_{ρ}) For every ρ ∈ Rep(G), α(ρ) has bounded orbits in (X(ρ), d_{ρ}). (4_{τ}) α(τ) has bounded orbits in (X(τ), d_{τ}).

Remark 8. It follows from the proof of Lemma 2 that here also

⟨F_{N}, Ξ⟩_{τ} + ⟨F_{N}^{ι}, Ξ⟩_{τ} = 0

for every φ-stable ℝ-filtration Ξ on V_{K}.

Proof. Obviously (1) ⇔ (1_{ρ}), (2_{τ}^{Δ}) ⇒ (2^{Δ}) and (x_{ρ}) ⇒ (x_{τ}) for x ∈ {1, 2, 3, 4}. We have already seen that (3_{x}) ⇔ (4_{x}) for all x ∈ {∅, ρ, τ} and Lemma 2 shows that (1_{ρ}) ⇔ (2_{ρ}^{Δ}) and (1_{τ}) ⇔ (2_{τ}^{Δ}). Set (2_{x}) = (2_{x}^{ℝ}) for x ∈ {∅, ρ, τ} and let (2′), (3′) and (4′) be (2), (3) and (4) for the base change (φ′, F′) of (φ, F) to an algebraic closure k′ of k. Then obviously (2_{x}) ⇒ (2_{x}^{Δ}), (2′) ⇒ (2) and (4) ⇔ (4′) – thus also (3′) ⇔ (3). The main theorem of [6] asserts that (2) ⇔ (3) provided that φ satisfies a certain decency condition, which holds true by the main result of [7] when k is algebraically closed. Therefore (2′) ⇔ (3′). We will show below that (2^{Δ}) ⇒ (2) (2.11), (2) ⇒ (2′) (2.12) and (4) ⇒ (4_{ρ}) (2.13). Since (3_{ρ}) ⇔ (2_{ρ}) and (3_{τ}) ⇔ (2_{τ}) are special cases of (3) ⇔ (3′) ⇔ (2′) ⇔ (2) (applied respectively to G = GL(V (ρ)) and G = GL(V (τ))), the theorem follows. □

Remark 9. Apart from the added generality of allowing Γ ≠ ℤ, the equivalence of most of these conditions is either well-known or trivial and can be found in various places. For instance (1) ⇔ (1_{τ}) follows from [21, 1.18] and (1_{τ}) ⇔ (3_{τ}) is Laffaille’s criterion for weak admissibility [18, 3.2]. Our only new contribution is (2) ⇔ (3), which is essentially our metric reformulation of Laffaille’s algebraic proof in [6].

2.11. (2^{Δ}) ⇒ (2). Suppose that (2) does not hold: there exists a φ-stable ℝfiltration Ξ on V_{K} such that ⟨F_{H}, Ξ⟩_{τ} > ⟨F_{N}, Ξ⟩_{τ}. By [5, 4.1.13], there exists a maximal split torus S of G_{K} with character group M = Hom_{K}(S, 𝔾_{m,K}) and a morphism f : M → ℝ such that for every ρ ∈ Rep(G) and γ ∈ ℝ,

Ξ(ρ)^{γ} = ⊕_{m∈M, f(m)≥γ}V (ρ)_{m} in V_{K}(ρ) = V (ρ_{K})

where V (ρ_{K}) = ⊕_{m∈M}V (ρ)_{m} is the weight decomposition of the restriction of ρ_{K} to S. The image of f is a finitely generated subgroup Q of ℝ. Since Δ is a nontrivial subgroup of ℝ, there exists a sequence of morphisms q_{n} : Q → Δ such that \frac{1}{n}q_{n} : Q → ℝ converges simply to the given embedding Q ↪ ℝ. The formula

Ξ_{n}(ρ)^{γ} = ⊕_{m∈M, q_{n}∘f(m)≥γ}V (ρ)_{m}

then defines a Δ-filtration Ξ_{n} ∈ F^{Δ}(G_{K}) with \frac{1}{n}Ξ_{n} → Ξ in (F^{ℝ}(G_{K}), d_{τ}), thus also

⟨F_{H}, Ξ_{n}⟩_{τ} = n ⟨F_{H}, \frac{1}{n}Ξ_{n}⟩_{τ} > n ⟨F_{N}, \frac{1}{n}Ξ_{n}⟩_{τ} = ⟨F_{N}, Ξ_{n}⟩_{τ}

for n ≫ 0. For any τ in Rep(G), (m_{1}, m_{2}) in {m ∈ M : V (ρ)_{m} ≠ 0} and n ≫ 0,

f(m_{1}) − f(m_{2}) = f(m_{1} − m_{2}) and q_{n} ∘ f(m_{1}) − q_{n} ∘ f(m_{2}) = q_{n} ∘ f(m_{1} − m_{2})

have the same sign in {0, ±1}, thus for every ρ in Rep(G) and n ≫ 0,

{Ξ_{n}(ρ)^{γ} : γ ∈ Δ} = {Ξ(ρ)^{γ} : γ ∈ ℝ}.

In particular, Ξ_{n}(τ) is fixed by φ(τ) for n ≫ 0. But then Ξ_{n} is a φ-stable Δfiltration on V_{K} by [5, 4.2.10 or 3.11.12], thus (2^{Δ}) also does not hold.

2.12. (2) ⇒ (2′). Suppose that (2′) does not hold. Then by [6, Theorem 1, (4)], there is a unique ξ in ∂Min(φ′) such that ⟨F_{H}, ξ⟩_{τ} + ⟨F_{N}^{ι}, ξ⟩_{τ} > 0 is maximal. This uniqueness implies that ξ ∈ F^{ℝ}(G_{K′}) is fixed by the group Gal(k′/k) of continuous automorphisms of K′/K. Therefore ξ ∈ F^{ℝ}(G_{K}) and (2) also does not hold.

2.13. (4) ⇒ (4_{ρ}). Suppose that (4) holds and fix ρ ∈ Rep(G) corresponding to a morphism f : G → H = GL(V (ρ)). By [5, 5.8.4], there is a finite Galois extension L of ℚ_{p} and for any factor K̃ of L ⊗ K, a map f : B^{e}(GK̃) → B^{e}(HK̃) such that

f(g · x) = f(g) · f(x), f(x + G) = f(x) + f(G) and f(θ · x) = θ · f(x)

for every x ∈ B^{e}(GK̃), g ∈ G(K̃), G ∈ F^{ℝ}(GK̃) and θ ∈ Aut(K̃/ℚ_{p}). Fix an extension σ̃ ∈ Aut(K̃/ℚ_{p}) of σ ∈ Aut(K/ℚ_{p}), let φ̃ = (b, σ̃) be the corresponding extension of φ = (b, σ), write α̃ and α̃(ρ) for the induced extensions of α and α(ρ) to B^{e}(GK̃) ⊃ B^{e}(G_{K}) and B^{e}(HK̃) ⊃ B^{e}(H_{K}). Then f(α̃(x)) = α̃(ρ)(f(x)) for every x ∈ B^{e}(GK̃). Since α has a fixed point x in B^{e}(G_{K}), f(x) is a fixed point of α̃(ρ) in B^{e}(HK̃), thus α̃(ρ) has bounded orbits, and so does α(ρ) = α̃(ρ)|B^{e}(H_{K}).

3. Mazur’s Inequality

In this section, we explain our variant of Mazur’s inequality (3.2) and prove its converse (Theorem 10, ν ≤ μ^{♯} ⟹ X_{φ}(μ) ≠ ∅ ), following the Fontaine-Rapoport strategy, i.e. using a covering of our affine Deligne-Lusztig sets (the X_{φ}(μ)’s) by our subsets of strongly divisible points (the X_{φ}(F)’s, see 3.1 and 3.3), whose non-emptiness is characterized by our variant of Laffaille’s theorem (Theorem 7).

3.1. Fix a Frobenius φ = b · σ ∈ G(K) ⋊ ⟨σ⟩. For μ ∈ C^{ℝ}(G_{K}), we define

X_{φ}(μ) = {x ∈ X : d(φ(x), x) = μ} .

For any x ∈ X_{φ}(μ), there is an F ∈ F^{ℝ}(G_{K}) of type μ such that x = φ(x) + F, i.e. x ∈ X_{φ}(F). Conversely, X_{φ}(F) ⊂ X_{φ}(μ) for any F of type μ, thus

X_{φ}(μ) = ∪_{F∈F^{ℝ}(G_{K}):t(F)=μ}X_{φ}(F) = ∪_{F∈Adm_{φ}(μ)}X_{φ}(F)

and X_{φ}(μ) ≠ ∅ ⇔ Adm_{φ}(μ) ≠ ∅ by the previous theorem, where

Adm_{φ}(μ) = {F ∈ F^{ℝ}(G_{K}) : (φ, F) is weakly admissible and t(F) = μ} .

3.2. Write ν = t(F_{N}). As F_{N} ∈ F^{ℚ}(G_{K}) is fixed by φ = b · σ and the type map t : F^{ℚ}(G_{K}) → C^{ℚ}(G_{K}) is G(K)-invariant, ν belongs to the fixed point set of σ in C^{ℚ}(G_{K}). Since C^{ℚ}(G_{K}) is a uniquely divisible commutative monoid on which ⟨σ⟩ acts with finite orbits, averaging over these orbits yields a retraction μ ↦ μ^{♯} onto the fixed point set of σ. When k is algebraically closed, [7, Théorème 6] shows that

X_{φ}(μ) ≠ ∅ ⟹ ν ≤ μ^{♯} in (C^{ℚ}(G_{K}), ≤)

where ≤ is the dominance partial order on C^{ℚ}(G_{K}). This is still true over any perfect field k, because if k′ is an algebraic closure of k and K′ = Frac(W (k′)), then C^{ℚ}(G_{K}) ↪ C^{ℚ}(G_{K′}) is a ⟨σ⟩-equivariant embedding of partially ordered commutative monoids. We will establish below that conversely, ν ≤ μ^{♯} implies X_{φ}(μ) ≠ ∅ when k is algebraically closed and G is unramified over ℚ_{p}. The latter assumption seems superfluous, but some variant of the former is really needed: if k = 𝔽_{p} and φ = σ (i.e. b = 1), then ν = 0 but X_{φ}(μ) ≠ ∅ ⟺ μ = 0.

3.3. The subset X_{φ}(μ) of X = B^{e}(G_{K}) is closed for the canonical topology of X, but it is typically not convex, also X_{φ}(ν) = Min(φ) is convex by [1, II.6.2]. On the other hand, X_{φ}(F) is always closed and convex by [2, Theorem 1.3]. The group

J = Aut^{⊗}(V_{K}, φ) = {g ∈ G(K) : φg = gφ} = {g ∈ G(K) : g ⋄ b = b}

acts on X_{φ}(μ) and Adm_{φ}(μ), and j ∈ J maps X_{φ}(F) to X_{φ}(j · F). The convex projection p : X ↠ Min(φ) is non-expanding, J-equivariant, and of formation compatible with algebraic base change on k. Indeed if k′ is a Galois extension of k and p′ : X′ ↠ Min(φ′) is the corresponding projection, then p′ is also Gal(k′/k)- equivariant. It thus maps X = X^{′Gal(k′/k)} [23, 2.6.1] to Min(φ′) ∩ X = Min(φ) [1, II.6.2], from which easily follows that p′|X equals p. The projection p restricts to a quasi-isometric J-equivariant embedding p : X_{φ}(μ) → Min(φ), which is even a J-equivariant quasi-isometry (as defined in [1, I.8.14]) when k is algebraically closed and X_{φ}(μ) ≠ ∅. Indeed, the restriction of p to X_{φ}(μ) has bounded fibers and J acts cocompactly on Min(φ) when k is algebraically closed by [7]. In this case, the boundary of X_{φ}(μ) equals that of Min(φ), i.e. it is the fixed point set of φ in the boundary of X. The boundary of X_{φ}(F) is described in [6, Theorem 1].

3.4. We now establish our second main result:

Theorem 10. If G is unramified over ℚ_{p} and k is algebraically closed, then

X_{φ}(μ) ≠ ∅ ⟺ Adm_{φ}(μ) ≠ ∅ ⟺ ν ≤ μ^{♯}.

Proof. It only remains to show that ν ≤ μ^{♯} implies Adm_{φ}(μ) ≠ ∅. By [5, 5.1.2],

ν ≤ μ^{♯} ⟺ ∀z ∈ C^{ℝ}(G_{K}) : ⟨μ^{♯}, z⟩_{τ}^{tr} ≤ ⟨ν, z⟩_{τ}^{tr}

where ⟨−, −⟩_{τ}^{tr} : C^{ℝ}(G_{K}) × C^{ℝ}(G_{K}) → ℝ is defined in [5, 4.2.5] and given by

⟨μ_{1}, μ_{2}⟩_{τ}^{tr} = inf {⟨F_{1}, F_{2}⟩_{τ} : (F_{1}, F_{2}) ∈ F^{ℝ}(G_{K})^{2}, t(F_{1}, F_{2}) = (μ_{1}, μ_{2})} .

This “scalar product” ⟨−, −⟩_{τ}^{tr} is σ-invariant and additive in both variables. It follows that ⟨μ^{♯}, z⟩_{τ}^{tr} = ⟨μ, z⟩_{τ}^{tr} for every z ∈ C^{ℝ}(G_{K}) fixed by σ, so that

ν ≤ μ^{♯} ⟹ ∀z ∈ C^{ℝ}(G_{K})^{σ=Id} : ⟨μ, z⟩_{τ}^{tr} ≤ ⟨ν, z⟩_{τ}^{tr} .

Suppose therefore that ν ≤ μ^{♯}. Then for any F_{H} ∈ F^{ℝ}(G_{K}) of type t(F_{H}) = μ and every φ-stable Ξ ∈ F^{ℝ}(G_{K}), since t(Ξ) = z is fixed by σ and t(F_{N}) = ν,

⟨t(F_{H}), t(Ξ)⟩_{τ}^{tr} ≤ ⟨t(F_{N}), t(Ξ)⟩_{τ}^{tr} ≤ ⟨F_{N}, Ξ⟩_{τ} .

Thus F_{H} will belong to Adm_{φ}(μ) if it happens to be in generic (= transverse) relative position [5, 4.2.5] with respect to every φ-stable Ξ ∈ F^{ℝ}(G_{K}), for then

⟨F_{H}, Ξ⟩_{τ} = ⟨t(F_{H}), t(Ξ)⟩_{τ}^{tr} ≤ ⟨F_{N}, Ξ⟩_{τ} .

It is therefore sufficient to show that the set of φ-stable ℝ-filtrations on V_{K} is a thin subset of F^{ℝ}(G_{K}) in the sense of [5, 4.1.19]. Since k is algebraically closed, we may assume that φ satisfies a decency equation as in [7, 2.8] for some integer s > 0, in which case every φ-stable filtration is also fixed by σ^{s}, i.e. defined over the fixed field ℚ_{p^{s}} of σ^{s} in K, a finite (unramified) extension of ℚ_{p}. Now since G is unramified over ℚ_{p}, it admits a reductive model over ℤ_{p}, which we also denote by G. We may now apply the thinness criterion of [5, 4.1.19], thus obtaining that the whole of F^{ℝ}(G_{ℚ_{p^{s}}}) is a thin subset of F^{ℝ}(G_{K}), which proves the theorem. □

4. Filtered Isocrystals

4.1. The hardest implication in Theorem 7 seems to be (2^{Δ}) ⇒ (1):

(2^{Δ}) ⇒ (2) ⇒ (2′) ⇒ (4′) ⇒ (4′_{τ}) ⇒ (3′_{τ}) ⇒ (2′_{τ}) ⇒ (2_{τ}) ⇒ (1_{τ}) ⇒ (1).

The key steps are: the fixed point theorem of [6], used in (2′) ⇒ (4′), and some functoriality of Bruhat-Tits buildings, used in (4′) ⇒ (4′_{τ}). However, the theory of Harder-Narasimhan filtrations provides an easier proof of a more general result.

4.2. Fix an extension L of K = W (k)[\frac{1}{p}] and denote by Fil_{L}^{Γ}Iso(k) the category of triples (V, φ, F) where (V, φ) is an isocrystal over k and F = F_{H} is a Γ-filtration on V_{L} = V ⊗_{K} L. It is an exact ⊗-category equipped with Harder-Narasimhan filtrations attached to the slope function which is defined (for V ≠ 0) by

μ(V, φ, F) = \frac{deg(F_{H}) − deg(F_{N})}{dim_{K} V} ∈ ℚ[Γ] ⊂ ℝ.

Here ℚ[Γ] is the ℚ-subspace of ℝ spanned by Γ. We now have four filtrations at our disposal: the Hodge Γ-filtration F = F_{H} on V_{L}, the φ-stable (opposed) Newton ℚ-filtrations F_{N} and F_{N}^{ι} on V = V_{K}, and the φ-stable Harder-Narasimhan ℚ[Γ]- filtration F_{HN} = F_{HN}(V, φ, F) on V = V_{K}.

Lemma 11. The following conditions are equivalent:

(1) (V, φ, F) is weakly admissible, i.e.: (a) deg(F_{H}) = deg(F_{N}) and

(b) deg(F_{H}|W) ≤ deg(F_{N}|W) for every φ-stable K-subspace W of V,

(2) For every φ-stable Δ-filtration Ξ on V, ⟨F_{H}, Ξ⟩ ≤ ⟨F_{N}, Ξ⟩.

(3) For every φ-stable Δ-filtration Ξ on V, ⟨F_{H}, Ξ⟩ + ⟨F_{N}^{ι}, Ξ⟩ ≤ 0.

(4) ⟨F_{H}, F_{HN}⟩ ≤ ⟨F_{N}, F_{HN}⟩.

(5) ⟨F_{H}, F_{HN}⟩ + ⟨F_{N}^{ι}, F_{HN}⟩ ≤ 0.

(6) F_{HN} = 0. In (2) and (3), Δ is any non-trivial subgroup of ℝ.

Proof. One proves (1) ⇔ (2) ⇔ (3) and (4) ⇔ (5) as in Lemma 2, moreover (1) ⇔ (6) by definition of F_{HN} and obviously (6) ⇒ (4), (5). Finally (4) ⇒ (6) because:

⟨F_{H}, F_{HN}⟩ − ⟨F_{N}, F_{HN}⟩ = ∑_{γ} γ · (deg Gr_{F_{HN}}^{γ} (F_{H}) − deg Gr_{F_{HN}}^{γ} (F_{N}))
= ∑_{γ} γ · dim_{K} (Gr_{F_{HN}}^{γ} ) · μ (Gr_{F_{HN}}^{γ} )
= ∑_{γ} γ^{2} · dim_{K} (Gr_{F_{HN}}^{γ} )
= ⟨F_{HN}, F_{HN}⟩

using the definition of F_{HN} for the third equality. □

4.3. For a reductive group G over ℚ_{p}, a Γ-filtered G-isocrystal over L is an exact ⊗-functor factorization (φ, F) of the fiber functor V_{K} : Rep(G) → Vect(K) trough the natural fiber functor Fil_{L}^{Γ}Iso(k) → Vect(K). It again induces four filtrations on the relevant fiber functors: the Hodge Γ-filtration F_{H} = F on V_{L}, the φ-stable (opposed) Newton ℚ-filtrations F_{N} and F_{N}^{ι} on V_{K}, and the φ-stable Harder-Narasimhan ℚ[Γ]-filtration F_{HN} = F_{HN}(φ, F) on V_{K} defined by

∀τ ∈ Rep(G) : F_{HN}(τ) = F_{HN}(V_{K}(τ), φ(τ), F(τ)).

It is not at all obvious that this formula indeed defines a filtration on V_{K}: it does yield a factorization F_{HN} : Rep(G) → Fil^{ℚ[Γ]}(K) of V_{K}, but we have to check that the latter is exact and compatible with the ⊗-products and their neutral objects. Since every exact sequence in Rep(G) is split (we are in characteristic 0), exactness here amounts to additivity, which is obvious. The issue is here the compatibility of Harder-Narasimhan filtrations with ⊗-products, where we need to use the main result of [24]. We say that (φ, F) : Rep(G) → Fil_{L}^{Γ}Iso(k) is weakly admissible if it factors through the full subcategory Fil_{L}^{Γ}Iso(k)^{wa} of weakly admissible objects.

4.4. Fix a faithful τ ∈ Rep(G) and some subgroup Δ ≠ 0 of ℝ. Then:

Theorem 12. The following conditions on (φ, F) are equivalent:

(1) (φ, F) is weakly admissible. (1_{ρ}) For every ρ ∈ Rep(G), (V_{K}(ρ), φ(ρ), F(ρ)) is weakly admissible. (1_{τ}) (V_{K}(τ), φ(τ), F(τ)) is weakly admissible. (2^{Δ}) For every φ-stable Δ-filtration Ξ on V_{K},

⟨F_{H}, Ξ⟩_{τ} ≤ ⟨F_{N}, Ξ⟩_{τ} .

(2_{ρ}^{Δ}) For every ρ ∈ Rep(G) and every φ-stable Δ-filtration Ξ on V_{K}(ρ),

⟨F_{H}(ρ), Ξ⟩ ≤ ⟨F_{N}(ρ), Ξ⟩ .

(2_{τ}^{Δ}) For every φ-stable Δ-filtration Ξ on V_{K}(τ),

⟨F_{H}(τ), Ξ⟩ ≤ ⟨F_{N}(τ), Ξ⟩ .

(5) F_{HN} = 0. (5_{ρ}) For every ρ ∈ Rep(G), F_{HN}(ρ) = 0. (5_{τ}) F_{HN}(τ) = 0.

Proof. The following implications are obvious, or given by the previous lemma:

[figure: (1); (1_{ρ}); (1_{τ}); (2_{ρ}); (2_{τ}); (2^{Δ}); (5); (5_{ρ}); (5_{τ})]

We have (2^{Δ}) ⇔ (2^{ℝ}) ⇔ (2^{ℚ[Γ]}) as in 2.11, (5_{τ}) ⇒ (5) by [5, 4.2.10 or 3.11.12] and finally (2^{ℚ[Γ]}) ⇒ (5_{τ}) since (2^{ℚ[Γ]}) with Ξ = F_{HN} gives

⟨F_{H}(τ), F_{HN}(τ)⟩ ≤ ⟨F_{N}(τ), F_{HN}(τ)⟩

which implies F_{HN}(τ) = 0 by the previous lemma. □

4.5. The above proof of (2^{Δ}) ⇒ (1) looks much easier, but it still uses Totaro’s theorem to the effect that the Harder-Narasimhan filtration is compatible with ⊗- products. Conversely, the above theorem implies that weakly admissible objects are stable under ⊗-product: apply (1_{τ}) ⇒ (1_{ρ}) with G = GL(V_{1})×GL(V_{2}), τ = ρ_{1} ⊞ρ_{2} and ρ = ρ_{1} ⊠ ρ_{2} with ρ_{i} the tautological representation of GL(V_{i}). In particular, Theorem 7 yields another proof of Totaro’s result when L = K, which was of course already known to Laffaille: if M_{i} is a strongly divisible lattice in V_{i}, then M_{1} ⊗ M_{2} is a strongly divisible lattice in V_{1} ⊗ V_{2}.

5. Weakly admissible filtered isocrystals

In this section, we define the Fargues ℚ-filtration on weakly admissible Γ-filtered isocrystals over L (5.1), show that it is compatible with tensor products (Theorem 15), and compute it as a convex projection (Lemma 13 and Proposition 16).

5.1. Weakly admissible Γ-filtered isocrystals over L have a Harder-Narasimhan filtration of their own, defined by Fargues in [10, §9] when Γ = ℤ, for the slope function μ = deg / dim, where the degree function is now given by

deg(V, φ, F) = − deg(F_{H}) = − deg(F_{N}) = deg(F_{N}^{ι}) ∈ ℚ.

An object (V, φ, F) in Fil_{L}^{Γ}Iso(k)^{wa} is thus again equipped with four filtrations: the Hodge Γ-filtration F = F_{H} of V_{L} by L-subspaces, the pair of opposed Newton ℚfiltrations (F_{N}, F_{N}^{ι}) of V by φ-stable K-subspaces, and the Fargues ℚ-filtration F_{F} of V by (φ-stable) weakly admissible K-subspaces. Note that the previous Harder-Narasimhan Γ-filtration F_{HN} on V by φ-stable K-subspaces is now trivial!

5.2. For a weakly admissible object (V, φ, F) in Fil_{L}^{Γ}Iso(k) and a subgroup Δ ≠ 0 of ℝ, we denote by F^{Δ}(V) the set of all Δ-filtrations on V, by F^{Δ}(V, φ) ⊂ F^{Δ}(V) its subset of Δ-filtrations by φ-stable K-subspaces, and by F^{Δ}(V, φ, F) ⊂ F^{Δ}(V, φ) its subset of Δ-filtrations by weakly admissible φ-stable K-subspaces. Moreover, we equip F^{ℝ}(V) with the φ-invariant CAT(0)-distance [5, 4.2.10] defined by

d(F_{1}, F_{2}) = \sqrt{∥F_{1}∥^{2} + ∥F_{2}∥^{2} − 2 ⟨F_{1}, F_{2}⟩}.

Then F^{ℝ}(V, φ) and F^{ℝ}(V, φ, F) are closed and convex subsets of F^{ℝ}(V). In fact:

Lemma 13. For (V, φ, F) ∈ Fil_{L}^{Γ}Iso(k)^{wa} and a subgroup 0 ≠ Δ ⊂ ℝ as above,

F^{Δ}(V, φ, F) = {Ξ ∈ F^{Δ}(V, φ) : ⟨F_{H}, Ξ⟩ = ⟨F_{N}, Ξ⟩} .

For Δ = ℝ, this is a closed convex subset of F^{ℝ}(V). Moreover F_{F} ∈ F^{ℚ}(V, φ, F) is the image of F_{N}^{ι} ∈ F^{ℚ}(V, φ) under the convex projection p : F^{ℝ}(V, φ) → F^{ℝ}(V, φ, F).

Proof. For Ξ ∈ F^{Δ}(V, φ), set {δ ∈ Δ : Gr_{Ξ}^{δ} ≠ 0} = {δ_{1} < ⋯ < δ_{n}}. Then Ξ^{δ_{i}} is a φ-stable K-subspace of V. Put d_{H}(i) = deg(F_{H}|Ξ_{K}^{δ_{i}}), d_{N}(i) = deg(F_{N}|Ξ^{δ_{i}}). Then

⟨F_{H}, Ξ⟩ − ⟨F_{N}, Ξ⟩ = ∑_{i=1}^{n} δ_{i} · (Δ_{H}(i) − Δ_{N}(i)) = ∑_{i=1}^{n} Δ_{i} · (d_{H}(i) − d_{N}(i))

where Δ_{⋆}(i) = d_{⋆}(i) − d_{⋆}(i + 1) for 1 ≤ i < n, Δ_{⋆}(n) = d_{⋆}(n), Δ_{i} = δ_{i} − δ_{i−1} for 1 < i ≤ n and Δ_{1} = δ_{1}, as in the proof of lemma 2. We now know that d_{H}(i) ≤ d_{N}(i) for all i with equality for i = 1. Since Δ_{i} > 0 for i > 1, we obtain

⟨F_{H}, Ξ⟩ = ⟨F_{N}, Ξ⟩ ⟺ ∀1 ≤ i ≤ n : d_{H}(i) = d_{N}(i).

The displayed equality immediately follows, which may also be written as

F^{Δ}(V, φ, F) = {Ξ ∈ F^{Δ}(V, φ) : ⟨F_{H}, Ξ⟩ + ⟨F_{N}^{ι}, Ξ⟩ = 0} ,
= {Ξ ∈ F^{Δ}(V, φ) : ⟨F_{H}, Ξ⟩ + ⟨F_{N}^{ι}, Ξ⟩ ≥ 0} .

Since the scalar product ⟨−, −⟩ is continuous and concave (see [5, 4.2.10]), this is (for Δ = ℝ) a closed and convex subset of the closed and convex subset F^{ℝ}(V, φ) of the CAT(0)-space F^{ℝ}(V). Take now Ξ = p(F_{N}^{ι}), the convex projection of F_{N}^{ι} to F^{ℝ}(V, φ, F). For 1 ≤ i ≤ n, a weakly admissible subspace W of Gr_{Ξ}^{δ_{i}} and a small ϵ > 0, let Ξ′ = Ξ(i, W, ϵ) be the φ-stable ℝ-filtration on V with Gr_{Ξ′}^{δ} = Gr_{Ξ}^{δ} unless δ = δ_{i} or δ_{i} + ϵ, where Gr_{Ξ′}^{δ_{i}} = Gr_{Ξ}^{δ_{i}}/W and Gr_{Ξ′}^{δ_{i}+ϵ} = W. By definition of Ξ,

⟨Ξ, Ξ⟩ − 2 ⟨F_{N}^{ι}, Ξ⟩ ≤ ⟨Ξ′, Ξ′⟩ − 2 ⟨F_{N}^{ι}, Ξ′⟩ ,

which easily unfolds to the following inequality, valid for small ϵ > 0:

ϵ^{2} dim_{K} W + 2ϵ (δ_{i} dim_{K} W − deg(F_{N}^{ι}|W)) ≥ 0.

It follows that δ_{i} ≥ μ(W). For W = Gr_{Ξ}^{δ_{i}}, we may also allow negative ϵ’s with small absolute values in the above argument, this time obtaining δ_{i} = μ(Gr_{Ξ}^{δ_{i}}). Therefore Gr_{Ξ}^{δ_{i}} is μ-semi-stable of slope δ_{i} for all 1 ≤ i ≤ n, and Ξ thus equals F_{F}. □

5.3. Let now G be a reductive group over ℚ_{p} and let

(φ, F) : Rep(G) → Fil_{L}^{Γ}Iso(k)^{wa}

be a weakly admissible Γ-filtered G-isocrystal over L. We denote by

F^{Δ}(G_{K}, φ, F) ⊂ F^{Δ}(G_{K}, φ) ⊂ F^{Δ}(G_{K})

the set of Δ-filtrations on the fiber functor V_{K} : Rep(G) → Vect(K) by respectively φ-stable and φ-stable weakly admissible K-subspaces. We also fix a faithful representation τ of G and equip F^{ℝ}(G_{K}) with the CAT(0)-metric defined by

d_{τ} (F_{1}, F_{2}) = \sqrt{∥F_{1}∥_{τ}^{2} + ∥F_{2}∥_{τ}^{2} − 2 ⟨F_{1}, F_{2}⟩_{τ}} .

Lemma 14. With notations as above,

F^{Δ}(G_{K}, φ, F) = {Ξ ∈ F^{Δ}(G_{K}, φ) : ⟨F_{H}, Ξ⟩_{τ} = ⟨F_{N}, Ξ⟩_{τ} } .

For Δ = ℝ, this is a closed convex subset of (F^{ℝ}(G_{K}), d_{τ}).

Proof. If Ξ ∈ F^{Δ}(G_{K}, φ, F), then Ξ(τ) ∈ F^{Δ}(V_{K}(τ), φ(τ), F(τ)), thus

⟨F_{H}, Ξ⟩_{τ} = ⟨F_{H}(τ), Ξ(τ)⟩ = ⟨F_{N}(τ), Ξ(τ)⟩ = ⟨F_{N}, Ξ⟩_{τ}

by the previous lemma. Suppose conversely that Ξ ∈ F^{Δ}(G_{K}, φ) satisfies this equation, choose a splitting of Ξ (which exists by [5, 3.11.3]) and use it to transport (Gr_{Ξ}(φ), Gr_{Ξ}(F)) back from the fiber functor Gr_{Ξ}(V_{K}) to (φ′, F′) on V_{K}. Since Fil_{L}^{Γ}Iso(k)^{wa} is a strictly full subcategory of Fil_{L}^{Γ}Iso(k) which is stable under extensions, it is sufficient to establish that (φ′, F′) is weakly admissible, and this now follows from (1_{τ}) ⇒ (1) of theorem 12 since (V_{K}(τ), φ′(τ), F′(τ)) is weakly admissible by the previous lemma. For Δ = ℝ, one shows as above that

F^{ℝ}(G_{K}, φ, F) = {Ξ ∈ F^{ℝ}(G_{K}, φ) : ⟨F_{H}, Ξ⟩_{τ} + ⟨F_{N}^{ι}, Ξ⟩_{τ} = 0}
= {Ξ ∈ F^{ℝ}(G_{K}, φ) : ⟨F_{H}, Ξ⟩_{τ} + ⟨F_{N}^{ι}, Ξ⟩_{τ} ≥ 0}

is a closed convex subset of F^{ℝ}(G_{K}, φ). □

5.4. We may now establish the analog of Totaro’s tensor product theorem for the Fargues filtration F_{F}. In the classical setting of p-adic Hodge theory where Γ = ℤ and L is a totally ramified finite algebraic extension of K, an entirely different proof is given in [10, §9.2], using Fontaine’s functor to work on the Galois side.

Theorem 15. The Fargues filtration defines a ⊗-functor

F_{F} : Fil_{L}^{Γ}Iso(k)^{wa} → Fil_{K}^{ℚ}.

Proof. The general formalism of Harder-Narasimhan filtrations yields the functoriality of F_{F} and reduces its compatibility with tensor products to the following statement: if (V_{i}, φ_{i}, F_{i}) ∈ Fil_{L}^{Γ}Iso(k)^{wa} is semi-stable of slope μ_{i} ∈ ℚ for i ∈ {1, 2},

(V, φ, F) = (V_{1} ⊗ V_{2}, φ_{1} ⊗ φ_{2}, F_{1} ⊗ F_{2}) ∈ Fil_{L}^{Γ}Iso(k)^{wa}

is semi-stable of slope μ = μ_{1} + μ_{2}, which means that for every weakly admissible subobject X of V, deg(F_{N}^{ι}|X) ≤ deg(V (μ)|X) with equality for X = V. Here V (ν) denotes the filtration on V such that Gr_{V (ν)}^{ν} = V. Exactly as in Lemma 2, this condition is itself equivalent to the following one: for every Ξ ∈ F^{ℝ}(V, φ, F),

⟨F_{N}^{ι}, Ξ⟩ ≤ ⟨V (μ), Ξ⟩ (or : ⟨F_{N}^{ι}, Ξ⟩ + ⟨V (−μ), Ξ⟩ ≤ 0).

We adapt Totaro’s proof to this setting. Applying the functor F^{ℝ}(−) to

GL(V_{1}) × GL(V_{2}) ↠ G = GL(V_{1}) × GL(V_{2})/Δ(𝔾_{m}) ↪ GL(V_{1} ⊗ V_{2})

where Δ is the diagonal embedding yields a factorization

F^{ℝ}(V_{1}) × F^{ℝ}(V_{2}) ↠ F^{ℝ}(G) ↪ F^{ℝ}(V_{1} ⊗ V_{2})

of the map (G_{1}, G_{2}) ↦ G_{1} ⊗ G_{2}, which identifies F^{ℝ}(G) with a closed and convex subset of F^{ℝ}(V_{1} ⊗ V_{2}). Let p : F^{ℝ}(V_{1} ⊗ V_{2}) ↠ F^{ℝ}(G) be the convex projection and pick (Ξ_{1}, Ξ_{2}) ∈ F^{ℝ}(V_{1}) × F^{ℝ}(V_{2}) mapping to p(Ξ) ∈ F^{ℝ}(G) (this is Kempf’s filtration of [24, §2]). Since p is φ-equivariant, p(Ξ) belongs to F^{ℝ}(G, φ) and then also Ξ_{i} ∈ F^{ℝ}(V_{i}, φ) for i ∈ {1, 2}. Moreover, for any (G_{1}, G_{2}) ∈ F^{ℝ}(V_{1,L})×F^{ℝ}(V_{2,L}),

(5.1)  ⟨G_{1} ⊗ G_{2}, Ξ⟩ ≤ ⟨G_{1} ⊗ G_{2}, p(Ξ)⟩

by [5, 5.5.12] since p is also the restriction to F^{ℝ}(V_{1} ⊗ V_{2}) of the convex projection from F^{ℝ}(V_{1,L} ⊗ V_{2,L}) onto F^{ℝ}(G_{L}). On the other hand, one checks easily that

⟨G_{1} ⊗ G_{2}, H_{1} ⊗ H_{2}⟩ = ⟨G_{1}, H_{1}⟩ · dim V_{2} + ⟨G_{2}, H_{2}⟩ · dim V_{1}

(5.2) + deg G_{1} · deg H_{2} + deg H_{1} · deg G_{2}

for every (G_{i}, H_{i}) ∈ F^{ℝ}(V_{i,L}). We will now apply these two formulas to

F_{N}^{ι}(on V) = F_{N}^{ι}(on V_{1}) ⊗ F_{N}^{ι}(on V_{2}),
F_{H}(on V_{L}) = F_{H}(on V_{1}) ⊗ F_{H}(on V_{2}),
V (−μ) = V_{1}(−μ_{1}) ⊗ V_{2}(−μ_{2}),
p(Ξ) = Ξ_{1} ⊗ Ξ_{2}.

First recall that since (V, φ, F) is weakly admissible (by Totaro’s theorem!),

⟨F_{H}, p(Ξ)⟩ + ⟨F_{N}^{ι}, p(Ξ)⟩ ≤ 0.

Since ⟨F_{H}, Ξ⟩ + ⟨F_{N}^{ι}, Ξ⟩ = 0 by assumption on Ξ and Lemma 13, actually

⟨F_{H}, p(Ξ)⟩ + ⟨F_{N}^{ι}, p(Ξ)⟩ = 0

by (5.1). Applying formula (5.2) twice and grouping terms, we obtain

0 = (⟨F_{H}, Ξ_{1}⟩ + ⟨F_{N}^{ι}, Ξ_{1}⟩) · dim V_{2}
+ (⟨F_{H}, Ξ_{2}⟩ + ⟨F_{N}^{ι}, Ξ_{2}⟩) · dim V_{1}
+ (deg(F_{H} on V_{1}) + deg(F_{N}^{ι} on V_{1})) · deg Ξ_{2}
+ (deg(F_{H} on V_{2}) + deg(F_{N}^{ι} on V_{2})) · deg Ξ_{1}.

Since V_{1} and V_{2} are weakly admissible, the first two terms are ≤ 0 and the last two trivial. Thus ⟨F_{H}, Ξ_{i}⟩ + ⟨F_{N}^{ι}, Ξ_{i}⟩ = 0 for i ∈ {1, 2}, i.e. Ξ_{i} ∈ F^{ℝ}(V_{i}, φ_{i}, F_{i}). Now applying our two formulas (5.1) and (5.2) twice again also gives that

⟨F_{N}^{ι}, Ξ⟩ + ⟨V (−μ), Ξ⟩ ≤ ⟨F_{N}^{ι}, p(Ξ)⟩ + ⟨V (−μ), p(Ξ)⟩
≤ (⟨F_{N}^{ι}, Ξ_{1}⟩ + ⟨V_{1}(−μ_{1}), Ξ_{1}⟩) · dim V_{2}
+ (⟨F_{N}^{ι}, Ξ_{2}⟩ + ⟨V_{2}(−μ_{1}), Ξ_{2}⟩) · dim V_{1}
+ (deg(F_{N}^{ι} on V_{1}) + deg(V_{1}(−μ_{1}))) · deg Ξ_{2}
+ (deg(F_{N}^{ι} on V_{2}) + deg(V_{2}(−μ_{2}))) · deg Ξ_{1}.

Since V_{i} is semi-stable of slope μ_{i} for i ∈ {1, 2}, the first two terms are ≤ 0 and the last two trivial. Thus ⟨F_{N}^{ι}, Ξ⟩ + ⟨V (−μ), Ξ⟩ ≤ 0 for all Ξ ∈ F^{ℝ}(V, φ, F) and (V, φ, F) is indeed semi-stable of slope μ. □

5.5. A weakly admissible Γ-filtered G-isocrystal (φ, F) over L thus again yields four filtrations: the Hodge Γ-filtration F_{H} = F ∈ F^{Γ}(G_{L}), the opposed Newton ℚfiltrations F_{N}, F_{N}^{ι} ∈ F^{ℚ}(G_{K}, φ) and the Fargues ℚ-filtration F_{F} ∈ F^{ℚ}(G_{K}, φ, F), which is obtained by composing the exact ⊗-functor (φ, F) with the ⊗-functor of the previous proposition (the resulting ⊗-functor is exact because it is plainly additive and every exact sequence in Rep(G) is split). The former Harder-Narasimhan filtration F_{HN} of section 4 is trivial. We claim that, just as in the GL(V)-case:

Proposition 16. The Fargues ℚ-filtration F_{F} is the convex projection of F_{N}^{ι}.

Proof. The map G ↦ G(τ) identifies F^{ℝ}(G_{K}) with a φ-stable closed convex subset of F^{ℝ}(V_{K}(τ)). Let p be the convex projection from F^{ℝ}(V_{K}(τ)) to F^{ℝ}(G_{K}). It is φ-equivariant and thus maps F^{ℝ}(V_{K}(τ), φ(τ)) to F^{ℝ}(G_{K}, φ). Moreover,

⟨H(τ), G⟩ ≤ ⟨H(τ), p(G)(τ)⟩ = ⟨H, p(G)⟩_{τ}

for every H ∈ F^{ℝ}(G_{K}) (or F^{ℝ}(G_{L})) and G ∈ F^{ℝ}(V_{K}(τ)), exactly as in the proof of the previous proposition. Suppose now that G belongs to F^{ℝ}(V_{K}(τ), φ(τ), F(τ)). Then p(G) also belongs to F^{ℝ}(G_{K}, φ, F). Indeed, since (φ, F) is weakly admissible,

⟨F_{H}, p(G)⟩_{τ} + ⟨F_{N}^{ι}, p(G)⟩_{τ} ≤ 0

but also ⟨F_{H}(τ), G⟩ + ⟨F_{N}^{ι}(τ), G⟩ = 0 thus actually

⟨F_{H}, p(G)⟩_{τ} + ⟨F_{N}^{ι}, p(G)⟩_{τ} = 0

i.e. p(G) belongs to F^{ℝ}(G_{K}, φ, F). Since p is non-expanding and F_{N}^{ι} ∈ F^{ℝ}(G_{K}),

d (F_{N}^{ι}(τ), G) ≥ d (F_{N}^{ι}(τ), p(G)(τ)) = d_{τ} (F_{N}^{ι}, p(G)) ≥ d_{τ} (F_{N}^{ι}, F′_{F}) = d(F_{N}^{ι}(τ), F′_{F} (τ))

where F′_{F} is the convex projection of F_{N}^{ι} ∈ F^{ℝ}(G_{K}, φ) to F^{ℝ}(G_{K}, φ, F). Since this holds true for any G ∈ F^{ℝ}(V_{K}(τ), φ(τ), F(τ)), it follows that F′_{F} (τ) = F_{F} (τ) by Lemma 13. But then F′_{F} = F_{F} since F^{ℝ}(G_{K}) ↪ F^{ℝ}(V_{K}(τ)) is injective. □

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CNRS, Sorbonne Université, Université Paris Diderot, Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG, F-75005, Paris, France.

E-mail address: christophe.cornut@imj-prg.fr
URL: https://www.imj-prg.fr/∼christophe.cornut/