Lefschetz operator and local Langlands mod ℓ : the limit case

Jean-François Dat

Abstract

Let K be a finite extension of ℚ_{p} with residue field 𝔽_{q}, and let ℓ be a prime such that q ≡ 1(mod ℓ). We investigate the cohomology of the Lubin-Tate towers of K with coefficients in \overline{𝔽}_{ℓ}, and we show how it encodes Vigneras’ Langlands correspondence for unipotent \overline{𝔽}_{ℓ}-representations.

Contents

1 Main results  1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 This paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Representation theory  5
2.1 The unipotent block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Elliptic unipotent representations . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Extensions between some simple H_{\overline{𝔽}_{ℓ}}(G, I)-modules . . . . . . . . . . . . . . . . 10
3 Cohomology and the Lefschetz operator  12
3.1 The unipotent part of the cohomology complex . . . . . . . . . . . . . . . . . . 12
3.2 The graded dimension of R_{π}^{∗} when π is elliptic unipotent. . . . . . . . . . . . . . 14
3.3 The description of the pair (R_{π}^{∗}, L_{π}^{∗}) . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4 Computation of (R_{π}^{red,∗}, L_{π}^{red}). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
A Proof of Theorem (2.3.5) and Proposition (2.3.7)  23
A.1 Construction of extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
A.2 Proof of Proposition (A.1.12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1 Main results

This paper is part of a project, outlined in [6], that aims at providing a geometric interpretation of the Vigneras correspondence for mod-ℓ representations of p-adic linear groups.

1.1 Motivation

(1.1.1) The problem. Let K be a local p-adic field, ℓ a prime distinct from p, and d ⩾ 1 an integer. Vigneras established in [16] a bijection between (classes of) irreducible smooth \overline{𝔽}_{ℓ}- representations of GL_{d}(K) and (classes of) d-dimensional Weil-Deligne \overline{𝔽}_{ℓ}-representations for K. On the one hand we have fairly natural “automorphic objects”, but on the other hand we get fairly unnatural “Galois objects”. Indeed, the nilpotent part of a Weil-Deligne \overline{𝔽}_{ℓ}-representation has no obvious Galois interpretation, in contrast with \overline{ℚ}_{ℓ}-representations, where it is related to the infinitesimal action of the tame inertia subgroup on some associated continuous \overline{ℚ}_{ℓ} representation of the Weil group. Therefore in the \overline{𝔽}_{ℓ} case, this nilpotent part appears as an“extra datum”, from the arithmetic point of view. In fact, Vigneras’ correspondence was obtained by purely representation-theoretic arguments (a classification theorem à la Zelevinski), and it is the aim of our project alluded to above to propose a geometric interpretation for it.

(1.1.2) The project. Since Carayol’s formulation of “non-abelian Lubin-Tate theory”, the cohomology of the Lubin-Tate tower (M_{LT,n})_{n∈ℕ} of height d is a natural place to look for a realization of any Langlands-type local correspondence. It has long been noticed by the author that in order to get something correct for non-supercuspidal representations, one should use the full “cohomology complex” RΓ_{c}(M_{LT}^{ca}, ℤ_{ℓ}), as an object in the derived category D^{b}(Rep_{ℤ_{ℓ}}^{∞}(GL_{d}(K))) endowed with an action of the Weil group W_{K} and of the unit group D^{×} of the division algebra with center K and invariant 1/d. Then for an irreducible representation π (over \overline{𝔽}_{ℓ} or \overline{ℚ}_{ℓ}), one considers the “derived π-coisotypical component”

R_{π}^{∗} := H^{∗} (R Hom_{ℤ_{ℓ}GL_{d}(K)}(RΓ_{c}(M_{LT}^{ca}, ℤ_{ℓ}), π)) [1 − d]

which is a finite dimensional graded smooth/continuous D^{×} × W_{K}-module supported in the range [1 − d, d − 1]. In [7], we proved the equality

[R_{π}^{∗}] = LJ(π) ⊗ σ^{ss}(π) (1.1.3)

in the Grothendieck group of smooth/continuous D^{×} × W_{K}-modules. Here LJ stands for the so-called “Langlands-Jacquet transfer” of [8] and σ^{ss}(π) is the Weil part (i.e. the semisimple part), of the Weil-Deligne representation σ(π) = (σ^{ss}(π), N_{π}) attached to π.

One would like to enrich R_{π}^{∗} with a nilpotent operator so that a similar formula holds in a suitable Grothendieck group of Weil-Deligne representations. The inspiration for this is Arthur’s second SL_{2} factor in the theory of automorphic forms. An hint that this may be useful for our purpose is the relation between switching the two SL_{2}-factors of a local A-parameter and the Zelevinski involution. Indeed, Vigneras’ correspondence is more a \overline{𝔽}_{ℓ}-analog of the “Zelevinski correspondence” than of the Langlands correspondence for \overline{ℚ}_{ℓ}-representations of GL_{d}(K).^{1}.

The origin of Arthur’s SL_{2} lies in the Lefschetz decomposition of the intersection cohomology of Shimura varieties. Building on this analogy, we defined in [6] a “Lefschetz operator”

L : RΓ_{c}(M_{LT}^{ca}, ℤ_{ℓ}) ⟶ RΓ_{c}(M_{LT}^{ca}, ℤ_{ℓ})[2](1)

1recall both correspondences are interwined by the Zelevinski involution.

as the cup product by the Chern class of a natural equivariant line bundle on the Lubin-Tate tower. In turn, this operator induces a graded equivariant map

L_{π}^{∗} : R_{π}^{∗} ⟶ R_{π}^{∗}[2](1).

In its roughest formulation, our hope is that the pair (R_{π}^{∗}, L_{π}^{∗}) encodes the Weil-Deligne representation associated to π by Vigneras’ correspondence. More precisely, one can define the class [R_{π}^{∗}, L_{π}^{∗}] = [R_{π}^{even}, L_{π}^{∗}] − [R_{π}^{odd}, L_{π}^{∗}] of (R_{π}^{∗}, L_{π}^{∗}) in the Grothendieck group of Weil-Deligne \overline{𝔽}_{ℓ}D^{×}-representations, as in [5, 2.2.2]. The best we can hope for is then the equality

[R_{π}^{∗}, L_{π}^{∗}] = LJ(π) ⊗ [σ^{ss}(π), N_{π}]. (1.1.4)

(1.1.5) Previous results. We proved such an equality in the following cases :

• when π is any irreductible \overline{ℚ}_{ℓ}-representation (in this case, by “Vigneras correspondence”, we just mean the Zelevinski correspondence), [6].

• when π is any unipotent irreducible \overline{𝔽}_{ℓ}-representation and the order of q in 𝔽_{ℓ}^{×} is at least d, [5].

In Vigneras’ terminology, a representation is “unipotent” if it belongs to the principal block of the category Rep_{\overline{𝔽}_{ℓ}}^{∞}(G), i.e. the unique block that contains the trivial representation. For an irreducible π, this is equivalent to π occuring as a subquotient of some Ind_{B}^{G} (χ) for χ an unramified character of a Borel subgroup B. Note that R_{π}^{∗} can be computed in greater generality, but the author is still unable to control L_{π}^{∗} when π is not unipotent.

In the second case above, the arguments are much more difficult in the “Coxeter congruence” case (when the order of q in 𝔽_{ℓ}^{×} is d) than in the “banal” case (when this order is greater than d), due to some representation-theoretic complications. A common feature of these two cases however is that one can still use enough of the theory of weights (on the Galois side) and exponents (on the GL_{d} side) so as to split the complex and compute explicitly Yoneda extensions. Moreover, in all the cases above, the formulas (1.1.3) and (1.1.4) involve no cancellation because R_{π}^{∗} turns out to be either oddly or evenly graded. In fact, the “énoncé optimiste” from [6, 1.3.3] holds true in these cases.

1.2 This paper

Here we study the case when q ≡ 1(mod ℓ) and ℓ > d. This is called the “limit case” in [2, Sec. 5]. From the point of view of weights or exponents, this is the most degenerate case, due to the congruence on q. For example, in this situation, the unipotent summand of the complex RΓ_{c}(M_{LT}^{ca}, ℤ_{ℓ}) can be shown to be indecomposable. Moreover, we have noted in [7, 2.2.7], that R_{π}^{∗} is generally not evenly nor oddly graded. Despite these bad news, computations are still feasible because of the additional assumption that ℓ > d, which simplifies much the representation theory, as indicated by Vigneras’ appendix to [2].

(1.2.1) The result of the computation. Let us describe (R_{π}^{∗}, L_{π}^{∗}) in a geometric way. We explain in (3.1.3) that for π unipotent, R_{π}^{∗} vanishes unless π is a subquotient of Ind_{B}^{G} (\overline{𝔽}_{ℓ}), or equivalently, unless σ^{ss}(π) is the trivial representation of W_{K} of dimension d. Then σ(π) = (σ^{ss}(π), N) is given by a nilpotent d × d matrix N, whose Jordan form has shape λ for some partition λ of d. Through Vigneras’ correspondence we thus get a parametrization λ ∈ P(d) ↦ π_{λ} of all π’s that occur as a subquotient of Ind_{B}^{G} (\overline{𝔽}_{ℓ}). We explain how to construct π_{λ} explicitly in (2.2.5).

Now, consider the Tate curve E = 𝔾_{m}/ϖ^{ℤ} with ϖ a uniformizer of O_{K}, and let A be the abelian variety E^{d}/E_{diag}. It carries a natural action of the symmetric group 𝔖_{d}. Since ℓ > d, we may decompose the cohomology H^{∗}(A^{ca}, \overline{𝔽}_{ℓ}) = ⨁_{λ} H_{λ}^{∗} by applying the central idempotents of \overline{𝔽}_{ℓ}[𝔖_{d}] associated to partitions of d. Further, let L^{∗} : H^{∗}(A^{ca}, \overline{𝔽}_{ℓ}) ⟶ H^{∗}(A^{ca}, \overline{𝔽}_{ℓ})[2](1) be the Lefschetz operator associated to some 𝔖_{d}-equivariant ample sheaf on A such that the Hard Lefschetz theorem holds over \overline{𝔽}_{ℓ} (for ℓ > d, one can always find such a sheaf). It also decomposes as a sum L^{∗} = ∑_{λ} L_{λ}^{∗}.

Theorem.– For λ ∈ P(d), the action of D^{×} on R_{π_{λ}}^{∗} is trivial and that of W_{K} is unipotent. Moreover, there is an isomorphism (R_{π_{λ}}^{∗}, L_{π_{λ}}^{∗})^{⊕d_{λ}} ≃ (H_{λ}^{∗}, L_{λ}^{∗}) of graded \overline{𝔽}_{ℓ}-vector spaces compatible with L operators.

Here d_{λ} is the dimension of the simple \overline{𝔽}_{ℓ}[𝔖_{d}]-module associated to λ. The geometric intuition behind this result is that the special fiber of the Tate formal model of A is closely related to the quotient of the special fiber of Deligne’s model of Drinfeld’s symmetric space by the action of the unipotent radical of B. In fact, some multiplicities make it difficult to compare vanishing cycles on both sides, but somehow these multiplicities disappear when q = 1 in the coefficients. Despite this intuition, our proof of this result is purely algebraic and does not provide a Galois-equivariant isomorphism, although we expect that there exists one.

By Representation Theory of 𝔖_{d} the theorem implies that R_{π_{λ}}^{∗} vanishes unless the Young tableau of λ is a hook or a double-hook. We explain in (2.2.7) that λ is a hook if and only if π_{λ} is elliptic, i.e. LJ(π_{λ}) ≠ 0. In this case we can make the theorem more explicit.

Corollary.– Assume λ = (d − j, 1^{(j)}) for some j ∈ {0, ⋯, d − 1} and put j′ = d − 1 − j. Then we have (R_{π_{λ}}^{∗}, L_{π_{λ}}^{∗}) ≃ (P_{j′})^{⊕j+1} ⊕ (P_{j′−1})^{⊕j} where P_{k} denotes the cohomology of a projective space of dimension k, shifted by −k, and with its tautological Lefschetz operator.

In particular the space R_{π_{λ}}^{∗} has total dimension 2jj′ + d. When jj′ ≠ 0, the pair (R_{π_{λ}}^{∗}, L_{π_{λ}}^{∗}) does not have the right dimension, and what is worse, it does not seem related to the Vigneras pair (σ^{ss}(π_{λ}), N_{λ}) in any reasonable Grothendieck group of Weil-Deligne representations. In other words, (1.1.4) fails in this case.

However, it is still true that it encodes the Vigneras pair, provided one uses extra structure.

(1.2.2) Main result. Observe that R_{π}^{∗} has the structure of a graded right module over the derived endomorphism algebra Ext_{\overline{𝔽}_{ℓ}G}^{∗}(π, π). Consider the subalgebra E_{π}^{∗} generated by extensions that “come from the boundary”, namely by the kernel of the map Ext_{\overline{𝔽}_{ℓ}G}^{1}(π, π) ⟶ Ext_{\overline{𝔽}_{ℓ}Ḡ}^{1}(π̄, π̄) where Ḡ = GL_{d}(𝔽_{q}) and π̄ = π^{1+ϖM_{d}(O)}. For a unipotent π, we’ll see that E_{π}^{∗} is also the image of a natural map Ext_{H}^{∗}(π^{I}, π^{I}) ⟶ Ext_{\overline{𝔽}_{ℓ}G}^{∗}(π, π) where I is an Iwahori subgroup and H is the corresponding Hecke algebra. This is a local graded algebra and we denote by E_{π}^{+} its maximal ideal. In the cases when (1.1.4) has been established, one also observes that either E_{π}^{+} = 0 or at least its action on R_{π}^{∗} vanishes. In contrast, in the limit case under study here, this action is non-zero and is somehow responsible for R_{π}^{∗} being “too big”. So, define R_{π}^{red} := R_{π}^{∗}/R_{π}^{∗}E_{π}^{+}. This is still a graded \overline{𝔽}_{ℓ}-representation of W_{K} × D^{×}, and L_{π}^{∗} induces an operator L_{π}^{red} : R_{π}^{red} ⟶ R_{π}^{red}[2](1). Let us finally denote by [R_{π}^{red}, L_{π}^{red}] the image of the pair (R_{π}^{red}, L_{π}^{red}) in the Grothendieck group of WD representations.

Theorem.– Let π be an elliptic unipotent irreducible \overline{𝔽}_{ℓ}-representation of GL_{d}(K). As above, let π = π_{λ} for λ = (d − j, 1^{(j)}). Then we have a \overline{𝔽}_{ℓ}-linear isomorphism (R_{π}^{red}, L_{π}^{red}) ≃ P_{j′} ⊕(P_{0})^{⊕j}[j′]. Hence in the Grothendieck group of Weil-Deligne \overline{𝔽}_{ℓ}D^{×}-representations we get

[R_{π}^{red}, L_{π}^{red}] = LJ(π) ⊗ [σ^{ss}(π), N_{π}].

When π is not elliptic but has a non-zero R_{π}^{∗} (i.e. π is associated to a double hook) we expect that [R_{π}^{red}, L_{π}^{red}] vanishes, but we don’t prove this here.

(1.2.3) Organization of the paper. Part 2 deals with most of representation-theoretic prerequisites. We recall and expand on Vigneras’ appendix to [2] to describe the unipotent block and the elliptic unipotent representations in our context. We then compute extensions of Iwahori-invariants of these representations. This involves representation theory of the symmetric group, and in particular some knowledge of decomposition of tensor products. We postpone to the appendix a delicate computation of cup-products in this context, which we use in the study of R_{π}^{red}. Part 3 deals with the cohomogical study. The main point is to show that some spectral sequences degenerate, which implies that the cohomology complex at Iwahori level is split. With this splitting property and our knowlegde of extensions and cup-products from part 2 in hand, the results listed above are quite easy computations. Theorem (1.2.1) is proved in section 3.3 and Theorem (1.2.2) in section 3.4.

2 Representation theory

2.1 The unipotent block

We put G := GL_{d}(K) and denote by Rep_{R}^{∞}(G) the abelian category of smooth representations of G with coefficients in the commutative ring R. Let b be the unique primitive idempotent of the center ℨ_{ℤ_{ℓ}}(G) of the category Rep_{ℤ_{ℓ}}^{∞}(G) which is non zero on the trivial representation. Denote by Rep_{b}^{∞}(G) the full subcategory of all objects on which b acts by the identity. This is a Serre subcategory, called the unipotent block of Rep_{ℤ_{ℓ}}^{∞}(G).

Let I be the standard Iwahori subgroup of G and let I^{ℓ} be the maximal prime-to-ℓ subgroup of I. This is a distinguished open subgroup of I and the quotient I/I^{ℓ} is isomorphic to the ℓ-Sylow subgroup Syl_{ℓ}(𝔽_{q}^{×})^{d} of (𝔽_{q}^{×})^{d}.

(2.1.1) Proposition.– The unipotent block is generated by the projective representation ℤ_{ℓ}[G/I^{ℓ}].

Proof. When ℤ_{ℓ} is replaced by \overline{𝔽}_{ℓ}, this is explained in [2, Appendix 1], and probably we could reduce our claim to this reference. Here is another argument relying on our description of the unipotent block in [5, Sec. 3.1]. Indeed, by [5, Prop 3.1.2], our claim reduces to a claim about the finite group Ḡ := GL_{d}(𝔽_{q}). Namely, let B̄^{ℓ} be the maximal prime-to-ℓ subgroup of the standard Borel subgroup B̄ of Ḡ, and let bḠ be the central idempotent in ℤ_{ℓ}[Ḡ] corresponding to the sum of all blocks that contain a unipotent rational series. Explicitly, bḠ is the sum, in \overline{ℚ}_{ℓ}[Ḡ] of all central primitive idempotents e_{π} corresponding to irreducible \overline{ℚ}_{ℓ}Ḡ-representations π whose semisimple part s_{π} in Lusztig’s classification is an ℓ-element. Then the claim is that P̄ := ℤ_{ℓ}[Ḡ/B̄^{ℓ}] is a projective generator of the category bḠRep_{ℤ_{ℓ}}(Ḡ).

It is indeed clear that P̄ is projective. Moreover, the Jordan-Hölder constituents of P̄ ⊗_{ℤ_{ℓ}} \overline{ℚ}_{ℓ} are all the irreducible representations whose semisimple part in Lusztig’s parametrization is an ℓ-element of a split torus of the dual group Ḡ^{∗}. But because of our assumption on ℓ, every semi-simple ℓ-element of Ḡ^{∗} lies in a split torus. The claim follows.

If H is an open compact subgroup of G, we denote by H_{R}(G, H) the Hecke algebra of left and right H-invariant measures on G with coefficients in R.

(2.1.2) Corollary.– The functor V ↦ V^{I^{ℓ}} induces an equivalence of categories

Rep_{b}^{∞}(G) ⟶^{∼} Mod(H_{ℤ_{ℓ}}(G, I^{ℓ}))

with quasi-inverse the functor M ↦ ℤ_{ℓ}[G/I^{ℓ}] ⊗_{H_{ℤ_{ℓ}}(G,I^{ℓ})} M.

The “intersection” Rep_{\overline{𝔽}_{ℓ}}^{∞}(G) ∩ Rep_{b}^{∞}(G) is Vignéras’ unipotent block. In particular, the irreducible \overline{𝔽}_{ℓ}-objects in Rep_{b}^{∞}(G) are the irreducible \overline{𝔽}_{ℓ}-representations which appear as subquotients of some representation Ind_{B}^{G} (χ), induced from an unramified \overline{𝔽}_{ℓ}-character χ of a Borel subgroup B, see [5, Prop. 3.1.3]. Via the functor of the above corollary, these irreducible objects are in bijection with simple H_{\overline{𝔽}_{ℓ}}(G, I^{ℓ})-modules.

Let 𝔪_{I} be the maximal ideal of the local subalgebra ℤ_{ℓ}[I/I^{ℓ}] of H_{ℤ_{ℓ}}(G, I^{ℓ}).

(2.1.3) Proposition.– The ideal 𝔪 := 𝔪_{I}H_{ℤ_{ℓ}}(G, I^{ℓ}) is two-sided, equal to H_{ℤ_{ℓ}}(G, I^{ℓ})𝔪_{I} and the map [I^{ℓ}gI^{ℓ}] ↦ [IgI] induces an isomorphism of 𝔽_{ℓ}-algebras

H_{ℤ_{ℓ}}(G, I^{ℓ})/𝔪 ⟶^{∼} H_{𝔽_{ℓ}}(G, I).

Before proving the proposition, we introduce some more notation. Let T be the diagonal torus in G and let N := N_{G}(T) be its normalizer. We denote by T^{0} the maximal compact subgroup of T, and by T^{ℓ} the maximal prime-to-ℓ subgroup of T^{0}. Both are normal subgroups of N.

Proof. Since T^{ℓ} ⊂ I^{ℓ}, any element w ∈ N/T^{ℓ} gives rise to a well-defined Hecke operator [I^{ℓ}wI^{ℓ}]. By the Iwahori decomposition, the Hecke operators [I^{ℓ}wI^{ℓ}], with w running on N/T^{ℓ}, form a ℤ_{ℓ}-basis of H_{ℤ_{ℓ}}(G, I^{ℓ}). Among them, the operators [I^{ℓ}tI^{ℓ}] with t ∈ T^{0}/T^{ℓ} form a basis of the subalgebra ℤ_{ℓ}[I/I^{ℓ}]. Then, the formula

[I^{ℓ}wI^{ℓ}] ∗ [I^{ℓ}tI^{ℓ}] = [I^{ℓ}wtw^{−1}I^{ℓ}] ∗ [I^{ℓ}wI^{ℓ}]

shows that 𝔪_{I}H_{ℤ_{ℓ}}(G, I^{ℓ}) = H_{ℤ_{ℓ}}(G, I^{ℓ})𝔪_{I} is a two-sided ideal, since 𝔪_{I} is generated by elements 1−[I^{ℓ}tI^{ℓ}], t ∈ T/T^{ℓ}. The same formula shows that the map of the proposition is an isomorphism of 𝔽_{ℓ}-vector spaces. That it is a morphism of algebras follows from the definition of convolution products on both sides, and the fact that for an element w ∈ N the obvious map

(I^{ℓ} ∩ wI^{ℓ}w^{−1})\I^{ℓ} ⟶ (I ∩ wIw^{−1})\I

is a bijection.

We note that if M is an H_{\overline{𝔽}_{ℓ}}(G, I^{ℓ})-module, then M/𝔪M identifies with the I-coinvariants M_{I}, where I acts through I/I^{ℓ}.

(2.1.4) Corollary.– Any simple H_{\overline{𝔽}_{ℓ}}(G, I^{ℓ})-module is killed by 𝔪, and the map of the previous proposition induces a bijection between simple H_{\overline{𝔽}_{ℓ}}(G, I)-modules and simple H_{\overline{𝔽}_{ℓ}}(G, I^{ℓ})- modules. Equivalently, for any irreducible \overline{𝔽}_{ℓ}-representations V of G, we have V^{I^{ℓ}} = V^{I} ⟶^{∼} V_{I} and the functor V ↦ V^{I} induces a bijection between irreducible \overline{𝔽}_{ℓ}-representations in Rep_{b}^{∞}(G) and simple H_{\overline{𝔽}_{ℓ}}(G, I)-modules.

2.2 Elliptic unipotent representations

We first recall the structure of the Iwahori-Hecke algebra, taking into account the fact that q = 1 in \overline{𝔽}_{ℓ}.

(2.2.1) Fact– The map w ↦ [IwI] is an isomorphism of \overline{𝔽}_{ℓ}-algebras

\overline{𝔽}_{ℓ}[N/T^{0}] ⟶^{∼} H_{\overline{𝔽}_{ℓ}}(G, I)

Proof. In general, this map induces an isomorphism of algebras T_{w} ↦ [IwI] from the Iwahori-Hecke algebra H_{q}(W̃) ⊗ \overline{𝔽}_{ℓ} with parameter q of the extended Weyl group W̃ = N/T^{0} of (G, T) to H_{\overline{𝔽}_{ℓ}}(G, I). The multiplication in H_{q}(W̃) is determined by the braid relation and the formula T_{s}^{2} = (q − 1)T_{s} + q for each simple reflection s associated to I. Specializing at q = 1 therefore gives H_{q}(W̃) ⊗ \overline{𝔽}_{ℓ} = \overline{𝔽}_{ℓ}[W̃].

We now revisit the classical relation between parabolic induction and compact induction in the context where q = 1 in \overline{𝔽}_{ℓ}. Let χ : T/T^{0} ⟶ \overline{𝔽}_{ℓ}^{×} be an unramified character of T and consider the (unnormalized) induction Ind_{B}^{G}(χ). Let us also identify the symmetric group 𝔖_{d} with the subgroup of permutation matrices of G in the usual way. Because of the double cosets decompositions G = ⨆_{w∈𝔖_{d}} IwB = ⨆_{w∈𝔖_{d}} I^{ℓ}wB, we see that

Ind_{B}^{G} (χ)^{I^{ℓ}} = Ind_{B}^{G} (χ)^{I} . (2.2.2)

In particular the action of H_{\overline{𝔽}_{ℓ}}(G, I^{ℓ}) on Ind_{B}^{G} (χ)^{I^{ℓ}} factors through H_{\overline{𝔽}_{ℓ}}(G, I). Through the previous isomorphism \overline{𝔽}_{ℓ}[N/T^{0}] ⟶^{∼} H_{\overline{𝔽}_{ℓ}}(G, I), this action is given as follows.

(2.2.3) Fact– For w ∈ 𝔖_{d}, let [IwB]_{χ} be the unique element of Ind_{B}^{G} (χ)^{I} that is supported on IwB and takes value 1 on w. Similarly, let [wT]_{χ} be the unique element of ind_{T}^{N} (χ) that is supported on wT and takes value 1 on w. Then the map [wT]_{χ} ↦ [IwB]_{χ} is an isomorphism of 𝔽_{ℓ}[N/T^{0}]-modules

ind_{T}^{N} (χ) ⟶^{∼} Ind_{B}^{G} (χ)^{I} .

Proof. The mixed Bruhat decomposition shows that ([IwB]_{χ})_{w∈𝔖_{d}} is a basis of ind_{B}^{G} (χ)^{I} over \overline{𝔽}_{ℓ}, therefore the map is an isomorphism of \overline{𝔽}_{ℓ}-vector spaces. It is elementary to check that [IwI] ∗ [IB]_{χ} = [IwB]_{χ} for all w ∈ 𝔖_{d}, showing that the map is 𝔖_{d}-equivariant. Moreover if t ∈ T dilates the unipotent radical of B, we see that [ItI] ∗ [IB]_{χ} = χ^{−1}(t)[IB]_{χ}. Since the semi-group T^{+} of all elements that dilate the radical of B generates the group T, this equality is true for all t ∈ T. The T -equivariance of the map follows.

As a particular case, we get Ind_{B}^{G} (\overline{𝔽}_{ℓ})^{I^{ℓ}} ≃ \overline{𝔽}_{ℓ}[N/T] = \overline{𝔽}_{ℓ}[𝔖_{d}]. Because of our assumption that ℓ > d, the right hand side is a semisimple H_{\overline{𝔽}_{ℓ}}(G, I^{ℓ})-module. We summarize this as follows.

(2.2.4) Corollary.– Ind_{B}^{G} (\overline{𝔽}_{ℓ}) is a semi-simple representation of G and the functor V ↦ V^{I} induces an isomorphism between the poset of subrepresentations of Ind_{B}^{G} (\overline{𝔽}_{ℓ}) and that of subrepresentations of the regular \overline{𝔽}_{ℓ}-representation of the symmetric group 𝔖_{d}.

(2.2.5) More notation. We put S := {1, ⋯, d − 1} and we think of S as the set of simple roots of T in the upper triangular matrices, row-wise numbered. To each subset J ⊂ S is associated a unique standard parabolic subgroup P_{J} which contains B and such that J is the set of simple roots of T in the upper triangular matrices of the Levi component L_{J} of P_{J}. Denote the Weyl group of L_{J} by 𝔖_{J}, a parabolic subgroup of the Weyl group 𝔖_{S} = 𝔖_{d} of G. Then we have an isomorphism of \overline{𝔽}_{ℓ}[N/T^{0}]-modules

Ind_{P_{J}}^{G} (\overline{𝔽}_{ℓ})^{I^{ℓ}} = Ind_{P_{J}}^{G} (\overline{𝔽}_{ℓ})^{I} ≃ \overline{𝔽}_{ℓ}[𝔖_{d}/𝔖_{J}].

In fact the image of the submodule Ind_{P_{J}}^{G} (\overline{𝔽}_{ℓ})^{I} of Ind_{B}^{G} (\overline{𝔽}_{ℓ})^{I} by the map of Corollary (2.2.4) is the submodule \overline{𝔽}_{ℓ}[𝔖_{d}/𝔖_{J}] of \overline{𝔽}_{ℓ}[𝔖_{d}]. As usual in this context, for any ring R we put

i_{J}(R) := Ind_{P_{J}}^{G} (R) and v_{J}(R) := i_{J}(R)/ ∑_{K⊃J} i_{K}(R).

Recall that Ind_{B}^{G} (\overline{ℚ}_{ℓ}) is multiplicity free, with pairwise distinct irreducible subquotients all v_{J}(\overline{ℚ}_{ℓ}), J ⊂ S. In contrast, Ind_{B}^{G}(\overline{𝔽}_{ℓ}) is not multiplicity free, and v_{J}(\overline{𝔽}_{ℓ}) need not be irreducible.

(2.2.6) Notation again. We denote by P (d) the set of partitions λ = (λ_{1} ⩾ λ_{2} ⩾ ⋯) of d. To such a partition is associated the parabolic subgroup 𝔖_{λ} := 𝔖_{λ_{1}} × 𝔖_{λ_{2}} × ⋯ of 𝔖_{d}, the permutation module M_{λ} := \overline{𝔽}_{ℓ}[𝔖_{d}/𝔖_{λ}], and the simple \overline{𝔽}_{ℓ}[𝔖_{d}]-module S_{λ}. The latter appears with multiplicity one in M_{λ} and may be inductively characterized by equalities M_{λ} = S_{λ} + ∑_{μ>λ} m_{λ,μ}S_{μ} in the Grothendieck group of \overline{𝔽}_{ℓ}[𝔖_{d}]-modules. We will denote by π_{λ} the unique irreducible \overline{𝔽}_{ℓ}-representation of G such that (π_{λ})^{I} ≃ S_{λ}.

To a subset J ⊂ S we associate the unique partition λ_{J} such that 𝔖_{λ_{J}} is conjugate to 𝔖_{J}. We then have i_{J}(\overline{𝔽}_{ℓ})^{I} ≃ M_{λ_{J}}, so that

i_{J}(\overline{𝔽}_{ℓ}) = π_{λ_{J}} + ∑_{μ>λ_{J}} m_{λ_{J} ,μ}π_{μ}

in the Grothendieck group of finite length \overline{𝔽}_{ℓ}-representations of G. We can also write

v_{J}(\overline{𝔽}_{ℓ}) = π_{λ_{J}} + ∑_{μ>λ_{J}} m′_{λ_{J} ,μ}π_{μ}

but in general m′_{λ_{J} ,μ} need not vanish.

(2.2.7) Elliptic unipotent representations. An irreducible representation of G is called elliptic if it is not a virtual sum of parabolically induced representations. We know from [8, Lemme 3.2.1] that up to unramified twist, an elliptic unipotent representation occurs as a subquotient of Ind_{B}^{G} (\overline{𝔽}_{ℓ}). However, in contrast with the regular case, not all such subquotients are elliptic.

Proposition.– The representation π_{λ} is elliptic if and only if λ is hook-shaped, that is if λ = (i, 1^{(d−i)}) for some i ∈ {1, ⋯, d}.

Proof. The set [M_{μ}]_{μ∈P (d)} is a basis of the Grothendieck group of \overline{𝔽}_{ℓ}-representations of 𝔖_{d} (recall that ℓ > d). Write [S_{λ}] = ∑_{μ⩾λ} a_{λ,μ}[M_{μ}]. By the above discussion, π_{λ} is elliptic if and only if a_{λ,(d)} ≠ 0. It is proved in [12, 2.3.17] that this is equivalent to λ being a hook.

Therefore, there are only d elliptic constituents in Ind_{B}^{G} (\overline{𝔽}_{ℓ}), in high contrast with the ℓ-adic or banal case (2^{d−1} of them) or the regular non-banal case (2^{d} − 1 of them).

There is a convenient realization of the modules S_{(i,1^{(n−i)})}. Denote by Std the standard (d − 1)-dimensional \overline{𝔽}_{ℓ}-representation of 𝔖_{d}. This is the subrepresentation of the permutation representation on \overline{𝔽}_{ℓ}^{d} on the subspace of vectors whose sum of coordinates vanish.

Fact.– For i = 1, ⋯ d, we have S_{(i,1^{(d−i)})} = ∧^{d−i}Std. In particular, S_{(d)} is the trivial representation, and S_{1^{(d)}} is the sign representation.

The next fact will be an important technical tool in the study of the unipotent part of the cohomology complex of the Lubin-Tate tower.

(2.2.8) Proposition.– For i ∈ {1, ⋯, d}, we have

v_{\{1,···,i−1\}}(\overline{𝔽}_{ℓ}) ≃ v_{\{d−i+1,···,d−1\}}(\overline{𝔽}_{ℓ}) ≃ π_{(i,1^{(d−i)})}.

Proof. Since π_{(i,1^{(d−i)})} is a Jordan-Hölder factor of both v_{\{1,···,i−1\}}(\overline{𝔽}_{ℓ}) and v_{\{d−i+1,···,d−1\}}(\overline{𝔽}_{ℓ}), it suffices to prove the following equalities of dimensions

dim_{\overline{𝔽}_{ℓ}} v_{\{1,···,i−1\}}(\overline{𝔽}_{ℓ})^{I} = dim_{\overline{𝔽}_{ℓ}} v_{\{d−i+1,···,d−1\}}(\overline{𝔽}_{ℓ})^{I} = dim_{\overline{𝔽}_{ℓ}} S_{(i,1^{(d−i)})}.

From the previous Fact or from the hook-length formula, we have dim_{\overline{𝔽}_{ℓ}} S_{(i,1^{(d−i)})} = \binom{d−1}{d−i}. On the other hand, for any subset J ⊂ S = \{1, ⋯, d − 1\}, we have by definition v_{J}(\overline{𝔽}_{ℓ}) = v_{J}(ℤ_{ℓ}) ⊗_{ℤ_{ℓ}} \overline{𝔽}_{ℓ} and we know from [14, Cor. 4.5] that v_{J}(ℤ_{ℓ}) is free over ℤ_{ℓ}. Therefore we have

dim_{\overline{𝔽}_{ℓ}} v_{J}(\overline{𝔽}_{ℓ})^{I} = dim_{\overline{𝔽}_{ℓ}} v_{J}(\overline{𝔽}_{ℓ})^{I^{ℓ}} = dim_{\overline{ℚ}_{ℓ}} v_{J}(\overline{ℚ}_{ℓ})^{I^{ℓ}} = dim_{\overline{ℚ}_{ℓ}} v_{J}(\overline{ℚ}_{ℓ})^{I}.

Denote by r_{B} the normalized Jacquet functor along B. Then Borel’s theorem on principal series representations tells us that for any J ⊂ S, we have

dim_{\overline{ℚ}_{ℓ}} v_{J}(\overline{ℚ}_{ℓ})^{I} = dim_{\overline{ℚ}_{ℓ}} r_{B}(v_{J}(\overline{ℚ}_{ℓ})).

Now, for w ∈ 𝔖_{d}, put J(w) := {j ∈ {1, ⋯, d − 1}, w(j) < w(j + 1)}. By Fact 2.1.1 of [5] and the subsequent paragraph in this reference, we have

dim_{\overline{ℚ}_{ℓ}} r_{B}(v_{J}(\overline{ℚ}_{ℓ})) = #\{w ∈ 𝔖_{d}, J(w) = J\}.

Observe that in the cases that J = {1, ⋯, i − 1} or J = {d − i, ⋯, d − 1}, the map w ↦ w_{|J} induces a bijection from {w ∈ 𝔖_{d}, J(w) = J} to the set of non-decreasing maps J ⟶ {1, ⋯, d − 1}. Therefore, in the same cases, the map w ↦ w(J) induces a bijection from {w ∈ 𝔖_{d}, J(w) = J} to the set of subsets of size |J| = i − 1 in S, whence the desired equalities

dim_{\overline{𝔽}_{ℓ}} v_{\{1,···,i−1\}}(\overline{𝔽}_{ℓ})^{I} = dim_{\overline{𝔽}_{ℓ}} v_{\{d−i+1,···,d−1\}}(\overline{𝔽}_{ℓ})^{I} = \binom{d − 1}{i − 1}.

2.3 Extensions between some simple H_{\overline{𝔽}_{ℓ}}(G, I)-modules

(2.3.1) Let G^{0} be the subgroup of G generated by compact elements. We have G^{0} = ker (| det |_{K} : G ⟶ ℝ_{+}^{×}). The isomorphism of Fact (2.2.1) restricts to an isomorphism

\overline{𝔽}_{ℓ}[(N ∩ G^{0})/T^{0}] ⟶^{∼} H_{\overline{𝔽}_{ℓ}}(G^{0}, I).

The group (N ∩ G^{0})/T^{0} is an extension

(T ∩ G^{0})/T^{0} ↪ (N ∩ G^{0})/T^{0} ↠ 𝔖_{d}.

where (T ∩ G^{0})/T^{0} is a free abelian group of rank d − 1 on which the conjugation action of N ∩ G^{0} factors through 𝔖_{d} and is the standard representation, namely

\overline{𝔽}_{ℓ} ⊗_{ℤ} ((T ∩ G^{0})/T^{0}) ≃_{𝔖_{d}} Std.

(2.3.2) Proposition.– Let A and B be two \overline{𝔽}_{ℓ}[𝔖_{d}]-modules, that we may see as \overline{𝔽}_{ℓ}[N/T^{0}]- modules via the projection N/T^{0} ↠ 𝔖_{d}.

i) There is a natural isomorphism

Ext_{\overline{𝔽}_{ℓ}[(N∩G^{0})/T^{0}]}^{∗}(A, B) ⟶^{∼} Hom_{\overline{𝔽}_{ℓ}[𝔖_{d}]}(A, B ⊗ ∧^{∗}Std)

functorial in A and B.

ii) If C is another \overline{𝔽}_{ℓ}[𝔖_{d}]-module, cup-products are given by the following compositions.

Ext_{\overline{𝔽}_{ℓ}[(N∩G^{0})/T^{0}]}^{k}(A, B) ⊗ Ext_{\overline{𝔽}_{ℓ}[(N∩G^{0})/T^{0}]}^{l}(B, C)
⟶^{∼} Hom_{\overline{𝔽}_{ℓ}[𝔖_{d}]}(A, B ⊗ ∧^{k}Std) ⊗ Hom_{\overline{𝔽}_{ℓ}[𝔖_{d}]}(B, C ⊗ ∧^{l}Std)
⟶ Hom_{\overline{𝔽}_{ℓ}[𝔖_{d}]}(A, C ⊗ ∧^{l}Std ⊗ ∧^{k}Std)
⟶ Hom_{\overline{𝔽}_{ℓ}[𝔖_{d}]}(A, C ⊗ ∧^{k+l}Std)
⟶^{∼} Ext_{\overline{𝔽}_{ℓ}[(N∩G^{0})/T^{0}]}^{k+l}(A, C)

where the second map is composition and the third one is induced by the exterior product.

Proof. As with any free abelian group of finite rank, there is a natural isomorphism of graded algebras

Ext_{\overline{𝔽}_{ℓ}[(T ∩G^{0})/T^{0}]}^{∗}(\overline{𝔽}_{ℓ}, \overline{𝔽}_{ℓ}) ≃ ⋀^{∗} (\overline{𝔽}_{ℓ} ⊗_{ℤ} ((T ∩ G^{0})/T^{0})) .

This isomorphism is compatible with automorphisms of the group (T ∩G^{0})/T^{0}, and in particular with the action of 𝔖_{d}. As already noted above the proposition, the RHS with its 𝔖_{d} action is ∧^{∗}Std. With A and B as in the proposition, we thus get

Ext_{\overline{𝔽}_{ℓ}[(N∩G^{0})/T^{0}]}^{∗}(A, B) ≃ Ext_{\overline{𝔽}_{ℓ}[(T ∩G^{0})/T^{0}]}^{∗}(A, B)^{𝔖_{d}}
≃ (Hom_{\overline{𝔽}_{ℓ}}(A, B) ⊗ ∧^{∗}Std))^{𝔖_{d}}
≃ Hom_{\overline{𝔽}_{ℓ}𝔖_{d}}(A, B ⊗ ∧^{∗}Std).

Here in the first line we have used that ℓ > d, so that 𝔖_{d} has no higher cohomology on \overline{𝔽}_{ℓ}[𝔖_{d}]- modules. This also shows statement ii) on the cup-products, since the algebra structure on Ext_{\overline{𝔽}_{ℓ}[(T ∩G^{0})/T^{0}]}^{∗}(\overline{𝔽}_{ℓ}, \overline{𝔽}_{ℓ}) is given by the exterior product.

We see in particular that the dimension of Ext_{\overline{𝔽}_{ℓ}[(N∩G^{0})/T^{0}]}^{k}(S_{λ}, S_{μ}) equals the multiplicity of S_{λ} as a constituent of S_{μ} ⊗ S_{(n−k,1^{(k)})}. Computing such multiplicities is a notoriously difficult problem and remains largely open. Fortunately, enough is known for our purposes in this paper.

(2.3.3) Corollary.– For i, j, k ∈ \{0, ⋯, d − 1\}, the dimension over \overline{𝔽}_{ℓ} of the extension space Ext_{\overline{𝔽}_{ℓ}[(N∩G^{0})/T^{0}]}^{k}(∧^{i}Std, ∧^{j}Std) is either 0 or 1. It is 1 if and only if the following inequalities hold :

i + j ⩾ k, j + k ⩾ i, k + i ⩾ j, i + j + k ⩽ 2d − 2.

Remark.– The symmetry of the above conditions should not be surprising since the dimension we are interested in is that of (∧^{i}Std ⊗ ∧^{j}Std ⊗ ∧^{k}Std)^{𝔖_{d}} by the last proposition and the self-duality of irreducible representations of 𝔖_{d}.

The above conditions are also invariant under the transformation (i, j) ↦ (i′, j′) := (d − 1 − i, d − 1 − j). This corresponds to the fact that ∧^{i′}Std ≃ ∧^{i}Std ⊗ ∧^{d−1}Std.

A less symmetric formulation of the inequalities of the Corollary, which is sometimes more convenient, is : |j − i| ⩽ k ⩽ min(i + j, i′ + j′). (2.3.4)

Proof. This is Theorem 2.1 of [13], formulated in a more symmetric way, and corrected. More precisely, assume, as we may from the above remark, that i ⩽ j and i + j ⩽ d − 1. Then our claim is that we have non vanishing (and multiplicity one) if and only if j − i ⩽ k ⩽ i + j. To match Remmel’s notation in [13], we put n := d, r := n−i and s := n−j. Then Remmel asserts that non-vanishing (and multiplicity one) holds if and only if j − i ⩽ k ⩽ i + j + 1, which seems incompatible with our claim, and which is obviously false when i = j = 0. However, there is a slight mistake in the proof, located in the third line on p. 113 of [13], where it is asserted that “there are two possibilities for the positions of the remaining green cells...”. Indeed, in the case that there are no green cells at all (i.e. when x = n − i − j) there is only one possibility. Once corrected, we get our claim.

In our study of the cohomology complex of the Lubin-Tate tower we will need some cupproducts between some non-vanishing Ext spaces of the above Corollary. In order to simplify the notation a bit, we will abbreviate Λ^{i} := ∧^{i}Std and write Ext^{k}(Λ^{i}, Λ^{j}) instead of Ext_{\overline{𝔽}_{ℓ}[(G^{0}∩T)/T^{0}]}^{k}(∧^{i}Std, ∧^{j}Std).

(2.3.5) Theorem.– For i = 1, ⋯, d − 2, we fix a generator ζ_{i,i}^{1} of Ext^{1}(Λ^{i}, Λ^{i}). Let i, j, k be integers such that Ext^{k}(Λ^{i}, Λ^{j}) is non-zero and Ext^{k+1}(Λ^{i}, Λ^{j}) is also non-zero. Then, both the cup-product maps

ξ_{i,i}^{1} ∪ − and − ∪ ξ_{j,j}^{1} : Ext^{k}(Λ^{i}, Λ^{j}) ⟶ Ext^{k+1}(Λ^{i}, Λ^{j})

are isomorphisms.

We postpone the proof of this Theorem to the appendix, in order to lighten the exposition a bit. Let us mention the following corollary.

(2.3.6) Corollary.– For i ∈ {1, ⋯, d − 1}, the self-extension algebra Ext^{∗}(Λ^{i}, Λ^{i}) is (graded) isomorphic to \overline{𝔽}_{ℓ}[X]/(X^{2 min(i,i′)}) via X ↦ ξ_{i,i}^{1}. Moreover for any other j ∈ \{1, ⋯, d − 1}, the graded space Ext^{∗}(Λ^{i}, Λ^{j}) is a (shifted) cyclic module over Ext^{∗}(Λ^{i}, Λ^{i}).

The proof of the next result is also postponed to the appendix, paragraph (A.1.4).

(2.3.7) Proposition.– Let i ⩽ j ∈ {0, ⋯, d−1} and fix a generator ξ_{j,i}^{j−i} of Ext^{j−i}(Λ^{j}, Λ^{i}). Then for any simple module S over \overline{𝔽}_{ℓ}[𝔖_{d}], the cup-product

ξ_{j,i}^{j−i} ∪ − : Ext^{j′}(Λ^{i}, S) ⟶ Ext^{i′}(Λ^{j}, S)

is an isomorphism (recall that i′ = d − 1 − i).

3 Cohomology and the Lefschetz operator

As in section 1, we denote by RΓ_{c}(M_{LT}^{ca}, ℤ_{ℓ}) the cohomology complex of the Lubin-Tate tower of height d of the field K. This is an object of the derived category D^{b}(Rep_{ℤ_{ℓ}}^{∞}(G)) with an action the Weil group W_{K} and of the unit group D^{×} of the division algebra with invariant 1/d over K. We refer to [4, Section 3.2] for a precise definition of this object.

We want to compute the graded \overline{𝔽}_{ℓ}[D^{×} × W_{K}]-module

R_{π}^{∗} := H^{∗} (R Hom_{ℤ_{ℓ}G}(RΓ_{c}(M_{LT}^{ca}, ℤ_{ℓ}), π)) [1 − d]

for π a unipotent irreducible \overline{𝔽}_{ℓ}-representation of G. Notice the shift by 1 − d which is here for convenience. Indeed, by [7, Prop 2.1.3], the graded space R_{π}^{∗} is supported in the range [1 − d, d − 1].

3.1 The unipotent part of the cohomology complex

Thanks to the equivalence of categories of Corollary (2.1.2) we have

R_{π}^{∗} ≃ R^{∗} Hom_{H_{ℤ_{ℓ}}(G,I^{ℓ})}(RΓ_{c}(M_{LT}^{ca}, ℤ_{ℓ})^{I^{ℓ}}, π^{I^{ℓ}})[1 − d].

Moreover, since the action of I on π^{I^{ℓ}} is trivial, Proposition (2.1.3) implies

R_{π}^{∗} ≃ H^{∗} (R Hom_{H_{𝔽_{ℓ}}(G,I)}(RΓ_{c}(M_{LT}^{ca}, ℤ_{ℓ})^{I^{ℓ}} ⊗_{H_{ℤ_{ℓ}}(G,I^{ℓ})}^{L} H_{𝔽_{ℓ}}(G, I), π^{I})) [1 − d]
≃ H^{∗} (R Hom_{H_{ℤ_{ℓ}}(G,I)}(L_{I}(RΓ_{c}(M_{LT}^{ca}, ℤ_{ℓ})), π^{I})) [1 − d]

where L_{I} denotes the left derived functor of the I-coinvariant functor. Let M_{LT,I} denote the Iwahori level of the Lubin-Tate tower (a quotient of the tame level M_{LT,1}). We have L_{I}(RΓ_{c}(M_{LT}^{ca}, ℤ_{ℓ})) ≃ RΓ_{c}(M_{LT,I}^{ca}, ℤ_{ℓ}). Now recall that the G-tower M_{LT} is induced from a G^{0}-tower M_{LT}^{(0)} (the notation we use is that of [4, 3.5.1]). The analytic space M_{LT,I}^{(0)} is the deformation space with Iwahori level structure of a formal O_{K}-module of height d over k̄. Finally we have obtained

(3.1.1) R_{π}^{∗} ≃ H^{∗} (R Hom_{H_{\overline{𝔽}_{ℓ}}(G^{0},I)}(RΓ_{c}(M_{LT,I}^{ca,(0)}, \overline{𝔽}_{ℓ}), π^{I})) [1 − d].

Let us abbreviate

C_{I} := RΓ_{c}(M_{LT,I}^{ca,(0)}, \overline{𝔽}_{ℓ})[d − 1] ∈ D^{b}(H_{\overline{𝔽}_{ℓ}}(G^{0}, I)).

Its cohomology is quite easy to describe, although the author does not know any elementary proof. Recall that any \overline{𝔽}_{ℓ}[𝔖_{d}]-module inflates to a H_{\overline{𝔽}_{ℓ}}(G^{0}, I)-module via the isomorphism

\overline{𝔽}_{ℓ}[(N ∩ G^{0})/T^{0}] ⟶^{∼} H_{\overline{𝔽}_{ℓ}}(G^{0}, I) and the projection (N ∩ G^{0})/T^{0} ↠ 𝔖_{d}.

(3.1.2) Proposition.– The cohomology of C_{I} vanishes outside the range {0, ⋯, d − 1}. For i ∈ {0, ⋯, d − 1}, we have H^{i}(C_{I}) ≃ ∧^{i′}Std where i′ := d − 1 − i.

Proof. The deformation ring R_{I} with Iwahori level structure of a formal O_{K}-module of height d is known to be isomorphic to Ŏ[[X_{1}, ⋯, X_{d}]]/(X_{1}X_{2} ⋯ X_{d} − ϖ) where Ŏ is the completed maximal unramified extension of O_{K} and ϖ is a uniformizer of O_{K}. A reference is [15], top of p. 483. It follows that the vanishing cycles Ψ^{i}(R_{I}, ℤ_{ℓ}) are isomorphic, as ℤ_{ℓ}-modules, to ∧^{i}(ℤ_{ℓ}^{d−1}). Since H^{i}(C_{I}) = H_{c}^{d−1+i}(M_{LT,I}^{ca,(0)}, \overline{𝔽}_{ℓ}) = Ψ^{d−1−i}(R_{I}, \overline{𝔽}_{ℓ})^{∨}, we get at least the expected dimension for these cohomology spaces.

Unfortunately, computing the action of the Hecke algebra is not so easy. However, here we observe that H_{c}^{d−1+i}(M_{LT,I}^{ca,(0)}, ℤ_{ℓ}) is torsion free, so that

H_{c}^{d−1+i}(M_{LT,I}^{ca,(0)}, \overline{𝔽}_{ℓ}) = H_{c}^{d−1+i}(M_{LT,I}^{ca,(0)}, ℤ_{ℓ}) ⊗ \overline{𝔽}_{ℓ}

and we may hope to deduce H_{c}^{d−1+i}(M_{LT,I}^{ca,(0)}, \overline{𝔽}_{ℓ}) by reduction modulo ℓ of H_{c}^{d−1+i}(M_{LT,I}^{ca,(0)}, \overline{ℚ}_{ℓ}). Indeed, by Proposition (2.2.8), it suffices to know that

H_{c}^{d−1+i}(M_{LT,I}^{ca,(0)}, \overline{ℚ}_{ℓ}) ≃ v_{\{1,···,i\}}(\overline{ℚ}_{ℓ})^{I}.

There are two ways to infer such an isomorphism. It follows from Boyer’s local theorem in [1], which uses global arguments, but remains “on the Lubin-Tate side”. It also follows by purely local arguments, from the Faltings-Fargues theorem [10] that the cohomology of the Lubin-Tate tower coincides with that of the Drinfeld tower, and the Schneider-Stuhler computation in [14] of the cohomology of the Drinfeld symmetric space.

(3.1.3) Corollary.– We have R_{π}^{∗} = 0 unless π is a subquotient of Ind_{B}^{G} (\overline{𝔽}_{ℓ}) and π^{I} = π_{λ} with λ a hook or double-hook partition. Moreover, the action of D^{×} is trivial and that of W_{K} is unipotent.

Proof. By the proposition we have a spectral sequence E_{2}^{p,q} := Ext_{H(G^{0},I)}^{p}(∧^{q′}Std, π^{I}) ⇒ R_{π}^{p−q}. So R_{π}^{∗} vanishes unless Ext_{H(G^{0},I)}^{p}(∧^{q′}Std, π^{I}) ≠ 0 for some p, q. In this case, π^{I} has to be trivial on (T ∩G^{0})/T^{0} ⊂ H(G^{0}, I), so that π is a subquotient of Ind_{B}^{G} (\overline{𝔽}_{ℓ}) and π^{I} comes from a simple \overline{𝔽}_{ℓ}[𝔖_{d}]-module. Then, by Proposition (2.3.2) this simple module occurs in ∧^{q′}Std ⊗ ∧^{p}Std. It follows from Remmel’s theorem in [13], that this simple module is associated to a double-hook or hook partition.

Let us turn to the actions of W_{K} and D^{×}. We know that W_{K} acts trivially on the cohomology of C_{I} (because of q = 1 in \overline{𝔽}_{ℓ}), therefore W_{K} acts unipotently on C_{I} hence also on R_{π}^{∗}. For the same reason, the action of D^{×} on C_{I} has to be unipotent. However, the center F^{×} of D^{×} acts on R_{π}^{∗} by the same character as F^{×} acts on π, i.e. the trivial character. Since ℓ does not divide the pro-order of D^{×}/F^{×}, we deduce that D^{×} acts trivially.

One consequence of the next paragraph will be the following theorem.

(3.1.4) Theorem.– The complex C_{I} is split in D^{b}(H_{\overline{𝔽}_{ℓ}}(G^{0}, I)). Namely, we have (noncanonically) C_{I} ≃ ⨁_{i=0}^{n−1} ∧^{i′}Std[−i] in D^{b}(H_{\overline{𝔽}_{ℓ}}(G^{0}, I)).

Proof. In the proof of Theorem (3.2.1) below, we get the following property on C_{I}. For all i = 0, ⋯, d − 1, the spectral sequence

E_{2}^{pq} = Ext_{H(G^{0},I)}^{p}(H^{q}(C_{I}), H^{i}(C_{I})) ⇒ Hom_{D^{b}(H(G^{0},I)}(C_{I}[q − p], H^{i}(C_{I}))

degenerates at E_{2}. But then, it follows from the proof of the implication i) ⇒ ii) of Proposition (1.2) of [9] (in fact a dual version of it as in Remark (1.4) of loc. cit.) that the complex C_{I} is split.

Remark.– In contrast, the complex b_{G}RΓ_{c}(M_{LT}, \overline{𝔽}_{ℓ}) is certainly not split in D_{\overline{𝔽}_{ℓ}}^{b}(G). Equivalently, C := RΓ_{c}(M_{LT}, \overline{𝔽}_{ℓ})^{I^{ℓ}} is not split in D^{b}(H(G, I^{ℓ})). Indeed, it is a perfect complex of \overline{𝔽}_{ℓ}[I/I^{ℓ}]-modules whose cohomology spaces are not of finite projective dimension since I acts trivially on them.

3.2 The graded dimension of R_{π}^{∗} when π is elliptic unipotent.

For j = 0, ⋯, d − 1, we put π_{j} := π_{(d−j,1^{(j)})}, so that (π_{j})^{I} ≃ ∧^{j}Std. As in Proposition (3.1.2), we put

j′ := d − 1 − j.

(3.2.1) Theorem.– The graded vector space R_{π_{j}}^{∗} is supported in the range [−j′, j′]. For k ∈ [−j′, j′] we have

dim_{\overline{𝔽}_{ℓ}}(R_{π_{j}}^{k}) = \{ \begin{matrix}j + 1 & if & k − j′ is even, \\ j & if & k − j′ is odd. \end{matrix}

Proof. We will prove equality by proving both inequalities. In order to bound above dim_{\overline{𝔽}_{ℓ}}(R_{π_{j}}^{k}), we use the spectral sequence

E_{2}^{p,q} := Ext_{H(G^{0},I)}^{p}(H^{q}(C_{I}), π_{j}^{I}) ⇒ R_{π_{j}}^{p−q}. (3.2.2)

Proposition (3.1.2) tells us that E_{2}^{p,q} = Ext_{H(G^{0},I)}^{p}(∧^{q′}Std, ∧^{j}Std) and Corollary (2.3.3) then insures that dim_{\overline{𝔽}_{ℓ}}(E_{2}^{p,q}) ⩽ 1 for all p, q and

dim_{\overline{𝔽}_{ℓ}}(E_{2}^{p,q}) = 1 ⇔ (−j′ ⩽ p − q ⩽ j′ and − j ⩽ p + q − (d − 1) ⩽ j)
⇔ (p, q) lies in the rectangle (0, j′), (j′, 0), (d − 1, j), (j, d − 1).

Note that the rectangle above is contained in the square [0, d − 1] × [0, d − 1] and its faces have slopes ±1. Since this spectral sequence has finite support, it converges and we have

dim_{\overline{𝔽}_{ℓ}}(R_{π_{j}}^{k}) = ∑_{i=0}^{d−1} dim(E_{∞}^{k+i,i}) ⩽ ∑_{i=0}^{d−1} dim(E_{2}^{k+i,i}).

In particular, we see that R_{π_{j}}^{k} vanishes unless −j′ ⩽ k ⩽ j′, in which case we get

dim_{\overline{𝔽}_{ℓ}}(R_{π_{j}}^{k}) ⩽ #\{i ∈ \{0, ⋯, d − 1\}, −j ⩽ k + 2i − (d − 1) ⩽ j\}
= #{i ∈ {0, ⋯, d − 1}, −j + k′ ⩽ 2i ⩽ j − k′}
= #{even integers in the range [−j + k′, j + k′]}.

For the last equality, we use that [−j + k′, j + k′] ⊂ [0, 2d − 2], which is indeed equivalent to −j′ ⩽ k ⩽ j′. Now the last expression in the RHS above is j + 1 if −j + k′ = j′ − k is even, and is j otherwise.

We now look for lower bounds on dim_{\overline{𝔽}_{ℓ}}(R_{π_{j}}^{k}). We will use the fact that, by Proposition (2.2.8), we have

π_{j} ≃ v_{\{1,···,j′\}}(\overline{𝔽}_{ℓ}) ≃ v_{\{j+1,···,d−1\}}(\overline{𝔽}_{ℓ}).

Denote by ℤ̆_{ℓ} the Witt vectors of \overline{𝔽}_{ℓ}. We put

ω_{j}^{+} := v_{{1,···,j′}}(ℤ̆_{ℓ}) and ω_{j}^{−} := v_{{j+1,···,d−1}}(ℤ̆_{ℓ}).

As recalled in the proof of Proposition (2.2.8), these are liftings of π_{j} over ℤ̆_{ℓ}, i.e. admissible free ℤ̆_{ℓ}-representations of G such that

ω_{j}^{±} ⊗ℤ̆_{ℓ} \overline{𝔽}_{ℓ} ≃ π_{j}.

Therefore we have universal coefficients exact sequences

R_{ω_{j}^{±}}^{k} ⊗ℤ̆_{ℓ} \overline{𝔽}_{ℓ} ↪ R_{π_{j}}^{k} ↠ R_{ω_{j}^{±}}^{k+1}[ℓ]

for all k ∈ ℤ, and where the [ℓ] denotes ℓ-torsion (kernel of multiplication by ℓ). Since the R_{ω_{j}^{±}}^{k}

are finitely generated ℤ̆_{ℓ}-modules, we have equalities

dim_{\overline{𝔽}_{ℓ}}(R_{ω_{j}^{±}}^{k} ⊗ℤ̆_{ℓ} \overline{𝔽}_{ℓ}) = dim_{\overline{𝔽}_{ℓ}}(R_{ω_{j}^{±}}^{k}[ℓ]) + dimℚ̆_{ℓ}(R_{ω_{j}^{±}}^{k}[1/ℓ])

where ℚ̆_{ℓ} = ℤ̆_{ℓ}[1/ℓ] is the fraction field of ℤ̆_{ℓ}. Therefore we get

(3.2.3) dim_{\overline{𝔽}_{ℓ}}(R_{π_{j}}^{k}) = dim_{\overline{𝔽}_{ℓ}}(R_{ω_{j}^{±}}^{k+1}[ℓ]) + dim_{\overline{𝔽}_{ℓ}}(R_{ω_{j}^{±}}^{k}[ℓ]) + dimℚ̆_{ℓ}(R_{ω_{j}^{±}}^{k}[1/ℓ]).

Since ω_{j}^{+}[1/ℓ] = v_{{1,···,j′}}(ℚ̆_{ℓ}) and ω_{j}^{−}[1/ℓ] = v_{{j+1,···,n−1}}(ℚ̆_{ℓ}), we have already computed the last summand of the RHS in [3] (see more precisely the display below Lemma 4.4.1 in [3]). This gives :

(3.2.4)  dimℚ̆_{ℓ}(R_{ω_{j}^{±}}^{k}[1/ℓ]) = 1 if −j′ < k < j′ and j′ − k is even
(3.2.5)  dimℚ̆_{ℓ}(R_{ω_{j}^{+}}^{−j′}[1/ℓ]) = j + 1 and dimℚ̆_{ℓ}(R_{ω_{j}^{+}}^{j′}[1/ℓ]) = 1
(3.2.6)  dimℚ̆_{ℓ}(R_{ω_{j}^{−}}^{−j′}[1/ℓ]) = 1 and dimℚ̆_{ℓ}(R_{ω_{j}^{−}}^{j′}[1/ℓ]) = j + 1
(3.2.7)  dimℚ̆_{ℓ}(R_{ω_{j}^{±}}^{k}[1/ℓ]) = 0  in all other cases.

Case k = −j′. In this case, the equality dim_{\overline{𝔽}_{ℓ}}(R_{π_{j}}^{−j′}) = j + 1 follows from (3.2.3) applied to ω_{j}^{+} in degree −j′, (3.2.5), and our previously obtained upper bound.

Case k = j′. Similarly, the equality dim_{\overline{𝔽}_{ℓ}}(R_{π_{j}}^{j′}) = j + 1 follows from (3.2.3) applied to ω_{j}^{−} in degree j′, (3.2.6), and our previously obtained upper bound.

Case −j′ < k < j′. For k in this range, we are going to prove the following statement :

\{ \begin{matrix}If j′ − k is even, then dim_{\overline{𝔽}_{ℓ}}(R_{ω_{j}^{+}}^{k}[ℓ]) = j and dim_{\overline{𝔽}_{ℓ}}(R_{ω_{j}^{−}}^{k}[ℓ]) = 0 \\ If j′ − k is odd, then dim_{\overline{𝔽}_{ℓ}}(R_{ω_{j}^{+}}^{k}[ℓ]) = 0 and dim_{\overline{𝔽}_{ℓ}}(R_{ω_{j}^{−}}^{k}[ℓ]) = j \end{matrix} (3.2.4)

Because of (3.2.3) and (3.2.4), this implies our desired equalities

dim_{\overline{𝔽}_{ℓ}}(R_{π_{j}}^{k}) = \{ \begin{matrix}j + 1 & if & j′ − k is even. \\ j & if & j′ − k is odd, \end{matrix}

We will prove (3.2.4) by induction on k. The first case is k = −j′ + 1. Equation (3.2.3) applied to ω_{j}^{−} in degree −j′ reads

j + 1 = dim_{\overline{𝔽}_{ℓ}}(R_{ω_{j}^{−}}^{−j′+1}[ℓ]) + dim_{\overline{𝔽}_{ℓ}}(R_{ω_{j}^{−}}^{−j′}[ℓ]) + 1.

The same equation in degree −j′ − 1 tells us that R_{ω_{j}^{−}}^{−j′}[ℓ] = 0, whence the desired equality

dim_{\overline{𝔽}_{ℓ}}(R_{ω_{j}^{−}}^{−j′+1}[ℓ]) = j. On the other hand, (3.2.3) applied to ω_{j}^{+} in degree −j′ immediatly

implies that dim_{\overline{𝔽}_{ℓ}}(R_{ω_{j}^{+}}^{−j′+1}[ℓ]) = 0.

We now assume that (3.2.4) has been proved up to k − 1 and we want to prove it for k. We distinguish two cases.

Suppose first that j′−k is even. Then our induction hypothesis tells us that dim_{\overline{𝔽}_{ℓ}}(R_{ω_{j}^{−}}^{k−1}[ℓ]) =

j so that the upper bound already obtained and (3.2.3) for ω_{j}^{−} in degree k − 1 imply that dim_{\overline{𝔽}_{ℓ}}(R_{ω_{j}^{−}}^{k}[ℓ]) = 0 and also that dim_{\overline{𝔽}_{ℓ}}(R_{π_{j}}^{k−1}) = j. Then, (3.2.3) for ω_{j}^{+} in degree k −1 together with the vanishing of R_{ω_{j}^{+}}^{k−1}[ℓ] (induction hypothesis) and (3.2.7) tell us that dim_{\overline{𝔽}_{ℓ}}(R_{ω_{j}^{+}}^{k}[ℓ]) = j, as desired.

Next, suppose that j′ − k is odd, and apply (3.2.3) to ω_{j}^{+} in degree k − 1. By the induction hypothesis, the upper bound, and (3.2.4), we get that dim_{\overline{𝔽}_{ℓ}}(R_{ω_{j}^{+}}^{k}[ℓ]) = 0 and also that

dim_{\overline{𝔽}_{ℓ}}(R_{π_{j}}^{k−1}) = j + 1. Apply then (3.2.3) to ω_{j}^{−} in degree k − 1. Again the induction hypothesis and (3.2.4) tell us that dim_{\overline{𝔽}_{ℓ}}(R_{ω_{j}^{−}}^{k}[ℓ]) = j, as desired.

Corollary. (Of the proof)– The spectral sequence (3.2.2) degenerates at E_{2}.

In particular, Theorem (3.1.4) is now proved. We may use it to recast the foregoing result in the following way.

(3.2.5) Corollary.– Any splitting ⨁_{q=0}^{d−1} ∧^{q′}Std[−q] ⟶^{∼} C_{I} as in Theorem (3.1.4) induces a graded isomorphism

R_{π_{j}}^{∗} ⟶^{∼} ⨁_{{−j′⩽p−q⩽j′ \atop −j⩽p+q+1−d⩽j}} Ext_{H_{\overline{𝔽}_{ℓ}}(G^{0},I)}^{p} (∧^{q′}Std, ∧^{j}Std) [q − p].

Moreover, each term of the above sum has dimension 1.

3.3 The description of the pair (R_{π}^{∗}, L_{π}^{∗})

In this section we prove Theorem (1.2.1) and Corollary (1.2.1).

(3.3.1) Let us write i_{B} := Ind_{B}^{G} (\overline{𝔽}_{ℓ}) and consider the graded \overline{𝔽}_{ℓ}-vector space R_{i_{B}}^{∗}. By (3.1.1) we have

R_{i_{B}}^{∗} ≃ H^{∗} (R Hom_{H_{\overline{𝔽}_{ℓ}}(G^{0},I)}(RΓ_{c}(M_{LT,I}^{ca,(0)}, \overline{𝔽}_{ℓ}), (i_{B})^{I})) [1 − d].

On the other hand we have on the left H_{\overline{𝔽}_{ℓ}}(G^{0}, I)-module (i_{B})^{I} = \overline{𝔽}_{ℓ}[𝔖_{d}] a right module structure over \overline{𝔽}_{ℓ}[𝔖_{d}] which induce a left module structure on R_{i_{B}}^{∗}. Now let λ ∈ P(d) and denote by ε_{λ} the central idempotent corresponding to the simple module S_{λ}, as well as d_{λ} := dim_{\overline{𝔽}_{ℓ}} S_{λ}. We then may recover R_{π_{λ}}^{∗} by applying ε_{λ} :

(R_{π_{λ}}^{∗})^{⊕d_{λ}} = ε_{λ}R_{i_{B}}^{∗}.

As in Corollary (3.2.5), the splitting property of C_{I} shows that for k = 0, ⋯, 2d − 2 we have

R_{i_{B}}^{k+1−d} ≃ ⨁_{p−q=k+1−d} Ext_{H_{\overline{𝔽}_{ℓ}}(G^{0},I)}^{p} (∧^{q′}Std, \overline{𝔽}_{ℓ}[𝔖_{d}]) .

Inserting Proposition (2.3.2) we get

R_{i_{B}}^{k+1−d} ≃ ⨁_{p+q′=k} Hom_{\overline{𝔽}_{ℓ}[𝔖_{d}]} (∧^{q′}Std, ∧^{p}Std ⊗ \overline{𝔽}_{ℓ}[𝔖_{d}]) .

By Frobenius reciprocity and self-duality we finally get a 𝔖_{d}-equivariant isomorphism

R_{i_{B}}^{k+1−d} ≃ ⨁_{p+q′=k} ∧^{q′}Std ⊗ ∧^{p}Std

which shows that, as a graded vector space R_{i_{B}}^{∗}[d − 1] is 𝔖_{d}-equivariantly isomorphic to the cohomology H^{∗} of the abelian variety considered in Theorem (1.2.1). So we have obtained half of this theorem and we now have to study compatibility with Lefschetz operators.

(3.3.2) We refer to [6] for the definition of the Lefschetz operator

L : RΓ_{c}(M_{LT}^{ca}, \overline{𝔽}_{ℓ}) ⟶ RΓ_{c}(M_{LT}^{ca}, \overline{𝔽}_{ℓ})[2](1)

on the cohomology complex of the Lubin-Tate tower. By functoriality, the latter induces a graded equivariant map L_{π}^{∗} : R_{π}^{∗} ⟶ R_{π}^{∗}[2](1) for any smooth \overline{𝔽}_{ℓ}-representation of G. Because of our assumption that q ≡ 1[ℓ], we may and will forget all Tate twists in the sequel. We will also denote by

L_{I} ∈ Hom_{D^{b}(H_{\overline{𝔽}_{ℓ}}(G^{0},I))} (C_{I}, C_{I}[2])

the morphism induced by L on the complex at Iwahori level. When π is in the unipotent block and π^{I} = π^{I^{ℓ}}, the morphism L_{π}^{∗} is induced by L_{I} through the identification (3.1.1). We also denote by L^{(k)} := L[2k − 2] ∘ ⋯ ∘ L : RΓ_{c} ⟶ RΓ_{c}[2k] the k^{th} iterate of L and similarly for (L_{I})^{(k)} or (L_{π}^{∗})^{(k)}.

(3.3.3) Theorem.– Let π be any unipotent irreducible \overline{𝔽}_{ℓ}-representation of G. Then (L_{π}^{∗})^{(k)} induces an isomorphism R_{π}^{−k} ⟶^{∼} R_{π}^{k} for any k ⩾ 0.

Proof. We know from Theorem (3.1.4) that C_{I} is a split complex. Let us choose a splitting C_{I} ⟶^{∼} ⨁_{i=0}^{d−1} H^{i}(C_{I})[−i]. As in Corollary (3.2.5), this induces an isomorphism from R_{π}^{∗} to the graded space associated to the bi-graded space

(p, q) ↦ E_{π}^{p,q} := Ext_{H_{\overline{𝔽}_{ℓ}}(G^{0},I)}^{p}(Λ^{q′}, π^{I}).

Moreover, this also induces an isomorphism

Hom_{D^{b}(H_{\overline{𝔽}_{ℓ}}(G^{0},I))} (C_{I}, C_{I}[2]) ≃ ⨁_{i,j=0}^{d−1} Ext^{i−j+2}(H^{i}(C_{I}), H^{j}(C_{I}))

according to which we may decompose L_{I} = ∑_{i,j=0}^{d−1} L_{I}^{i,j}. By Proposition (3.1.2) we have Ext^{i−j+2}(H^{i}(C_{I}), H^{j}(C_{I})) ≃ Ext^{i−j+2}(Λ^{i′}, Λ^{j′}) and by Corollary (2.3.3) (see (2.3.4)), the latter has dimension 1 if |i − j| ⩽ i − j + 2 ⩽ min(i + j, i′ + j′) and vanishes otherwise. In particular, when it does not vanish, we have i − j + 2 ⩾ 1 with equality if and only if j = i + 1. Now, each L_{I}^{i,j} acts on the bi-graded space E_{π} by a map of degree (i − j + 2, i − j). Since (i − j + 2) + (i − j) ⩾ 0, it follows that L_{I} preserves the decreasing filtration on E_{π} defined by Fil_{r}E_{π} := ⨁_{p+q⩾r} E_{π}^{p,q}, hence that L_{π}^{∗} preserves the filtration induced on R_{π}^{∗}. In particular, we may check the expected property of (L_{π}^{∗})^{k} on the associated graded space. Concretely, this means that it suffices to prove that for for all 0 ⩽ p ⩽ q ⩽ d − 1, the map

E_{π}^{p,q} ↪ E_{π} ⟶^{(L_{I})^{(q−p)}} E_{π} ↠ E_{π}^{q,p}

is an isomorphism. This map is the composition

E^{p,q} ⟶^{L_{I}^{p,p+1}} E^{p+1,q−1} ⟶^{L_{I}^{p+1,p+2}} ⋯ ⟶^{L_{I}^{q−1,q}} E^{q,p}

and is given by cup-product :

(L_{I}^{p,p+1} ∪ ⋯ ∪ L_{I}^{q−1,q}) ∪ − : Ext^{p}(Λ^{q′}, π^{I}) ⟶ Ext^{q}(Λ^{p′}, π^{I}).

But Proposition (2.3.7) and the next Lemma imply that the latter map is an isomorphism.

Lemma.– The element L_{I}^{p,p+1} ∪ ⋯ ∪ L_{I}^{q−1,q} is non-zero in Ext^{q−p}(Λ^{p′}, Λ^{q′}).

Proof. Clearly, it suffices to prove this for p = 0 and q = d − 1. Let us consider the space Hom_{D^{b}(H_{\overline{𝔽}_{ℓ}}(G^{0},I))} (C_{I}, C_{I}[2d − 2]). It is isomorphic to ⨁_{i,j=0}^{d−1} Ext^{i−j+2d−2}(H^{i}(C_{I}), H^{j}(C_{I})) via our splitting of C_{I}. However, for i, j in {0, ⋯, d − 1} the space Ext^{i−j+2d−2}(Λ^{i′}, Λ^{j′}) vanishes unless i − j + 2d − 2 ⩽ d − 1, which happens only when i = 0 and j = d − 1. In other words, we have

Hom_{D^{b}(H_{\overline{𝔽}_{ℓ}}(G^{0},I))} (C_{I}, C_{I}[2d − 2]) ≃ Ext^{d−1}(H^{0}(C_{I}), H^{d−1}(C_{I})) = Ext^{d−1}(Λ^{d−1}, Λ^{0}).

Moreover, through this identification we have

(L_{I})^{(d−1)} = L_{I}^{0,1} ∪ L_{I}^{1,2} ∪ ⋯ ∪ L_{I}^{d−2,d−1}.

So we are left to show that (L_{I})^{(d−1)} is non-zero.

Now, in the case that π = π_{0} is the unit representation of G over \overline{𝔽}_{ℓ}, the very definition of L gives us the following description of the pair (R_{π_{0}}^{∗}, L_{π_{0}}^{∗}), see the first paragraph of the proof of Thm 4.2.2 in [5].

Fact.– We have an isomorphism R_{π_{0}}^{∗} ≃ H^{∗}(ℙ^{d−1,ca}, \overline{𝔽}_{ℓ})[d − 1] = ⨁_{i=0}^{d−1} \overline{𝔽}_{ℓ}[d − 1 − 2i] with L_{π_{0}}^{∗} corresponding to the Chern class of the tautological sheaf on ℙ^{d−1}.

In particular, (L_{π_{0}}^{∗})^{(d−1)} is non-zero, and hence so is (L_{I})^{(d−1)}.

(3.3.4) Theorem (3.3.3) tells us that (R_{i_{B}}^{∗}, L_{i_{B}}^{∗}) satisfies the “Hard Lefschetz theorem”. Therefore, by the discussion in (3.3.1) it is \overline{𝔽}_{ℓ}[𝔖_{d}]-equivariantly isomorphic to the pair (H^{∗}, L^{∗}) shifted by 1 − d of Theorem (1.2.1), which finishes the proof of the latter theorem.

(3.3.5) For π elliptic, we will now describe precisely the isomorphism class of the pair (R_{π}^{∗}, L_{π}^{∗}) in the category of finite dimensional graded vector spaces endowed with a degree 2 endomorphism. This category is abelian artinian and its indecomposable objects are isomorphic, up to schift, to some

P_{k} := (⨁_{i=0}^{k} \overline{𝔽}_{ℓ}[k − 2i], L_{k}) with L_{k} the unique map of degree 2 and rank k.

Note also that

P_{k} ≃ (H^{∗}(ℙ^{k}, \overline{𝔽}_{ℓ})[k], c(O_{ℙ^{k}(1)})) ,

the shifted cohomology of a projective space of dimension k with its tautological Lefschetz operator.

Corollary.– For j = 0, ⋯, d − 1, we have (R_{π_{j}}^{∗}, L_{π_{j}}^{∗}) ≃ (P_{j′})^{⊕j+1} ⊕ (P_{j′−1})^{⊕j}.

Proof. By Theorem (3.2.1), the graded pieces R_{π_{j}}^{−j′}, R_{π_{j}}^{−j′+2}, ⋯, R_{π_{j}}^{j′} all have dimension j + 1. By Theorem (3.3.3) we see that L_{π_{j}}^{∗} induces isomorphisms

R_{π_{j}}^{−j′} ⟶^{∼} R_{π_{j}}^{−j′+2} ⟶^{∼} ⋯ ⟶^{∼} R_{π_{j}}^{j′}

whence the summand (P_{j′})^{⊕j+1}. Similarly, L_{π_{j}}^{∗} induces isomorphisms

R_{π_{j}}^{−j′+1} ⟶^{∼} R_{π_{j}}^{−j′+3} ⟶^{∼} ⋯ ⟶^{∼} R_{π_{j}}^{j′−1}

between all these j-dimensional spaces, whence the summand (P_{j′−1})^{⊕j}.

3.4 Computation of (R_{π}^{red,∗}, L_{π}^{red}).

We now assume that the unipotent representation π is elliptic. So it is of the form π = π_{j} for some j ∈ {0, ⋯, d − 1}. The particular case j = 0 corresponds to the trivial representation π_{j} ≃ \overline{𝔽}_{ℓ}. In this case we find that R_{π_{0}}^{∗} = ⨁_{i=0}^{d−1} \overline{𝔽}_{ℓ}[−2i + d − 1] so that the total dimension of R_{π_{0}}^{∗} is d. In the other extreme case, when j = d − 1 so that π_{j} is the Steinberg representation, we get that R_{π_{d−1}}^{∗} is concentrated in degree 0 and has dimension d. These are however the only cases where the total dimension is d, in contrast with the other situations we have studied in previous papers (ℓ-adic, banal, and regular cases).

(3.4.1) In order to recover a d-dimensional vector space, we consider the following “reduced” version of R_{π}^{∗}. We put

E_{π^{I}}^{∗} := R^{∗}End_{H_{\overline{𝔽}_{ℓ}}(G^{0},I)}(π^{I}) = ⨁_{k⩾0} Ext_{H_{\overline{𝔽}_{ℓ}}(G^{0},I)}^{k}(π^{I}, π^{I}),

the self-extension algebra of the H_{\overline{𝔽}_{ℓ}}(G^{0}, I)-module π^{I}. This is a positively graded algebra and we denote by E_{π^{I}}^{+} := ⨁_{k>0} Ext_{H_{\overline{𝔽}_{ℓ}}(G^{0},I)}^{k}(π^{I}, π^{I}) its augmentation ideal. Via the isomorphism

(3.1.1), the graded \overline{𝔽}_{ℓ}-vector space R_{π}^{∗} carries a graded right-module structure over E_{π^{I}}^{∗}. We then put

R_{π}^{red,∗} := R_{π}^{∗}/R_{π}^{∗}E_{π^{I}}^{+}.

For π elliptic, we will show that this graded vector space has total dimension d, as desired.

(3.4.2) Theorem.– Choose a splitting ⨁_{q=0}^{d−1} ∧^{q′}Std[−q] ⟶^{∼} C_{I} as in Theorem (3.1.4). Then, through the isomorphism of Corollary (3.2.5), we have

R_{π_{j}}^{∗}E_{π_{j}^{I}}^{+} ⟶^{∼} ⨁_{{−j′<p−q⩽j′ \atop −j<p+q+1−d⩽j}} Ext_{H_{\overline{𝔽}_{ℓ}}(G^{0},I)}^{p} (∧^{q′}Std, ∧^{j}Std) [q − p].

Therefore we also get an isomorphism

R_{π_{j}}^{red} ≃ ⨁_{q=0}^{j′} Ext_{H_{\overline{𝔽}_{ℓ}}(G^{0},I)}^{j′−q} (Λ^{q′}, Λ^{j}) [−j′ + 2q] ⊕ ⨁_{q=j′+1}^{d−1} Ext_{H_{\overline{𝔽}_{ℓ}}(G^{0},I)}^{q−j′} (Λ^{q′}, Λ^{j}) [j′].

In particular R_{π_{j}}^{red} has total dimension d and its graded dimension is given by

dim_{\overline{𝔽}_{ℓ}} R_{π_{j}}^{red,k} = \{ \begin{matrix}j + 1 & if k = −j′ \\ 1 & if k = −j′ + 2i with 0 < i ⩽ j′ \\ 0 & else. \end{matrix}

Proof. Let D be the square defined by inequalities −j′ ⩽ p−q ⩽ j′ and −j ⩽ p+q−d+1 ⩽ j and let D^{+} be the complement in D of the left-side edges defined by p−q = −j′ and p+q = −j+d−1. Corollary (3.2.5) expresses R_{π_{j}}^{∗} as the graded vector space associated to the double graded vector space (p, q) ↦ E^{pq} := Ext^{p}(Λ^{q′}, Λ^{j}), whose support is D. For l > 0, an element of E_{π^{I}}^{l} = Ext^{l}(Λ^{j}, Λ^{j}) acts through a bigraded map of degree (l, 0). Therefore, R_{π_{j}}^{∗}E_{π_{j}^{I}}^{+} is the graded space associated to a bigraded subspace of E^{∗•} whose support is contained in D^{+}. In other words, the isomorphism of Corollary (3.2.5) takes R_{π_{j}}E_{π_{j}^{I}}^{+} in the RHS of our statement.

To get the other inclusion, we need to understand the action of a generator ξ_{j,j}^{1} of E_{π_{j}}^{1} = Ext^{1}(Λ^{j}, Λ^{j}), which is given by the cup-product

E^{p,q} = Ext_{H(G^{0},I)}^{p}(Λ^{q′}, Λ^{j}) ⟶^{− ∪ ξ_{j,j}^{1}} E^{p+1,q} = Ext_{H(G^{0},I)}^{p+1}(Λ^{q′}, Λ^{j}).

But Theorem (2.3.5) tells us that this is an isomorphism, whenever both sides are non-zero. It follows that E^{p+1,q} ⊂ R_{π_{j}}^{∗}E_{π_{j}^{I}}^{+}, and finally that the isomorphism of Corollary (3.2.5) induces the

claimed isomorphism. The claim on R_{π_{j}}^{red} is a direct a consequence of the latter.

By definition the graded map L_{π}^{∗} commutes with E_{π^{I}}^{+}, so L_{π}^{∗} induces a graded linear map L_{π}^{red} : R_{π}^{red} ⟶ R_{π}^{red}[2].

(3.4.3) Corollary.– For j = 0, ⋯, d − 1, we have (R_{π_{j}}^{red}, L_{π_{j}}^{red}) ≃ P_{j′} ⊕ (P_{0})^{⊕j}[j′]. In particular we have the equality

[R_{π_{j}}^{red}, L_{π_{j}}^{red}] = LJ(π_{j}) ⊗ (σ^{ss}(π_{j}), N_{π_{j}})

in the Grothendieck group of Weil-Deligne \overline{𝔽}_{ℓ}D^{×}-representations.

Proof. By Corollary (3.3.5) (and its proof) and the very definition of R_{π_{j}}^{red} as a quotient of R_{π_{j}}^{∗}, we see that L_{π_{j}}^{red} induces a surjective map R_{π_{j}}^{red,k} ↠ R_{π_{j}}^{red,k+2} for all k ⩾ −j′. So our first claim follows from the description of the graded dimension of R_{π_{j}}^{∗} in the above theorem. The second claim follows from the equalities

• LJ(π_{j}) = (−1)^{j}LJ(π_{0}) = (−1)^{j′}LJ(π_{d−1}) = (−1)^{j′}[1_{D^{×}}].

• σ^{ss}(π_{j}) = (1_{W_{K}})^{⊕d} and N_{π_{j}} has Jordan type λ = (d − j, 1^{(j)}).

The latter equalities are seen e.g. by reducing modulo ℓ the corresponding equalities for v_{\{1,···,j′\}}(\overline{ℚ}_{ℓ}), since we have π_{j} = v_{\{1,···,j′\}}(\overline{𝔽}_{ℓ}).

(3.4.4) Remark.– There are dual versions of the theorem and corollary. Instead of considering R_{π}^{red,∗}, one could consider

R_{π}^{cored,∗} := R_{π}^{∗}[E_{π^{I}}^{+}]

the graded sub-vector space of R_{π}^{∗} which is annihilated by the augmentation ideal E_{π^{I}}^{+}. Then a very similar argument to the proof of the above theorem shows that the isomorphism of Corollary (3.2.5) induces the following one.

R_{π_{j}}^{cored} ≃ ⨁_{q=0}^{j−1} Ext_{H_{\overline{𝔽}_{ℓ}}(G^{0},I)}^{j′+q} (Λ^{q′}, Λ^{j}) [j′] ⊕ ⨁_{q=j}^{d−1} Ext_{H_{\overline{𝔽}_{ℓ}}(G^{0},I)}^{j+q′} (Λ^{q′}, Λ^{j}) [j′ − 2q].

In particular R_{π_{j}}^{cored,∗} has total dimension d and graded dimension given by

dim_{\overline{𝔽}_{ℓ}} R_{π_{j}}^{red,k} = \{ \begin{matrix}j + 1 & if k = j′ \\ 1 & if k = j′ − 2i with 0 < i ⩽ j′ \\ 0 & else. \end{matrix}

Similar arguments as in last Corollary show that L_{π}^{∗} takes R_{π}^{cored} to itself, and that for j = 0, ⋯, d − 1, we have (R_{π_{j}}^{cored}, L_{π_{j}}^{cored}) ≃ P_{j′} ⊕ (P_{0})^{⊕j}[−j′], leading to the same conclusion that [R_{π_{j}}^{cored}, L_{π_{j}}^{cored}] = LJ(π_{j}) ⊗ (σ^{ss}(π_{j}), N_{π_{j}}).

(3.4.5) We now give the relation beween the algebra E_{π^{I}}^{∗} used here and that used in the statement of the main theorem in (1.2.2). The inflation functor from H_{\overline{𝔽}_{ℓ}}(G^{0}, I)-modules to H_{\overline{𝔽}_{ℓ}}(G^{0}, I^{ℓ})-modules along the map of Proposition (2.1.3) yields a morphism of graded algebras

E_{π^{I}}^{∗} ⟶ Ext_{\overline{𝔽}_{ℓ}G^{0}}^{∗}(π, π)

and the right action of E_{π^{I}}^{∗} on R_{π}^{∗} that one deduces from this morphism coincides with the right-action used above, which was obtained from the expression (3.1.1). We claim that the image of the above map is the algebra E_{π}^{∗} defined in (1.2.2). Since E_{π^{I}}^{∗} is generated in degree 1, we only need to see that the following sequence is exact

Ext_{H_{\overline{𝔽}_{ℓ}}(G^{0},I)}^{1}(π^{I}, π^{I}) ⟶ Ext_{\overline{𝔽}_{ℓ}G^{0}}^{1}(π, π) ⟶ Ext_{\overline{𝔽}_{ℓ}Ḡ}^{1}(π̄, π̄).

But through the equivalence of Proposition (2.1.2) this sequence reads

Ext_{H_{\overline{𝔽}_{ℓ}}(G^{0},I)}^{1}(π^{I}, π^{I}) ⟶ Ext_{H_{\overline{𝔽}_{ℓ}}(G^{0},I^{ℓ})}^{1}(π^{I}, π^{I}) ⟶ Ext_{H_{\overline{𝔽}_{ℓ}}(GL_{d}(O),I^{ℓ})}^{1}(π^{I}, π^{I}).

On the other hand, since ℓ > d, the RHS coincides with Ext_{\overline{𝔽}_{ℓ}[I^{ℓ}/I]}^{1}(π^{I}, π^{I}). Therefore the exactness is equivalent to the fact that any extension E of π^{I} by π^{I} as H(G^{0}, I^{ℓ})-modules is a H(G^{0}, I)-module if and only if it splits as an extension of I^{ℓ}/I-modules. But this is clear.

A Proof of Theorem (2.3.5) and Proposition (2.3.7)

In order to compute enough cup-products between the non-vanishing Ext spaces of Corollary (2.3.3), we need to exhibit explicit generators of some of these Ext spaces.

A.1 Construction of extensions

Recall that we have abbreviated Λ^{i} := ∧^{i}Std and Ext_{\overline{𝔽}_{ℓ}[(G^{0}∩T)/T^{0}]}^{k}(∧^{i}Std, ∧^{j}Std) by Ext^{k}(Λ^{i}, Λ^{j}).

(A.1.1) Further notation. The standard representation Λ^{1} = Std can be presented as the quotient of the permutation module \overline{𝔽}_{ℓ}^{d} by the line generated by (1, 1, ⋯, 1). Let e_{1}, ⋯, e_{d} be the image of the canonical basis of \overline{𝔽}_{ℓ}^{d} in Λ^{1}. This set of vectors enjoys the following properties:

• it is a generating set with “only” linear relation ∑_{r=1}^{d} e_{r} = 0,

• the action of 𝔖_{d} is given by σ(e_{r}) = e_{σ(r)}, where we see 𝔖_{d} has the permutation group of {1, ⋯, d}.

More generally, if I is a subset of {1, ⋯, d} of size |I| = i we put

e_{I} := e_{r_{1}} ∧ ⋯ ∧ e_{r_{i}} ∈ Λ^{i}

where I = {r_{1}, r_{2}, ⋯, r_{i}} and r_{1} < r_{2} < ⋯ < r_{i}. For a collection I_{1}, ⋯, I_{n} of subsets of {1, ⋯, d} we define ε(I_{1}, ⋯, I_{n}) by the following equality :

e_{I_{1}} ∧ ⋯ ∧ e_{I_{n}} = ε(I_{1}, ⋯, I_{n})e_{I_{1}∪···∪I_{n}}.

Thus ε(I_{1}, ⋯, I_{n}) is 0 unless all I_{t}’s are pairwise disjoint, in which case it is a sign. Now, if we fix i, the set of vectors {e_{I}, I ⊂ {1, ⋯, d}, |I| = i} enjoys the following properties :

• it is a generating set whose space of linear relations is generated by the following ones : ∑_{r=1}^{d} ε(J, {r})e_{J∪{r}} = 0 for each J ⊂ {1, ⋯, d} of size |J| = i − 1. In particular, the subset of all e_{I}’s for I contained in {1, ⋯, d − 1} is a basis of Λ^{i}.

• the action of 𝔖_{d} is given by σ(e_{I}) = sgn(σ_{|I})e_{σ(I)}, where σ_{|I} is considered as a permutation of {1, ⋯, i} via the orderings on I and σ(I) inheritated from that of {1, ⋯, d}.

When i = i_{1} + ⋯ + i_{n} is a composition series of i, we have a canonical map (exterior product) Λ^{i_{1}} ⊗ ⋯ ⊗ Λ^{i_{n}} ⟶^{prod} Λ^{i}. We will later need the quasi-section Λ^{i} ⟶^{can} Λ^{i_{1}} ⊗ ⋯ ⊗ Λ^{i_{n}} of this map defined by

can(v_{1} ∧ ⋯ ∧ v_{i}) = ∑_{τ∈[𝔖_{i}/𝔖_{i_{1}}×···×𝔖_{in}]} sgn(τ)(v_{τ(1)} ∧ ⋯ ∧ v_{τ(i_{1})}) ⊗ ⋯ ⊗ (v_{τ(i−i_{n}+1)} ∧ ⋯ ∧ v_{τ(i)})

where the index set stands for permutations τ that are increasing on each [∑_{s<t} i_{s} +1, ∑_{s⩽t} i_{s}]. Note that prod ∘ can is the multiplication by \frac{i!}{i_{1}!···i_{n}!}, which is non-zero in \overline{𝔽}_{ℓ}, due to our assumption that ℓ > d. Note also that can is 𝔖_{d}-equivariant and the evaluation on e_{I} is given by

can(e_{I}) = ∑_{I_{1}⊔···⊔I_{n}=I,|I_{t}|=i_{t}} ε(I_{1}, ⋯, I_{n})e_{I_{1}} ⊗ ⋯ ⊗ e_{I_{n}}. (A.1.2)

(A.1.3) Definition.– For d − 1 ⩾ i ⩾ j ⩾ 0, we denote by ξ_{i,j}^{i−j} the generator of Ext^{i−j}(Λ^{i}, Λ^{j}) provided by the non-zero 𝔖_{d}-equivariant map Λ^{i} ⟶^{can} Λ^{j} ⊗ Λ^{i−j} via the isomorphism of Proposition (2.3.2) i).

(A.1.4) Proof of Proposition (2.3.7). Since the regular representation \overline{𝔽}_{ℓ}[𝔖_{d}] is semisimple, it is sufficient to prove that for all i ⩽ j the cup-product

(A.1.5) ξ_{j,i}^{j−i} ∪ − : Ext^{j′}(Λ^{i}, \overline{𝔽}_{ℓ}[𝔖_{d}]) ⟶ Ext^{i′}(Λ^{j}, \overline{𝔽}_{ℓ}[𝔖_{d}])

is an isomorphism. Note that, as a \overline{𝔽}_{ℓ}[(N ∩ G^{0})/T^{0}]-module, we have \overline{𝔽}_{ℓ}[𝔖_{d}] = ind_{T ∩G^{0}}^{N∩G^{0}}(\overline{𝔽}_{ℓ}), so Frobenius reciprocity tells us that

Ext_{\overline{𝔽}_{ℓ}[(N∩G^{0})/T^{0}]}^{j′}(Λ^{i}, \overline{𝔽}_{ℓ}[𝔖_{d}]) ≃ Ext_{\overline{𝔽}_{ℓ}[(T ∩G^{0})/T^{0}]}^{j′}(Λ^{i}, \overline{𝔽}_{ℓ}) ≃ Hom_{\overline{𝔽}_{ℓ}}(Λ^{i}, Λ^{j′}).

Now, by the formula for cup-products in (2.3.2) ii) and the expression of ξ_{j,i}^{j−i} in Definition (A.1.3), we see that the map (A.1.5) is isomorphic to the map

(A.1.6)  Hom_{\overline{𝔽}_{ℓ}}(Λ^{i}, Λ^{j′}) → Hom_{\overline{𝔽}_{ℓ}}(Λ^{j}, Λ^{i′}) φ ↦ ψ : (Λ^{j} ⟶^{can} Λ^{i} ⊗ Λ^{j−i} ⟶^{φ ⊗ Id} Λ^{j′} ⊗ Λ^{i′−j′} ⟶^{∧} Λ^{i′}) .

That the latter map is an isomorphism is an avatar of the Lefschetz decomposition of the exterior algebra of an hermitian space (see for example [11, Ch. 0.7]). Since our field of coefficients \overline{𝔽}_{ℓ} has positive characteristic we briefly review the argument in order to insure that it is still valid in our context.

Let us consider the graded space H^{∗} := Hom_{\overline{𝔽}_{ℓ}}(Λ^{∗}, Λ^{∗}) = Λ̌^{∗} ⊗ Λ^{∗} where Λ̌ is the \overline{𝔽}_{ℓ}-linear dual space of Λ. We endow it with an operator of degree 2 which on the (p, q) part is given by

L : Hom_{\overline{𝔽}_{ℓ}}(Λ^{p}, Λ^{q}) → Hom_{\overline{𝔽}_{ℓ}}(Λ^{p+1}, Λ^{q+1}) φ ↦ Lφ : (Λ^{p+1} ⟶^{can} Λ^{p} ⊗ Λ ⟶^{φ ⊗ Id} Λ^{q} ⊗ Λ ⟶^{∧} Λ^{q+1}) .

and an operator of degree −2 which on the (p, q)-part is given by

tL : Hom_{\overline{𝔽}_{ℓ}}(Λ^{p}, Λ^{q}) → Hom_{\overline{𝔽}_{ℓ}}(Λ^{p−1}, Λ^{q−1}) φ ↦ ^{t}Lφ : (Λ^{p−1} ⟶^{∧} Λ^{p} ⊗ Λ̌ ⟶^{φ ⊗ Id} Λ^{q} ⊗ Λ̌ ⟶^{can} Λ^{q−1}) .

Here the last map is the composition of can with the evaluation map Λ ⊗ Λ̌ ⟶ \overline{𝔽}_{ℓ}. Explicitly, denoting by e_{1}^{∗}, ⋯, e_{d−1}^{∗} the dual basis of e_{1}, ⋯, e_{d−1}, we have Lφ = φ ∧ (∑_{i} e_{i}^{∗} ⊗ e_{i}), or even more explicitly we have for I, J ⊂ {1, ⋯, d − 1} with |I| = p and |J| = q, and denoting by I^{c} the complementary subset,

L(e_{I}^{∗} ⊗ e_{J}) = ∑_{k∈I^{c}∩J^{c}} ε(I, k)ε(J, k)e_{I∪{k}}^{∗} ⊗ e_{J∪{k}}

while

tL(e_{I}^{∗} ⊗ e_{J}) = ∑_{k∈I∩J} ε(I \ {k}, k)ε(J \ {k}, k)e_{I\{k}}^{∗} ⊗ e_{J\{k}}

Then a simple computation shows that the commutator [L,^{t}L] acts on Hom_{\overline{𝔽}_{ℓ}}(Λ^{p}, Λ^{q}) by multiplication by p + q − d + 1, hence it acts on H^{k} by multiplication by k − d + 1. It follows that the triple (L, [L,^{t}L],^{t}L) is a 𝔰𝔩_{2}-triple, i.e. is the image of the canonical basis (E_{12}, E_{11} − E_{22}, E_{21}) of 𝔰𝔩_{2}(\overline{𝔽}_{ℓ}) by a unique structure of 𝔰𝔩_{2}(\overline{𝔽}_{ℓ})-module on H^{∗}. Moreover, the weights are 1 − d, 2 − d, ⋯, d − 1 and the eigenspace of weight 1 − d + k is H^{k}. Now, we use our assumption that ℓ > d. This means that the simple 𝔰𝔩_{2}(\overline{𝔽}_{ℓ})-modules with weights in the above range are constructed in the same way as for 𝔰𝔩_{2}(ℂ). In particular, for each λ ∈ {0, ⋯, d − 1} there is a unique simple module (V_{λ}, r_{λ}) of highest weight λ, its weights are −λ, λ + 2, ⋯, λ and r_{λ}(E_{12}) induces an isomorphism between eigenspaces V_{λ}^{k} ⟶^{∼} V_{λ}^{k+2} for −λ ⩽ k < λ. Now, taking a filtration of H^{∗} with simple subquotients, we see that L^{k} induces an isomorphism H^{d−1−k} ⟶^{∼} H^{d−1+k} for all k = 0, ⋯, d − 1, and consequently an isomorphism

L^{k} : Hom_{\overline{𝔽}_{ℓ}}(Λ^{p}, Λ^{q}) ⟶^{∼} Hom_{\overline{𝔽}_{ℓ}}(Λ^{p+k}, Λ^{q+k})

whenever p + q = d − 1 − k. But we have L^{k}φ = φ ∧ (k! ∑_{|K|=k} e_{K}^{∗} ⊗ e_{K}), so that L^{k} is also given by

L^{k}φ = k!(Λ^{p+k} ⟶^{can} Λ^{p} ⊗ Λ^{k} ⟶^{φ ⊗ Id} Λ^{q} ⊗ Λ^{k} ⟶^{∧} Λ^{q+k}).

Now, taking p = i, k = j − i = i′ − j′ and q = j′ we get that (A.1.6) is an isomorphism, as desired. □

(A.1.7) Lemma.– For i = 1, ⋯, d − 1, we define inductively an element F_{i} ∈ Λ^{i} ⊗ Λ^{i} by setting

F_{1} := \frac{1}{d} ∑_{r=1}^{d} e_{r} ⊗ e_{r}, and F_{i} = \frac{1}{d} ∑_{r=1}^{d} e_{r} ∧ F_{i−1} ∧ e_{r}, for i > 1. (A.1.8)

Then, F_{i} is a generator of (Λ^{i} ⊗ Λ^{i})^{𝔖_{d}} and we have the formula

F_{i} = (−1)^{i(i−1)/2} \frac{i!}{d^{i}} ∑_{I⊂\{1,···,d\},|I|=i} e_{I} ⊗ e_{I}. (A.1.9)

Proof. Formula A.1.9 is easily checked by induction, using the fact that

e_{r} ∧ e_{I} ⊗ e_{I} ∧ e_{r} = ε({r}, I)ε(I, {r})e_{I∪{r}} ⊗ e_{I∪{r}}

is 0 unless r ∈ {1 ⋯ d}\I, in which case we have ε({r}, I)ε(I, {r}) = (−1)^{|I|}. Now, F_{i} is clearly 𝔖_{d}-invariant, so it only remains to check it is non-zero. Consider the element

E_{i} := e_{\{i+1,···,d−1\}} ∧ F_{i} ∧ e_{\{i+2,···,d\}} ∈ Λ^{d−1} ⊗ Λ^{d−1} ≃ \overline{𝔽}_{ℓ}.

In our formula for F_{i}, the only subset I that will contribute to E_{i} is I = {1, ⋯, i}, so that we get

E_{i} = c_{i} · e_{{1,···,d}\{d−1}} ⊗ e_{{1,···,d}\{i+1}}

with c_{i} = ±i!/d^{i} non-zero. Hence E_{i} ≠ 0 and therefore F_{i} ≠ 0.

The element F_{i} defines a non-zero morphism ξ_{0,i}^{i} : Λ^{0} = \overline{𝔽}_{ℓ} ⟶ Λ^{i} ⊗ Λ^{i}, and therefore provides a generator ξ_{0,i}^{i} for Ext^{i}(Λ^{0}, Λ^{i}). More generally, for i ⩽ j, the map Λ^{i} ⟶ Λ^{j} ⊗ Λ^{j−i}, v ↦ v ∧ F_{j−i} is a 𝔖_{d}-equivariant map and the above proof shows that it takes non-zero value on v = e_{{d−i,···,d−1}} for example.

(A.1.10) Definition.– For 0 ⩽ i ⩽ j ⩽ d − 1, we denote by ξ_{i,j}^{j−i} the generator of Ext^{j−i}(Λ^{i}, Λ^{j}) provided by the non-zero 𝔖_{d}-equivariant map v ↦ v ∧ F_{j−i}, Λ^{i} ⟶^{can} Λ^{j} ⊗ Λ^{j−i} via the isomorphism of Proposition (2.3.2) i).

(A.1.11) Lemma.– For 0 < i < d − 1, the mapping e_{r} ↦ e_{r} ∧ F_{i−1} ∧ e_{r} − F_{i} extends uniquely to a 𝔖_{d}-equivariant \overline{𝔽}_{ℓ}-linear non-zero map

ξ_{1,i}^{i} : Λ^{1} ⟶ Λ^{i} ⊗ Λ^{i}.

Proof. Since the e_{r}’s generate Λ^{1}, the extension is unique. For its existence we need to check compatibility with the linear relation ∑_{r} e_{r} = 0. But this follows from the recursive definition A.1.8. Equivariance is clear, given the invariance of F_{i} and F_{i−1}. It remains to check nonvanishing. But we have

e_{{i+1,···,d−1}} ∧ ξ_{1,i}^{i}(e_{d}) ∧ e_{{i+2,···,d}} = e_{{i+1,···,d−1}} ∧ (e_{d} ∧ F_{i−1} ∧ e_{d} − F_{i}) ∧ e_{{i+2,···,d}}
= −e_{{i+1,···,d−1}} ∧ F_{i} ∧ e_{{i+2,···,d}}

which was shown to be non-zero in the previous proof.

The homomorphism ξ_{1,i}^{i} therefore provides a generator of Ext^{i}(Λ, Λ^{i}). We will now construct generators of some other non-zero Ext spaces.

(A.1.12) Explicit basis of Ext spaces. In this paragraph we fix integers i, j, k ∈ {0, ⋯, d − 1} and we assume that i ⩽ j and i + j ⩽ d − 1. Under these assumptions, Corollary (2.3.3) tells us that Ext^{k}(Λ^{i}, Λ^{j}) has dimension 1 exactly when j − i ⩽ k ⩽ j + i.

Definition.– In this context, if k is of the form k = j − i + 2l, we define

\begin{matrix}ξ_{i,j}^{k} : Λ^{i} ⟶^{can} Λ^{i−l} ⊗ Λ^{l} & → & Λ^{j} ⊗ Λ^{k} \\ v ⊗ w & ↦ & v ∧ F_{k−l} ∧ w \end{matrix} .

If k is of the form k = j − i + 2l + 1, we define

\begin{matrix}ξ_{i,j}^{k} : Λ^{i} ⟶^{can} Λ^{i−l−1} ⊗ Λ^{1} ⊗ Λ^{l} & → & Λ^{j} ⊗ Λ^{k} \\ v ⊗ u ⊗ w & ↦ & v ∧ ξ_{1,k−l}^{k−l}(u) ∧ w \end{matrix} .

By construction, the map ξ_{i,j}^{k} is 𝔖_{d}-equivariant, and we denote by the same symbol the corresponding element of Ext^{k}(Λ^{i}, Λ^{j}) via Proposition (2.3.2) i).

Remark.– This definition is consistent with Definitions (A.1.10) and (A.1.12). Moreover, by Proposition (2.3.2) ii), we see that by construction we have a factorization ξ_{i,j}^{k} = ξ_{i,i−l}^{l} ∪ξ_{i−l,j}^{k−l} with l as in the definition above.

The following proposition is the key to Theorem (2.3.5).

Proposition.– For all k such that j − i ⩽ k ⩽ j + i, the element ξ_{i,j}^{k} is a generator of Ext^{k}(Λ^{i}, Λ^{j}). Moreover, if k < j + i, we have

i) ξ_{i,i}^{1} ∪ ξ_{i,j}^{k} ∈ \overline{𝔽_{ℓ}}^{×} · ξ_{i,j}^{k+1},

ii) ξ_{i,j}^{k} ∪ ξ_{j,j}^{1} ∈ \overline{𝔽_{ℓ}}^{×} · ξ_{i,j}^{k+1}.

Proof. The proofs of i) and ii) are somewhat lengthy and complicated computations, and are postponed to section A.2 in order to lighten a bit this section.

Let us deduce that ξ_{i,j}^{k} is a generator of Ext^{k}(Λ^{i}, Λ^{j}). We know that this is equivalent to ξ_{i,j}^{k} being non zero. By either point i) or ii), it is enough to prove that ξ_{i,j}^{i+j} is non zero. By definition, the homomorphism ξ_{i,j}^{i+j} : Λ^{i} ⟶ Λ^{j} ⊗ Λ^{i+j} is given by v ↦ F_{j} ∧ v. But the proof of Lemma A.1.9 shows that e.g. F_{j} ∧ e_{{d−i+1,···,d}} is non-zero (note that j + 2 ⩽ d − i + 1).

(A.1.13) Proof of Theorem (2.3.5). The last proposition implies the claim in the case where i ⩽ j and i + j ⩽ d − 1. Using duality we get the case where i ⩾ j and i + j ⩽ d − 1. Here by duality we mean contragredient A ↦ A^{∗} = Hom_{\overline{𝔽}_{ℓ}}(A, \overline{𝔽}_{ℓ}) of \overline{𝔽}_{ℓ}-representations of 𝔖_{d}. Indeed, for such representations we have functorial isomorphisms Ext_{\overline{𝔽}_{ℓ}[(N∩G^{0})/T^{0}]}^{∗}(A, B) ⟶^{∼} Ext_{\overline{𝔽}_{ℓ}[(N∩G^{0})/T^{0}]}^{∗}(B^{∗}, A^{∗}) that are compatible with cup-products in the obvious sense, and on the other hand we have (Λ^{i})^{∗} ≃ Λ^{i} and (Λ^{j})^{∗} ≃ Λ^{j}. So the claim is now proved for i+j ⩽ d−1.

In order to get the case i + j ⩾ d − 1 we use the endo-equivalence of the category of \overline{𝔽}_{ℓ}-representations of 𝔖_{d} given by A ↦ A ⊗ Λ^{d−1}. Again we have functorial isomorphisms

Ext_{\overline{𝔽}_{ℓ}[(N∩G^{0})/T^{0}]}^{∗}(A, B) ⟶^{∼} Ext_{\overline{𝔽}_{ℓ}[(N∩G^{0})/T^{0}]}^{∗}(A ⊗ Λ^{d−1}, B ⊗ Λ^{d−1}) that are compatible with cup-

products. On the other hand we have Λ^{i} ⊗ Λ^{d−1} ≃ Λ^{d−1−i} and Λ^{j} ⊗ Λ^{d−1} ≃ Λ^{d−1−j} and (d − 1 − i) + (d − 1 − j) ⩽ d − 1 whenever i + j ⩾ d − 1.

A.2 Proof of Proposition (A.1.12)

Here we prove items i) and ii) of Proposition (A.1.12) by brute force. We will evaluate both sides of the claimed equalities on elements e_{I} for I ⊂ {1, ⋯, d} of size i. Inserting (A.1.2) in our definitions, we get

ξ_{i,i}^{1}(e_{I}) = ∑_{r∈I} ε(I \ {r}, {r}) e_{I\{r}} ∧ (e_{r} ⊗ e_{r} − F_{1})
= ∑_{r∈I} e_{I} ⊗ e_{r} − ∑_{r∈I} ε(I \ {r}, {r}) e_{I\{r}} ∧ F_{1}.

Similarly, when k = j − i + 2l we get

ξ_{i,j}^{k}(e_{I}) = ∑_{I=I_{i−l}⊔I_{l}} ε(I_{i−l}, I_{l}) e_{I_{i−l}} ∧ F_{k−l} ∧ e_{I_{l}}

and when k = j − i + 2l + 1 we get

ξ_{i,j}^{k}(e_{I}) = ∑_{I=I_{i−l−1}⊔{r}⊔I_{l}} ε(I_{i−l−1}, {r}, I_{l}) e_{I_{i−l−1}} ∧ (e_{r} ∧ F_{k−l−1} ∧ e_{r} − F_{k−l}) ∧ e_{I_{l}}.

(A.2.1) Proof of i) in the case k = j − i + 2l. We assume that k = j − i + 2l. From the two expressions above and from the dictionnary for cup-products of Proposition (2.3.2) ii), we get

(ξ_{i,i}^{1} ∪ ξ_{i,j}^{k})(e_{I}) = A − B

with

A = ∑_{r∈I} ξ_{i,j}^{k}(e_{I}) ∧ e_{r} = ∑_{r∈I} ∑_{I=I_{i−l}⊔I_{l}} ε(I_{i−l}, I_{l}) e_{I_{i−l}} ∧ F_{k−l} ∧ e_{I_{l}} ∧ e_{r}

and

B = \frac{1}{d} ∑_{r∈I} ε(I \ \{r\}, \{r\}) ∑_{s=1}^{d} ξ_{i,j}^{k}(e_{I\\{r\}} ∧ e_{s}) ∧ e_{s}.

Now in the expression A, each summand is 0 unless r ∈ I_{i−l}, in which case we have

ε(I_{i−l}, I_{l}) e_{I_{i−l}} ∧ F_{k−l} ∧ e_{I_{l}} ∧ e_{r}
= (−1)^{l}ε(I_{i−l}, I_{l})ε(I_{i−l} \ {r}, {r}) e_{I_{i−l}\{r}} ∧ (e_{r} ∧ F_{k−l} ∧ e_{r}) ∧ e_{I_{l}}
= (−1)^{l}ε(I_{i−l} \ {r}, {r}, I_{l}) e_{I_{i−l}\{r}} ∧ (e_{r} ∧ F_{k−l} ∧ e_{r}) ∧ e_{I_{l}},

so that eventually we may rewrite

A = (−1)^{l} ∑_{I=I_{i−l−1}⊔{r}⊔I_{l}} ε(I_{i−l−1}, {r}, I_{l}) e_{I_{i−l−1}} ∧ (e_{r} ∧ F_{k−l} ∧ e_{r}) ∧ e_{I_{l}}

and thus recognize one summand of the expression of ξ_{i,j}^{k+1}(e_{I}). Moreover, in expression B each summand is 0 unless s is outside I \ {r}, in which case, putting I^{−r+s} := I \ {r} ⊔ {s}, we may write

ξ_{i,j}^{k}(e_{I\{r}} ∧ e_{s}) ∧ e_{s} = ε(I \ {r}, {s})ξ_{i,j}^{k}(e_{I^{−r+s}}) ∧ e_{s}
= ε(I \ {r}, {s}) ∑_{I^{−r+s}=I_{i−l}^{−r+s}⊔I_{l}^{−r+s}} ε(I_{i−l}^{−r+s}, I_{l}^{−r+s}) e_{I_{i−l}^{−r+s}} ∧ F_{k−l} ∧ e_{I_{l}^{−r+s}} ∧ e_{s}.

Now in the last expression, each summand vanish unless s ∈ I_{i−l}^{−r+s} and, as above, we have

ε(I_{i−l}^{−r+s}, I_{l}^{−r+s}) e_{I_{i−l}^{−r+s}} ∧ F_{k−l} ∧ e_{I_{l}^{−r+s}} ∧ e_{s}
= (−1)^{l}ε(I_{i−l}^{−r+s} \ {s}, {s}, I_{l}^{−r+s}) e_{I_{i−l}^{−r+s}\{s}} ∧ (e_{s} ∧ F_{k−l} ∧ e_{s}) ∧ e_{I_{l}^{−r+s}}.

Hence we get

ξ_{i,j}^{k}(e_{I\{r}} ∧ e_{s}) ∧ e_{s}
= (−1)^{l}ε(I \ {r}, {s}) ∑_{I\{r}=I_{i−l−1}⊔I_{l}} ε(I_{i−l−1}, {s}, I_{l}) e_{I_{i−l−1}} ∧ (e_{s} ∧ F_{k−l} ∧ e_{s}) ∧ e_{I_{l}}.

Because of the equality

ε(I_{i−l−1}, {s}, I_{l}) = (−1)^{l}ε(I_{i−l−1}, I_{l}, {s}) = (−1)^{l}ε(I_{i−l−1}, I_{l})ε(I \ {r}, {s})),

we have

ε(I \ {r}, {s})ε(I_{i−l−1}, {s}, I_{l}) = ε(I \ {r}, {r})ε(I_{i−l−1}, {r}, I_{l}),

so that eventually

B = \frac{(−1)^{l}}{d} ∑_{s=1}^{d} ∑_{I=I_{i−l−1}⊔\{r\}⊔I_{l}} ε(I_{i−l−1}, \{r\}, I_{l}) e_{I_{i−l−1}} ∧ (e_{s} ∧ F_{k−l} ∧ e_{s}) ∧ e_{I_{l}}
= (−1)^{l} ∑_{I=I_{i−l−1}⊔{r}⊔I_{l}} ε(I_{i−l−1}, {r}, I_{l}) e_{I_{i−l−1}} ∧ F_{k−l+1} ∧ e_{I_{l}}

which gives the second summand in the expression of ξ_{i,j}^{k+1}(e_{I}). Namely we have shown that ξ_{i,i}^{1} ∪ ξ_{i,j}^{k} = (−1)^{l}ξ_{i,j}^{k+1} in this case.

(A.2.2) Proof of i) in the case k = j − i + 2l + 1. We now assume that k = j − i + 2l + 1. Again we can write

(ξ_{i,i}^{1} ∪ ξ_{i,j}^{k})(e_{I}) = A − B

with

A = ∑_{r∈I} ξ_{i,j}^{k}(e_{I}) ∧ e_{r}

and

B = \frac{1}{d} ∑_{r∈I} ε(I \ \{r\}, \{r\}) ∑_{s=1}^{d} ξ_{i,j}^{k}(e_{I\\{r\}} ∧ e_{s}) ∧ e_{s}.

Inserting the expression of ξ_{i,j}^{k} in A we may decompose A = A_{1} + A_{2} with

A_{1} = ∑_{r∈I} ∑_{I=I_{i−l−1}⊔{t}⊔I_{l}} ε(I_{i−l−1}, t, I_{l}) e_{I_{i−l−1}} ∧ e_{t} ∧ F_{k−l−1} ∧ e_{t} ∧ e_{I_{l}} ∧ e_{r}

and

A_{2} = − ∑_{r∈I} ∑_{I=I_{i−l−1}⊔{t}⊔I_{l}} ε(I_{i−l−1}, t, I_{l}) e_{I_{i−l−1}} ∧ F_{k−l} ∧ e_{I_{l}} ∧ e_{r}.

Each summand of A_{1} vanishes unless r ∈ I_{i−l−1}. This remark allows us to rewrite the sum in the following way :

A_{1} = ∑_{I=I_{i−l}⊔I_{l}} ε(I_{i−l}, I_{l}) ∑_{r,t∈I_{i−l}} e_{I_{i−l}} ∧ F_{k−l−1} ∧ e_{t} ∧ e_{I_{l}} ∧ e_{r} = 0.

Indeed, the sum vanishes because the summands are antisymmetric in (r, t). As for A_{2}, we split it further A_{2} = A_{21} + A_{22} according to r = t or r ≠ t. Namely we have

A_{21} = − ∑_{I=I_{i−l−1}⊔{t}⊔I_{l}} ε(I_{i−l−1}, t, I_{l}) e_{I_{i−l−1}} ∧ F_{k−l} ∧ e_{I_{l}} ∧ e_{t}.

and

A_{22} = − ∑_{I=I_{i−l−1}⊔{t}⊔I_{l}} ∑_{r≠t} ε(I_{i−l−1}, t, I_{l}) e_{I_{i−l−1}} ∧ F_{k−l} ∧ e_{I_{l}} ∧ e_{r}.

We are then pleased to see that

A_{21} = −(−1)^{l} ∑_{I=I_{i−l−1}⊔{t}⊔I_{l}} ε(I_{i−l−1}, I_{l} ∪ {t}) e_{I_{i−l−1}} ∧ F_{k−l} ∧ e_{I_{l}∪{t}}
= (−1)^{l+1}(l + 1) ∑_{I=I_{i−l−1}⊔I_{l+1}} ε(I_{i−l−1}, I_{l+1}) e_{I_{i−l−1}} ∧ F_{k−l} ∧ e_{I_{l+1}}
= (−1)^{l+1}(l + 1)ξ_{i,j}^{k+1}(e_{I}).

The term A_{22} will be cancelled by a term occurring in B, so we now turn to B, and begin to decompose it in B = B_{1} + B_{2} with

B_{1} = ∑_{r∈I} ε(I \ r, r) \frac{1}{d} ∑_{s=1}^{d} ε(I \ r, s)

∑_{I^{s}=I_{i−l−1}^{s}⊔{t}⊔I_{l}^{s}} ε(I_{i−l−1}^{s}, {t}, I_{l}^{s})e_{I_{i−l−1}^{s}} ∧ (e_{t} ∧ F_{k−l−1} ∧ e_{t}) ∧ e_{I_{l}^{s}} ∧ e_{s}

and

B_{2} = − ∑_{r∈I} ε(I \ r, r)\frac{1}{d} ∑_{s=1}^{d} ε(I \ r, s) ∑_{I^{s}=I_{i−l−1}^{s}⊔\{t\}⊔I_{l}^{s}} ε(I_{i−l−1}^{s}, \{t\}, I_{l}^{s})e_{I_{i−l−1}^{s}} ∧ F_{k−l} ∧ e_{I_{l}^{s}} ∧ e_{s}

where we have written I^{s} := I \ {r} ∪ {s}. We may split further B_{2} = B_{21} + B_{22} according to t = s or t ≠ s. Then we have

B_{21} = − ∑_{r∈I} ε(I \ r, r)\frac{1}{d} ∑_{s=1}^{d} ε(I \ r, s) ∑_{I\\{r\}=I_{i−l−1}⊔I_{l}} ε(I_{i−l−1}, \{s\}, I_{l})e_{I_{i−l−1}} ∧ F_{k−l} ∧ e_{I_{l}} ∧ e_{s}

= − ∑_{r∈I} ε(I \ r, r)\frac{1}{d} ∑_{s=1}^{d} ∑_{I\\{r\}=I_{i−l−1}⊔I_{l}}(−1)^{l}ε(I_{i−l−1}, I_{l})e_{I_{i−l−1}} ∧ F_{k−l} ∧ e_{I_{l}} ∧ e_{s}

= − ∑_{r∈I} ε(I \ r, r) ∑_{I\\{r\}=I_{i−l−1}⊔I_{l}}(−1)^{l}ε(I_{i−l−1}, I_{l})e_{I_{i−l−1}} ∧ F_{k−l} ∧ e_{I_{l}} ∧ (\frac{1}{d} ∑_{s=1}^{d} e_{s})

= 0 (since ∑_{s=1}^{d} e_{s} = 0).

As for B_{22}, we note that each summand vanishes unless s ∈ I_{i−l−1}^{s}. In this case, we may write e_{I_{i−l−1}^{s}} = ε(I_{i−l−2}, s)e_{I_{i−l−2}} ∧ e_{s} for I_{i−l−2} a subset of I \ {r, t}, and we note that I_{l}^{s} is also a subset of I \ {r, s}, so we simply denote it by I_{l}. Therefore we may rearrange the sum in the following way

B_{22} = − ∑_{r,t∈I,r≠t} ∑_{I\\{r,t\}=I_{i−l−2}⊔I_{l}} \frac{1}{d} ∑_{s=1}^{d} sign(r, s, t, I_{i−l−2}, I_{l})e_{I_{i−l−2}} ∧ e_{s} ∧ F_{k−l} ∧ e_{I_{l}} ∧ e_{s}

where

sign(r, s, t, I_{i−l−2}, I_{l}) = ε(I \ r, r)ε(I \ r, s)ε(I_{i−l−2} ∪ {s}, {t}, I_{l})ε(I_{i−l−2}, {s})
= ε(I \ r, r)ε(I \ r, s)ε(I_{i−l−2}, {s}, {t}, I_{l})
= (−1)^{l+1}ε(I \ r, r)ε(I \ r, s)ε(I_{i−l−2}, {t}, I_{l}, {s})
= (−1)^{l+1}ε(I \ r, r)ε(I_{i−l−2}, {t}, I_{l})
= (−1)^{l+1}ε(I_{i−l−2}, {t}, I_{l}, {r})
= ε(I_{i−l−2}, {r}, {t}, I_{l}).

But since ε(I_{i−l−2}, {r}, {t}, I_{l}) = −ε(I_{i−l−2}, {t}, {r}, I_{l}), we see that B_{22} = 0. It remains to deal with B_{1}. As in the case of B_{22}, we see that each summand vanishes unless s ∈ I_{i−l−1}^{s} so that we may rearrange the sum in the following way :

B_{1} = ∑_{r,t∈I,r≠t} ∑_{I\\{r,t\}=I_{i−l−2}⊔I_{l}} \frac{1}{d} ∑_{s=1}^{d} ε(I_{i−l−2}, \{r\}, \{t\}, I_{l})e_{I_{i−l−2}} ∧ e_{s} ∧ (e_{t} ∧ F_{k−l−1} ∧ e_{t}) ∧ e_{I_{l}} ∧ e_{s}
= (−1)^{l} ∑_{r,t∈I,r≠t} ∑_{I\{r,t}=I_{i−l−2}⊔I_{l}} ε(I_{i−l−2}, {r}, {t}, I_{l})e_{I_{i−l−2}} ∧ (e_{t} ∧ F_{k−l} ∧ e_{t}) ∧ e_{I_{l}}
= − ∑_{r,t∈I,r≠t} ∑_{I\{r,t}=I_{i−l−2}⊔I_{l}} ε(I_{i−l−2} ∪ {t}, {r}, I_{l})e_{I_{i−l−2}∪{t}} ∧ F_{k−l} ∧ e_{I_{l}} ∧ e_{t}
= − ∑_{I=I_{i−l−1}⊔{r}⊔I_{l}} ∑_{t≠r} ε(I_{i−l−1}, r, I_{l}) e_{I_{i−l−1}} ∧ F_{k−l} ∧ e_{I_{l}} ∧ e_{t}
= A_{22}.

Finally we have proved that in the case k = j − i + 2l + 1,

(ξ_{i,i}^{1} ∪ ξ_{i,j}^{k})(e_{I}) = A_{1} + A_{21} + A_{22} − B_{1} − B_{21} − B_{22} = A_{21} = (−1)^{l+1}(l + 1)ξ_{i,j}^{k+1}(e_{I}).

(A.2.3) Proof of ii). It is certainly possible to do a direct computation as above, but it seems easier to use the case i) and prove ii) by induction on k. Indeed, if we assume that ξ_{i,j}^{k−1} ∪ ξ_{j,j}^{1} ∈ \overline{𝔽}_{ℓ}^{×} · ξ_{i,j}^{k}, then we get thanks to i)

ξ_{i,j}^{k} ∪ ξ_{j,j}^{1} ∈ \overline{𝔽}_{ℓ}^{×} · ξ_{i,i}^{1} ∪ ξ_{i,j}^{k−1} ∪ ξ_{j,j}^{1} ∈ \overline{𝔽}_{ℓ}^{×} · ξ_{i,i}^{1} ∪ ξ_{i,j}^{k} ∈ \overline{𝔽}_{ℓ}^{×} · ξ_{i,j}^{k+1}.

Therefore all we have to do is initializing the induction by proving that ξ_{i,j}^{j−i} ∪ ξ_{j,j}^{1} ∈ \overline{𝔽}_{ℓ}^{×} · ξ_{i,j}^{j−i+1}. But since we already know that ξ_{j,j}^{1} ∪ ξ_{j,j}^{1} ∈ \overline{𝔽}_{ℓ}^{×}ξ_{j,j}^{2}, it will suffice to check that ξ_{i,j}^{j−i} ∪ ξ_{j,j}^{2} ∈ \overline{𝔽}_{ℓ}^{×} · ξ_{i,j}^{j−i+2} (this involves less computation). Again we evaluate both sides on elements e_{I}. Let us write

ξ_{i,j}^{j−i}(e_{I}) = e_{I} ∧ F_{j−i} = C_{j−i} ∑_{|K|=j−i} e_{I} ∧ e_{K} ⊗ e_{K}

with C_{j−i} = (−1)^{(j−i)(j−i−1)/2} \frac{(j−i)!}{d^{j−i}}. We then have

(ξ_{i,j}^{j−i} ∪ ξ_{j,j}^{2})(e_{I}) = C_{j−i} ∑_{|K|=j−i} ε(I, K)ξ_{j,j}^{2}(e_{I∪K}) ∧ e_{K}

Inserting the expression (valid for I and K disjoint)

ξ_{j,j}^{2}(e_{I∪K}) = ∑_{r∈I∪K} ε(I ∪ K \ r, r)e_{I∪K\r} ∧ F_{1} ∧ e_{r},

we get

(ξ_{i,j}^{j−i} ∪ ξ_{j,j}^{2})(e_{I}) = C_{j−i} ∑_{|K|=j−i} ∑_{r∈I∪K} ε(I, K)ε(I ∪ K \ r, r)e_{I∪K\r} ∧ F_{1} ∧ e_{r} ∧ e_{K}.

We see that those summands where r ∈ K vanish. Hence we may restrict the second sum to I.

(ξ_{i,j}^{j−i} ∪ ξ_{j,j}^{2})(e_{I}) = C_{j−i} ∑_{|K|=j−i} ∑_{r∈I} ε(I, K)ε(I \ r ∪ K, r)e_{I\r∪K} ∧ F_{1} ∧ e_{r} ∧ e_{K}
= C_{j−i} ∑_{|K|=j−i} ∑_{r∈I} μ(I, K, r)e_{I\r} ∧ e_{K} ∧ F_{1} ∧ e_{K} ∧ e_{r}

with

μ(I, K, r) = ε(I, K)ε(I \ r ∪ K, r)ε(I \ r, K)(−1)^{j−i}
= ε(I, K)ε(I \ r, K, r)(−1)^{j−i}
= ε(I, K)ε(I \ r, r, K)
= ε(I \ r, r)

Eventually we get

(ξ_{i,j}^{j−i} ∪ ξ_{j,j}^{2})(e_{I}) = C_{j−i} ∑_{|K|=j−i} ∑_{r∈I} ε(I \ r, r)e_{I\r} ∧ e_{K} ∧ F_{1} ∧ e_{K} ∧ e_{r}

= ∑_{r∈I} ε(I \ r, r)e_{I\r} ∧ (C_{j−i} ∑_{|K|=j−i} e_{K} ∧ F_{1} ∧ e_{K} ) ∧ e_{r}
= ∑_{r∈I} ε(I \ r, r)e_{I\r} ∧ F_{j−i+1} ∧ e_{r}
= ξ_{i,j}^{j−i+2}(e_{I})

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