Galois irreducibility implies cohomology freeness for KHT Shimura varieties

Given a KHT Shimura variety provided with an action of its unramified Hecke algebra $\mathbb T$, we proved in a previous work, see also the work of Caraiani-Scholze for other PEL Shimura varieties, that its localized cohomology groups at a generic maximal ideal $\mathfrak m$ of $\mathbb T$, appear to be free. In this work, we obtain the same result for $\mathfrak m$ such that its associated Galois $\overline{\mathbb F}_l$-representation $\overline{\rho_{\mathfrak m}}$ is irreducible, under the hypothesis that $[F(\exp(2i\pi/l):F]>d$ where $F$ is the reflex field, $d$ the dimension of the KHT Shimura variety and $l$ the residual characteristic.


Introduction
From Matsushima's formula and computations of (G, K ∞ )-cohomology, we know that tempered automorphic representations contributions in the cohomology of Shimura varieties with complex coefficients, is concentrated in middle degree.If one considers cohomology with coefficients in a very regular local system, then only tempered representations can contribute so that all of the cohomology is concentrated in middle degree.
For Z ℓ -coefficients and Shimura varieties of Kottwitz-Harris-Taylor types, we proved in [9], whatever the weight of the coefficients is, when the level is large enough at ℓ, there are always nontrivial torsion cohomology classes, so that the F ℓ -cohomology can not be concentrated in middle degree.Thus if one wants an F ℓ -analog of the previous Q ℓ -statement, one must cut off some part of the cohomology.
In [7] for KHT Shimura varieties, and more generally in [12] for any PEL proper Shimura variety, we obtain such a result under some genericity hypothesis which can be stated as follows.Let (Sh K ) K⊂G(A ∞ ) be a tower, indexed by open compact subgroups K of G(A ∞ ), of compact Shimura varieties of Kottwitz type associated to some similitude group G and of relative dimension d − 1 over its reflex field F = EF + where F + is totally real and E/Q is an imaginary quadratic extension.Let then m be a system of Hecke eigenvalues appearing in H n0 (Sh K × F F , F ℓ ).By the main result of [20], one can attach to such an m, a mod ℓ Galois representation From [12,Def. 19], we say that m is generic (resp.decomposed generic) at some split p in E, if for all places v of F dividing p, the set {λ 1 , . . ., λ d } of eigenvalues of ρ m (Frob v ) satisfies λ i /λ j ̸ ∈ {q ±1  v } for all i ̸ = j (resp.and are pairwise distinct), where q v is the cardinal of the residue field at v. Then under the hypothesis (1) that there exists such a p with m generic at p, the integer n 0 above is necessarily equal to the relative dimension of Sh K .In particular the H i (Sh K × F F , Z ℓ ) m are all torsion-free.
In this work we consider the particular case of Kottwitz-Harris-Taylor Shimura varieties Sh K of [16] associated to inner forms of GL d .Exploiting the fact, which is particular to these Shimura varieties, that the non supersingular Newton strata are geometrically induced, we are then able to prove the following result which happens to be useful at least for our approach of Ihara's lemma, cf.[10].
Theorem.-We suppose that [F (exp(2iπ/ℓ) : F ] > d.Let m be a system of Hecke eigenvalues such that ρ m is irreducible, then the localized cohomology groups of Sh K with coefficients in any Z ℓ -local system V ξ , are all free.

Remarks
-From [4, §4], we know that outside the middle degree the irreducible Galois subquotients of H i (Sh K × F F , Z ℓ ) m ⊗ Z ℓ Q ℓ are of dimension strictly less than d, so that over Q ℓ , the cohomology, localized at m, is concentrated in middle degree.The previous theorem then tells us it is the same for the F ℓ -cohomology.
-Note that Koshikawa, cf.[18], starting from [7] and using techniques from group theory, proved a similar result in low dimension. (1)In their new preprint, Caraiani and Scholze explained that, from an observation of Koshikawa, one can replace decomposed generic by simply generic, in their main statement.

J.É.P. -M., 2023, tome 10
Lemma.-There exist infinitely many places v of F such that q v , the order of the residue field of F at v, is of order strictly greater than d in (Z/ℓZ) × .
Proof.-The hypothesis [F (exp(2iπ/ℓ) : F ] > d is equivalent to the fact that the minimal polynomial of exp(2iπ/ℓ) over F is of degree δ > d.By Cebotarev's theorem, there exists then a set of places v of strictly positive density such that the minimal polynomial of exp(2iπ/ℓ) over the residue field at v is also of degree δ.Recall that the roots of this minimal polynomial are exp(2iπ/ℓ) q k v with k = 0, . . ., δ − 1 with exp(2iπ/ℓ) q δ v = 1 in the residue field.We then deduce that the order of q v modulo ℓ divides δ but, as exp(2iπ/ℓ) q k v ̸ = 1 in the residue field for 0 ⩽ k < δ, then q v is of order δ > d modulo ℓ. □ This property is used at three places in the proof.
-For a place v as above, there is no irreducible cuspidal representation π v of GL g (F v ) with g > 1 such that its reduction modulo ℓ has a supercuspidal support made of characters, cf. the remark after Notation 1.1.4.This simplification is completely harmless and if one wants to take care of these cuspidal representations, it suffices to use [11,Prop. 2.4.1].
-With this hypothesis we also note that the pro-order of GL d (O v ) is invertible modulo ℓ so that, concerning torsion cohomology classes, we can easily pass from infinite to maximal level at v, cf. for example Lemma 3.1.15.
-Finally in the last section, arguing by contradiction, we are able to construct a sequence of intervals {λ, λq v , . . ., λq r v } contained in the set of eigenvalues of ρ m (Frob v ) so that at the end we obtain the full set {λq n v ∈ F ℓ | n ∈ Z} which is of order the order of q v modulo ℓ, which is trivially absurd if this order is strictly greater than the dimension d of ρ m .
The proof takes place in four main steps.
(1) First, in Section 3.1, we analyze the torsion in the cohomology of Harris-Taylor perverse sheaves at some place v and with infinite level at v, and we focus on ℓ-torsion cohomology classes with maximal non degeneracy level, cf.Lemma 3.1.17.
(2) Recall, cf.[6], that one can define an exhaustive filtration of stratification Fill • (Ψ v ) of the nearby cycles perverse sheaf whose graded parts are perverse sheaves )), cf.Section 2.2 for more details about p and p+ perverse structures and bimorphisms.The main point is that the constructions of [6] is of geometric nature and so works whatever the coefficients are.
Then on can compute the cohomology of the Shimura variety through the spectral sequence associated to the filtration Fill • (Ψ v ), whose E 1 terms are given by the cohomology groups of the Harris-Taylor perverse Z ℓ -sheaves.The main point, cf.Lemma 3.2.5, is that the ℓ-torsion of the cohomology of the Shimura variety with infinite level at v does not have an irreducible subquotient whose cuspidal support is made of characters.
(3) We then consider levels which are of Iwahori type at v and infinite at a place w verifying the same hypothesis as v above.We can then resume the previous result for w, i.e., the cohomology of the Shimura variety with infinite level at w does not have an irreducible subquotient whose cuspidal support is made of characters.We then focus on the various lattices of where m ⊂ m can be seen as a near equivalence class Π m in the sense of of [22].The ones given by the Z ℓ -cohomology, are only slightly modified from the ones given by the cohomology of the Harris-Taylor perverse sheaves, in the sense that the ℓ-torsion of the cokernel measuring the difference between two such lattices, as a representation of GL d (F w ), does not have any irreducible generic subquotient with cuspidal support made of characters, cf.Proposition 3.3.17.Moreover if the torsion sub-module of were not trivial, we prove, using the geometrical induced structure of the Newton strata, that it would exist m such that the lattices mentioned above, are not isomorphic, cf.Proposition 3.3.18.
(4) Finally in Section 3.4, for any m ⊂ m, As ρ m is supposed to be irreducible, then this lattice is isomorphic to a tensor product of a stable lattice of (Π ∞ m ) K by a stable lattice of ρ m .Then the idea is to start from the filtration of the free quotient of given by the filtration of the nearby cycles perverse sheaf, so that, using repeatedly diagrams as (3.4.2), we arrive at Γ m .In the process we are able to construct an increasing sequence of intervals contained in the set of eigenvalues of ρ m (Frob v) so that at the end we obtain a full set {λq n v | n ∈ Z} which is of order the order of q v modulo ℓ which is, by hypothesis, strictly greater than the dimension of ρ m , which is absurd.
We refer the reader to the introduction of Section 3 for more details.
1. Reminder from [7] 1.1.Representations of GL d (K).-Let K/Q p be a finite extension with O K its ring of integers, ϖ an uniformizer, and κ its residue field with order q.We denote by The Zelevinsky line associated to π is by definition the set {π{n} | n ∈ Z}.
Notations 1.1.1.-For π 1 and π 2 representations of respectively GL n1 (K) and GL n2 (K), we will denote by , the normalized parabolic induced representation where for any sequence we write P r for the standard parabolic subgroup of GL d with Levi Recall that a representation ϱ of GL d (K) is called cuspidal (resp.supercuspidal) if it is not a subspace (resp.subquotient) of a proper parabolic induced representation.When the field of coefficients is of characteristic zero then these two notions coincides, but this is no more true for F ℓ .Definition 1.1.2(see [24, §9] and [4, §1.4]).-Let g be a divisor of d = sg and π an irreducible cuspidal Q ℓ -representation of GL g (K).The induced representation admits a unique irreducible quotient (resp.subspace) denoted St s (π) (resp.Speh s (π)); it is a generalized Steinberg (resp.Speh) representation.
The supercuspidal support of the reduction modulo ℓ of a cuspidal representation, is a segment associated to some irreducible F ℓ -supercuspidal representation ϱ of GL g−1(ϱ) (F v ) with g = g −1 (ϱ)t, where t is either equal to 1 or of the following shape t = m(ϱ)ℓ u with u ⩾ 0 and where m(ϱ) is defined as follows.
Remark.-When ϱ is the trivial representation then m(1 v ) is either the order of q modulo ℓ when it is > 1, otherwise m(1 v ) = ℓ.We say that such a π v is of ϱ-type u with u ⩾ −1.
Notation 1.1.5.-For ϱ an irreducible F ℓ -supercuspidal representation, we denote by Cusp ϱ (resp.Cusp ϱ (u) for some u ⩾ −1) the set of equivalence classes of irreducible Q ℓ -cuspidal representations whose reduction modulo ℓ has for supercuspidal support a segment associated to ϱ (resp. of ϱ-type u).

P. Boyer
We now recall the notion of level of non degeneracy from [1, §4].The mirabolic subgroup M d (K) of GL d (K) is the sub-group of matrices with last row (0, . . ., 0, 1): we denote by its unipotent radical.We fix a nontrivial character ψ of K and let θ be the character of For G = GL r (K) or M r (K), we denote by Alg(G) the abelian category of algebraic representations of G and, following [1], we introduce defined by We also introduce the normalized compact induced functor -Ψ − (resp.Φ + ) is left adjoint to Ψ + (resp.Φ − ) and the following adjunction maps are isomorphisms, with an exact sequence is called the k-th derivative of τ .If τ (k) ̸ = 0 and τ (m) = 0 for all m > k, then τ (k) is called the highest derivative of τ .
is called the level of non-degeneracy of π and denoted by λ(π).We can also iterate the construction so that at the end we obtain a partition λ(π) of d.Note moreover that the reduction modulo ℓ of any irreducible generic representation admits a unique generic irreducible subquotient.
Recall the well-known following lemma.-Let F = F + E be a CM field where E/Q is quadratic imaginary and F + /Q is totally real with a fixed real embedding τ : F + → R. For a place v of F , we will denote by Let B be a division algebra with center F , of dimension d 2 such that at every place x of F , either B x is split or a local division algebra and suppose B provided with an involution of second kind * such that * |F is the complex conjugation.For any β ∈ B * =−1 , denote by ♯ β the involution x → x ♯ β = βx * β −1 and let G/Q be the group of similitudes, denoted by G τ in [16], defined for every Q-algebra R by where, identifying places of F + over x with places of F over y, x = i z i in F + .
Convention. -For a place x = yy c of Q split in E and z a place of F over y, we shall make throughout the text the following abuse of notation by denoting G(F z ) in place of the factor (B op z ) × in the formula (1.2.1).
In [16], the authors justify the existence of some G like above such that moreover for the embedding τ and (0, d) for the others.
As in [16, bottom of p. 90], a compact open subgroup U of G(A ∞ ) is said to be small enough if there exists a place x such that the projection from U v to G(Q x ) does not contain any element of finite order except identity.
For each I ∈ I, we write Spl(I) for the subset of Spl of places which does not divide I.
In the sequel, v and w will denote places of F in Spl.For such a place v, the scheme Sh I,η has a projective model Sh I,v over Spec O v with special fiber Sh I,sv .For I going through I, the projective system (Sh I,v ) I∈I is naturally equipped with an action of G(A ∞ ) × Z such that any w v in the Weil group W v of F v acts by − deg(w v ) ∈ Z, where deg = val • Art −1 and Art −1 : W ab v ≃ F × v is the isomorphism of Artin sending the geometric Frobenius to a uniformizer.Notations 1.2.4.-For I ∈ I, the Newton stratification of the geometric special fiber Sh I,sv is denoted by where Sh =h I,sv := Sh ⩾h I,sv − Sh ⩾h+1 I,sv is an affine scheme, smooth of pure dimension d − h built up by the geometric points whose connected part of their Barsotti-Tate group is of rank h.For each 1 ⩽ h < d, write i h : Sh ⩾h I,sv −→ Sh ⩾1 I,sv , j ⩾h : Sh =h I,sv −→ Sh ⩾h I,sv , and j =h = i h • j ⩾h .Let σ 0 : E → Q ℓ be a fixed embedding and write Φ for the set of embeddings σ : F → Q ℓ whose restriction to E equals σ 0 .There exists then, cf.[16, p. 97], an explicit J.É.P. -M., 2023, tome 10 bijection between irreducible algebraic representations ξ of G over Q ℓ and (d + 1)uples a 0 , ( − → a σ ) σ∈Φ , where a 0 ∈ Z and for all σ ∈ Φ, we have ).We then denote by V ξ,Z ℓ the associated Z ℓ -local system on Sh I .Recall that an irreducible automorphic representation Π is said ξ-cohomological if there exists an integer i such that where U is a maximal open compact subgroup modulo the center of G(R).Let d i ξ (Π ∞ ) be the dimension of this cohomology group.

Cohomology of the Newton strata
, We moreover consider the action of , its unipotent radical acts trivially.
From [7,Prop. 3.6], for any irreducible tempered automorphic representation Π of G(A) and for every i ̸ = 0, the Π ∞,v -isotypic component of [H i (h, ξ)] and [H i !(h, ξ)] are zero.About the case i = 0, for Π an irreducible automorphic tempered ξ-cohomological representation, its local component at v is generic and so looks like [7,Prop. 3.9]).-With the previous notations, we order the representations π v,i such that the first r ones are unramified characters.Then the where J.É.P. -M., 2023, tome 10 Remark.-With the notations of [7], we only consider Harris-Taylor perverse sheaves associated to the trivial representation which then selects the unramified characters to describe its cohomology groups.
Note also that if [H 0 (h, ξ)] has nontrivial invariant vectors under some open compact subgroup Example.-For w ∈ Spl, we have where T w,i is the characteristic function of where the limit is taken over the ideals I which are maximal at each place outside S.
For an open compact subgroup I maximal at each place outside S, we will also denote by Let state some remarks about these Hecke algebras.
-The torsion cohomology classes give also sets of Satake parameters and we can ask if they correspond to maximal ideals of T S ξ which we choose to define through the Q ℓ -coefficients.In [7], we proved that the torsion cohomology classes of H i (Sh I,η , V ξ,Z ℓ ) with I maximal at each place outside S, can be lifted in characteristic zero.More precisely, for any place v ′ ∈ Spl not in S, there exists a maximal ideal m of (T ) m , such that the set of Satake parameters of our torsion cohomology class is associated to m, where J.É.P. -M., 2023, tome 10 -As explained in the introduction, we will consider maximal ideals m such that ρ m is irreducible, so that the Q ℓ -cohomology groups are all concentrated in middle degree, i.e., in degree 0 if we deal with perverse sheaves.
-With the notations of Section 2.2 about the Harris-Taylor local systems, in [4], we proved that, except for the GL d (F v )-action, the irreducible subquotients of the The minimal prime ideals of T S ξ are the prime ideals above the zero ideal of Z ℓ and are then in bijection with the prime ideals of T S ξ ⊗ Z ℓ Q ℓ .To such an ideal, which corresponds to giving a collection of Satake parameters, is then associated a unique near equivalence class in the sense of [22], denoted by Π m , which is the finite set of irreducible automorphic cohomological representations whose multi-set of Satake parameters at each place x ∈ Unr(I), is given by the multi-set S m (x) of roots of the Hecke polynomial = 0 .Thanks to [16] and [22], we denote by the Galois representation associated to any Π ∈ Π m .Recall that the reduction modulo ℓ of ρ m depends only of m, and was denoted above by ρ m .For every w ∈ Spl(I), we also denote by S m (w) the multi-set of Satake parameters modulo ℓ at w given as the multi-set of roots of

About the nearby cycles perverse sheaf
Our strategy to compute the cohomology of the KHT-Shimura variety Sh I,η with coefficients in V ξ,Z ℓ , is to realize it as the outcome of the nearby cycles spectral sequence at some place v ∈ Spl.
Note that the role of the local system V ξ,Z ℓ associated to ξ is completely harmless when dealing with sheaves: one just has to add a tensor product with it to all the statements without the index ξ.In the following we will sometimes not mention the index ξ in the statements to make formulas more readable.Of course, when looking at the cohomology groups, the role of V ξ,Z ℓ is crucial as it selects the automorphic representations which contribute to the cohomology.

2.1.
The case where the level at v is maximal.-By the smooth base change theorem, we have ) is zero for i < 0 and free for i = 0. Using this property and the following short exact sequence of free perverse sheaves where i h+1→h : Sh ⩾h+1 I,sv → Sh ⩾h I,sv , we then obtain for every i > 0 an exact sequence (2.1.1) , arguing by induction from h = d to h = 1, we prove that for a maximal ideal m of T S ξ such that S m (v) does not contain any subset of the form {α, q v α}, all the cohomology groups H i (Sh ⩾h I,sv , V ξ,Z ℓ ) m are free: note that in order to deal with i ⩾ 0, one has to use Grothendieck-Verdier duality.
Without this hypothesis, arguing similarly, we conclude that any torsion cohomology class comes from a non strict map In particular it lifts in characteristic zero to some free subquotient of Main assumption.-We argue by contradiction and we suppose there exists a finite level I maximal at v such that there exist nontrivial torsion cohomology classes in the m-localized cohomology of Sh I,η v with coefficients in V ξ,Z ℓ .We then fix such a finite level I. , V ξ,Z ℓ ) m,tor ̸ = (0).Then we have the following properties: Remark.-Note that any system of Hecke eigenvalues m of T S ξ inside the torsion of some H i (Sh I,η v , V ξ,Z ℓ ) lifts in characteristic zero, i.e., is associated to a minimal prime ideal m of T S∪{v} ξ : the level of m outside v can still be taken to be I v , and, at v, using J.É.P. -M., 2023, tome 10 2.2.Harris-Taylor perverse sheaves over Z ℓ .-Consider now the ideals and exchanged by the action of the pure Newton stratum defined as the image of Let then denote by m v the multiset of Hecke eigenvalues given by m but outside v and introduce for any representation Π h of GL h (F v ): , where g ∈ GL h (F v ) acts on Π h as well as on H i (Sh ⩾h Note moreover that the unipotent radical of P h,d (F v ) acts trivially on these cohomology groups.We then introduce their induced version More generally, with the notations of [3], replace now the trivial representation by an irreducible cuspidal representation π v of GL g (F v ) for some 1 ⩽ g ⩽ d.
Notations 2.2.2.-Let 1 ⩽ t ⩽ s := ⌊d/g⌋ and let Π t be any representation of GL tg (F v ).We then denote by the Z ℓ -Harris-Taylor local system on the Newton stratum Sh =tg I,sv,1tg , where We also introduce the induced version where the unipotent radical of P tg,d (F v ) acts trivially and the action of We also introduce and the perverse sheaf and their induced version, HT (π v , Π t ) Z ℓ and P (t, π v ) Z ℓ , where and L ∨ , the dual of L, is the local Langlands correspondence which sends geometric frobenii to uniformizers.Finally we will also use the index ξ in the notations, for example HT ξ (π v , Π t ) Z ℓ , when we twist the sheaf with V ξ,Z ℓ .
With the previous notations, from (1.1.6),we deduce the following equality in the Grothendieck group of Hecke-equivariant local systems We now focus on the perverse Harris-Taylor sheaves.Note first, cf.[17, (2.2-2.4)], that over Z ℓ , there are two notions of intermediate extension associated to the two classical t-structures p and p+: essentially they come from the choice about F ℓ as a sheaves over the point, represented by the complex Z ℓ ×ℓ −→ Z ℓ and where one decides to put the zero grading on the second factor, which corresponds to the p-structure, or on the first one which gives the p+-structure.So for every π v ∈ Cusp ϱ of GL g (F v ) and 1 ⩽ t ⩽ d/g, we can define: the symbol → → + meaning bimorphism, i.e., both a monomorphism and an epimorphism, so that the cokernel for the t-structure p (resp. the kernel for p+) has support in Sh ⩾tg+1 I,sv .When π v is a character, i.e., when g = 1, the associated bimorphisms are isomorphisms, as explained in the following lemma, but in general they are not.Lemma 2.2.5.-With the previous notations, we have an isomorphism J.É.P. -M., 2023, tome 10 Proof.-Recall that Sh ⩾h I,sv,1 h is smooth over Spec F p .Up to a modification of the action of the fundamental group through the character χ v , we have -One of the main results of [11, Prop.2.4.1] is the fact that the previous lemma holds for any π v ∈ Cusp ϱ (−1).As explained in the introduction, with the hypothesis on the order of q v modulo ℓ, which is supposed to be strictly greater than d, for ϱ = 1 v being the trivial character, we do not need to bother about the representations π v ∈ Cusp 1v (u) for u ⩾ 0, cf. the remark after Notation 1.1.4.

2.3.
Filtrations of the nearby cycles perverse sheaf.-Let us denote by (3)   Ψ the nearby cycles autodual free perverse sheaf on the geometric special fiber Sh I,sv of Sh I .We also set we prove the following splitting: (2.3.1) with the property that the irreducible subquotients of are exactly the perverse Harris-Taylor sheaves, of level I, associated to an irreducible cuspidal Q ℓ -representation of some GL g (F v ) such that the supercuspidal support of the reduction modulo ℓ of π v is a segment associated to the inertial class ϱ.This splitting relies on the various filtrations defined over Z ℓ , of Ψ v constructed by means of the Newton stratification, cf.[8].Using the adjunction morphism j =t !j =t, * → Id as in [6], we then define a filtration of Ψ ξ,ϱ where the symbol −|→ means a monomorphism such that the cokernel is torsionfree, which here means that Fil t !(Ψ ξ,ϱ ) is the saturated image of j =t !j =t, * Ψ ξ,ϱ → Ψ ξ,ϱ .We then denote by gr k !(Ψ ξ,ϱ ) the graded parts and we have a spectral sequence: Remark.-Over Q ℓ , in [6] we proved that where N is the monodromy operator at v.
We can refine this filtration to define a filtration Fil )) coincides with the filtration by iterated images of N , i.e., gr r (gr k !(Ψ ξ,πv )) = Im N r ∩ Ker N k , so that we recover the usual monodromy bifiltration of [3].
We then obtain, cf.[6], an exhaustive filtration of stratification Fill whose graded parts are free, isomorphic to some free perverse Harris-Taylor sheaves.
-In [11] we proved that, following the previous process, then all the previous graded parts of Ψ ϱ are isomorphic to p-intermediate extensions.In the following we will only consider the case where ϱ is the trivial character so that as the order of q v being supposed to be > d, then Cusp ϱ is made of characters in which case, cf.Lemma 2.2.5, the p and p+ intermediate extensions coincide.Note that in the following we will not use the results of [11].

Irreducibility implies freeness
Recall, cf. the main assumption in Section 2.1, that we argue by contradiction assuming there exist nontrivial cohomology classes in some of the The strategy is then to choose a place v ∈ Spl I such that the order of q v modulo ℓ is strictly greater than d and to allow ramification at v, either infinitely with I v (∞) as denoted in the next paragraph, or of Iwahori type.
In the arguments we need to consider another place w ̸ = v verifying the same hypothesis as v, i.e., w ∈ Spl I and such that the order of q w modulo ℓ is strictly greater than d.We will also allow to increase infinitely the level at w.As the order of both q v and q w is supposed to be strictly greater than d, then the functors of invariants by any open compact subgroups either at v or w, are exact.In particular as there exist nontrivial torsion classes in level I in some degree i, when I is maximal at v and w, there also exist nontrivial torsion classes in degree i, whatever the level J J.É.P. -M., 2023, tome 10 such that J v,w = I v,w is.In particular when the level at v is infinite, from the splitting there also exist nontrivial torsion classes in Remark.
-From now on, localization at m means that we prescribe the Satake parameters modulo ℓ as usual, but outside {v, w}.
Let now explain the main steps of the following sections.(a) Following the arguments of the previous section, we first analyze the torsion cohomology classes of Harris-Taylor perverse sheaves with infinite level at v, and we deduce, cf.Lemma 3.1.17,that, as F ℓ -representations of GL d (F v ), irreducible subquotients of the ℓ-torsion of their cohomology in infinite level at v, with highest non degeneracy level, appear in degrees 0, 1.
(b) In Section 3.2, considering always infinite level at v, we analyze the torsion cohomology classes of the graded parts gr t !(Ψ ϱ ) of the filtration of stratification Fil • !(Ψ ϱ ) and more specifically when ϱ = 1 v is the trivial character.We then deduce, cf.Lemma 3.2.5, that the ℓ-torsion of H i (Sh I v (∞),sv , V ξ,Z ℓ ) m does not have, as an F ℓ -representation of GL d (F v ), any irreducible generic subquotient whose supercuspidal support is made of characters.
(c) In Section 3.3, we obtain two fundamental results.
-First, cf.Lemma 3.3.6,under the hypothesis that there exist nontrivial torsion cohomology classes, we show that the graded pieces Γ k of the filtration given by the spectral sequence of vanishing cycles, of the free quotient of H 0 (Sh I v (∞),η v , V ξ,Z ℓ ) m are not always given by the lattice Γ 0 of given by the integral cohomology of P ξ (t, χ v ) Z ℓ .Roughly, there exist some k and a short exact sequence Γ 0 → Γ k → → T , where T is nontrivial and torsion.
-We then play with the action of GL d (F w ) by allowing infinite level at w.The main observation at the end of the section, cf.Proposition 3.3.13, is that, as an F ℓ -representation of GL d (F w ), all the irreducible subquotients of the ℓ-torsion of the cokernels T between two lattices in the previous point, up to multiplicities, are also subquotients of the ℓ-torsion of the cohomology of the Shimura variety.In particular, as v and w are playing symmetric roles, these subquotients are not generic, cf.Corollary 3.3.14.
(d) In Section 3.4, the last step is to prove, under the absurd hypothesis that there exist nontrivial torsion cohomology classes while ρ m being irreducible, that S m (v) contains a full Zelevinsky line modulo ℓ {λq n v ∈ F ℓ | n ∈ Z} which is of order the order of q v modulo ℓ.As this order is supposed to be strictly greater than d, this is a contradiction.For more insight on the strategy to prove this fact using the previous properties about lattices, we refer to the introduction of Section 3.4.

3.1.
Torsion classes for Harris-Taylor perverse sheaves.-We focus on the torsion in the cohomology groups of the Harris-Taylor perverse sheaves P ξ (χ v , t) Z ℓ when the level at v is infinite, and as explained above, cf. the main assumption of Section 2.1, especially when the reduction modulo ℓ of χ v is the trivial character.Notation 3.1.1.-We will denote by I v ∈ I a finite level outside v, and we also denote by m the maximal ideal of T S∪{v} ξ associated to m, i.e., we do not prescribe the Satake parameters modulo ℓ at v. Let us also set which can be viewed as a Z ℓ [GL d (F v )]-module.Then, morally, I v (∞) is a finite level outside v and infinite at v. Proposition 3.1.2(cf.[11], second global result of the introduction) We have the following resolution of -as this resolution is equivalent to the computation of the sheaves cohomology groups of p j =h !* HT (χ v , St h (χ v )) Z ℓ as explained for example in [11, Prop.B.1.5],then, over Q ℓ , the proposition follows from the main results of [3].
-Over Z ℓ , as every terms are free perverse sheaves, then all maps are necessary strict.
-This resolution, for a a general supercuspidal representation with supercuspidal reduction modulo ℓ, is one of the main result of [11, §2.3].However, in the case of a character χ v the arguments are much easier.
Consider the finite level I v (n) = I v I v,n , where are smooth, then, cf. the proof of Lemma 2.2.5, the constant sheaf, up to shift, is perverse and so equal to the intermediate extension of the constant sheaf, shifted by d − h, on Sh =h I v (n),sv,1 h .In particular, as a constant sheaf, its sheaf cohomology groups are well-known, so, over Sh ⩾h I v (n),sv,1 h and so for Sh ⩾h The stated resolution is then simply the induced version of the resolution of p j =h 1 h ,! * HT (χ v , St h (χ v )) 1,Z ℓ : recall that a direct sum of intermediate extensions is still an intermediate extension.

J.É.P. -M., 2023, tome 10
By the adjunction property, the map We then have Indeed one can compute p i h+1,! j =h !HT (χ v , Π h ) Z ℓ by means of the spectral sequence associated to the exhaustive filtration of stratification of graded parts, using Lemma 2.2.5 and [6], ).As remarked before, the sheaf cohomology groups of )) Z ℓ are torsion-free, so, by Grothendieck-Verdier duality, the same is true for The statement follows then from the fact that, over Q ℓ , the previous spectral sequence degenerates at E 1 .
Remark.-This property is also true when we replace the character χ v by any irreducible cuspidal representation π v , cf. [11].
Fact. -In particular, up to homothety, the map (3.1.6),and so those of (3.1.5),is unique.Finally, as the maps of (3.1.3)are strict, the given maps (3.1.4)are uniquely determined, that is, if we forget the infinitesimal parts these maps are independent of the chosen t in (3.1.3).
We now copy the arguments of Section 2.1.
Remark.-By duality, as p j =h !* = p+ j =h !* for Harris-Taylor local systems associated to a character, note that when i I (h, χ v ) is finite then i I v (h, χ v ) ⩽ 0. Notation 3.1.9.-Suppose there exists I ∈ I such that there exists h with 1 ⩽ h ⩽ d and with i I v (h, χ v ) finite, and denote by h 0 (I v , χ v ) the biggest such h.
Proof.-Note first that for every h such that h 0 (I v , χ v ) ⩽ h ⩽ s, the cohomology groups of j =h !HT ξ (χ v , Π h ) are torsion-free.The spectral sequence associated to the filtration (3.1.7),localized at m, is then concentrated in middle degree and is torsionfree.
Consider then the spectral sequence associated to the resolution (3.1.3):its E 1 terms are torsion-free and it degenerates at E 2 .As, by hypothesis, the abutment of this spectral sequence is free and is equal to only one E 2 terms, we deduce that all the maps m are strict.Then from the previous fact stressed after (3.1.6),this property remains true when we consider the associated spectral sequence for 1 ⩽ h ′ ⩽ h 0 (I v , χ v ).
Consider now h = h 0 (I v , χ v ), where we know the torsion to be nontrivial.From what was observed above we then deduce that the map (3.1.12) Finally for any 1 ⩽ h ⩽ h 0 (I v , χ v ), the map like (3.1.12)for h

cokernel has nontrivial torsion, which gives then a nontrivial torsion class in
-The integers h 0 (I v , χ v ) and i I v (h, χ v ) only depend on the reduction modulo ℓ of χ v .
Proof.-For P a torsion-free Z ℓ -perverse sheaf, recall the well-known short exact sequence J.É.P. -M., 2023, tome 10 We apply it to X = Sh I v ,sv and P = P ξ (h 0 (I v , χ v ), χ v ) Z ℓ .Recall that, thanks to our hypothesis on m, the Q ℓ -cohomology of P , localized at m, is concentrated in degree 0. The same is true for ) is a local system on the stratum Sh ⩾h0(I v ,χv) I v ,sv so that is simply the reduction modulo ℓ of this local system.From the definition of h 0 (I v , χ v ), the cohomology of P ξ (h 0 (I v , χ v ), χ v ) Z ℓ has torsion in degrees 0 and 1 so that its F ℓcohomology, localized at m, is concentrated in degrees −1, 0, 1: The same is then true for the F ℓ -cohomology, localized at m, of P ξ (, h 0 (I v , χ v ), χ ′ v ) Z ℓ so that its Z ℓ -cohomology, localized at m, must have torsion in degrees 0 and 1.
the result then follows from the previous lemma.□ From the main assumption of Section 2.1 and as explained in the introduction of this section, we focus on Ψ ξ,ϱ when ϱ = 1 v is the trivial character.We are then interested in the characters χ v congruent to the trivial character 1 v modulo ℓ.
Notation 3.1.14.-Following the notation of Proposition 2.1.4,we will denote h 0 (I v ) Proof.-Consider the previous map (3.1.12)by replacing h 0 (I v ) by h 0 (I).As by hypothesis the order of q v modulo ℓ is strictly greater than d, then the pro-order of the local component I v of I at v is invertible modulo ℓ, so that the functor of invariants under I v is exact.Note then that, as the I v -invariants of the map (3.1.12)when replacing h 0 (I v ) by h 0 (I), has a cokernel which is not free, then the cokernel of (3.1.12),for h 0 (I), is also not free.□ From the previous proof, we also deduce that all cohomology classes of any of the H i (Sh I v (∞),sv , P ξ (t, χ v ) Z ℓ ) m come from the non strictness of some map (3.1.12)with Π v := St t (χ v ).In the following we will focus on . More precisely we are interested in irreducible such subquotients which have maximal non-degeneracy level at v. Notation 3.1.16.-Fix such a non degeneracy level λ for GL d (F v ) in the sense of Notation 1.1.9,which is maximal for torsion classes in H 0 (Sh , for i ̸ = 0, 1, have a level of non degeneracy strictly less than λ. Proof.-It easily follows from the observation that the level of non degeneracy of the reduction modulo ℓ of Speh h (χ v ) ≃ χ v is strictly less than those of the reduction modulo ℓ of St h (χ v ) which, cf.[5], is irreducible as the order of q v modulo ℓ is strictly greater than d and so strictly greater than h.□ 3.2.Global torsion and genericity.-Recall that v ∈ Spl is such that the order of q v modulo ℓ is strictly greater than d.Let us denote by I v the component of I outside v.We then simply denote by Ψ v and Ψ v,ξ , the inductive system of perverse sheaves indexed by the finite level Remark.-In the following, we will be mainly concerned with the case where π v is a character χ v whose reduction modulo ℓ is the trivial character.We will then write the main statement in this case.

Recall the following resolution of
which is proved in [6] over Q ℓ .I claim it is also true over Z ℓ .Indeed, using Lemma 2.2.5, it is equivalent to the fact the sheaf cohomology of Fil 1 !,χv (Ψ ϱ ) is torsion-free, which follows then from [19], the comparison theorem of Faltings-Fargues cf.[15] and the main theorem of [14].
Remark.-In [11], we prove the same resolution for any irreducible cuspidal representation π v in place of χ v .

More generally for gr
Finally all the torsion cohomology classes of the H i (Sh I v (∞),sv , gr t !,χv (Ψ ϱ )) m come from the non strictness of the maps where We can then copy the proof of Lemma 3.1.10which gives us the following statement.
J.É.P. -M., 2023, tome 10 Lemma 3.2.4.-For every h such that 1 ⩽ h ⩽ h 0 (I v ), the number i I (h) = h − h 0 (I v ) of Notation 3.1.8,is also the lowest integer i such that the torsion of -For every i, the ℓ-torsion of ]-module, does not have an irreducible generic subquotient whose cuspidal support is made of characters.
Remark.-Note that when the order of q v modulo ℓ is strictly greater than d, then there is no difference between cuspidal or supercuspidal support made of characters.
Proof.-Recall first that, as by hypothesis ρ m is irreducible, the Q ℓ -version of the spectral sequence (2.3.2) degenerates at E 1 so that in particular all the torsion cohomology classes appear in the E 1 terms.As we are only interested in representations with cuspidal support made of characters, we only have to deal with the perverse sheaves P (t, χ v ) Z ℓ so that the result follows from the previous maps (3.2.3) and the fact that for any r > 0, the reduction modulo ℓ of LT χv (t − 1, r) does not admit any irreducible generic subquotient.□

Torsion and modified lattices.
-Recall that we argue by contradiction, assuming there exists a finite level I unramified at the place v, such that the torsion of some of the H i (Sh I,η v , V ξ,Z ℓ ) m is non zero.We then increase the level at v to infinity and define the index h 0 (I v ) which might be greater than the index h 0 (I) defined in level I, cf.Notation 3.1.14, of the first Harris-Taylor perverse sheaf associated to a character χ v congruent to 1 v modulo ℓ, with nontrivial torsion cohomology class.
We now come back to a finite level at v with two main objectives: first we want to keep the torsion in the cohomology of P ξ (χ v , h 0 ) Z ℓ and secondly we intend to simplify the spectral sequence of vanishing cycles.
Remark.-The main reason to go to infinite level at v is to be able to use the notion of level of non degeneracy.
To be able to deal with representations, we fix a place w ̸ = v with w ∈ Spl(I) and verifying the same hypothesis as v, i.e., q w modulo ℓ is of order strictly greater than d.Notation 3.3.1.-We then denote as before by I w (∞) when the level is infinite at w and I w,v (∞) when the level is infinite at v and w.We also denote h 0 for h 0 (I w ), the highest index when torsion appear in the cohomology of a Harris-Taylor perverse sheaf in infinite level at w and maximal level at v.
-Dealing with infinite level at w allows to talk about representations of GL d (F w ), while Iwahori type subgroup allows to simplify the spectral sequence.Indeed from Lemma 1.1.12,and using the definition of LT χv (t, s) through an induced representation, cf.Definition 1.1.2,we note that for h ⩾ h 0 +2, then LT χv (h, d−h−1) does not have any nontrivial vector invariant by Iw v (d − h 0 , 1, . . ., 1).Moreover for an irre- the examples following Lemma 1.1.12.Similar simplifications also appear in Lemma 3.3.9.
We then focus on the free quotient of H 0 (Sh ) m by means of the spectral sequence of vanishing cycles.From (2.3.1),we are then lead to study the cohomology of Ψ ξ,1v by means of its filtration of stratification and so we first focus on the cohomology of gr h0 !,χv (Ψ ξ,1v ) for a character χ v ∈ Cusp −1 (1 v ) which is by definition a quotient of gr h0 !(Ψ ξ,1v ).To do so, consider first the filtration constructed in [6]: )), with successive free graded parts gr i (gr h0 !,χv (Ψ ξ,1v )) which, for i ⩾ 0, is a Z ℓ -structure of the Q ℓ -perverse sheaf -The first one denoted by Γ ξ,χv,m (I, h 0 + i) is given by the free Z ℓ -cohomology: recall that as the order of q v modulo ℓ is supposed to be strictly greater than d then, cf.[5], the reduction modulo ℓ of St h0+i (χ v ) and that of χ v [t] D , remains irreducible so that, up to homothety, there is a unique stable lattice of P ξ (h 0 + i, χ v ) Z ℓ .
-The spectral sequence associated to the previous filtration of gr h0 !,χv (Ψ ξ,1v ) provides a filtration of H 0 free (Sh I w (h0),sv , gr h0 !,χv (Ψ ξ,1v )) m , and Γ ξ,χv,!,m (I, h 0 +i, h 0 ) is then the lattice of the subquotient in this filtration corresponding to (3.3.4).Γ ξ,χv,m (I, h 0 + 1) −→ Γ ξ,χv,!,m (I, h 0 + 1, h 0 ), but the cokernel of torsion might be nontrivial due to torsion in the remaining of the E ∞ terms.The main point is first to show, under the absurd hypothesis of Section 2.1, that this cokernel is nontrivial and then, to prove some property verified by its ℓ-torsion and finally achieve to a contradiction.We then first focus on the first step about nontriviality.
Lemma 3.3.6.-The cokernel T of (3.3.5): is non zero and every irreducible subquotient of its ℓ-torsion as a (T S ξ,m ×GL d (F w ))⊗ Z ℓ F ℓ -module, can be obtained as a subquotient of the torsion submodule of the cokernel of Proof.-The idea is to compute the cohomology of gr h0 !(Ψ ξ,χv ) in two different ways, first by means of the spectral sequence associated to (3.2.2) and secondly by means of its filtration of stratification with graded parts the Harris-Taylor perverse sheaves.
To argue we will rest on the level of non degeneracy at v so that we pass to I w,v (∞)level: as q v modulo ℓ is of order > d taking invariant under any sub-group of GL d (O v ) is an exact functor and it will be easy to go down to level I w (h 0 ).
First note that the I w,v (∞)-version of (3.3.7) is non strict if and only if the same is true for its non induced version in the next formula, whatever a representation Π h0 of GL h0 (F v ) is: is the disjoint union of the pure strata, cf.Notation 2.2.1, (ii) For h > h 0 , whatever are the representations Π h and Π h+1 of respectively GL h (F v ) and GL h+1 (F v ), the cokernel of 3), the cokernel of (3.3.10)does not have any non zero invariant vector under Iw v (d − h 0 , 1, . . ., 1).
Proof.-The integer h 0 is chosen so that, by imposing Π h0 to be unramified, using also the fact that q v modulo ℓ is of order > d so that the functor of P h0,d (O v )invariants is exact, then the cokernel of (3.3.8) has nontrivial vectors invariant under P h0,d (O v ).Then when Π h0 = St h0 (χ v ), by modifying the factor GL d−h0 (O v ) by its classical Iwahori subgroup, we then deduce (i).
(ii) It follows from the definition of h 0 and the fact that the functor of GL d (O v )invariants is exact.
(1) Following the proof of 3.1.10with the spectral sequence associated to (3.2.2) and neglecting torsion classes which do not have nontrivial vectors invariant by Iw v (d − h 0 , 1, . . ., 1), we then deduce that H i tor (Sh I w,v (∞),sv , gr h0 !(Ψ ξ,χv )) m does not have any nontrivial vector invariant under Iw v (d − h 0 , 1, . . ., 1) if i ̸ = 0, 1, while for i = 0 the torsion is nontrivial and the vectors invariant by Iw v (d − h 0 , 1, . . ., 1) are given by the non strictness of (3.3.11)H 0 (Sh I w,v (∞),sv , j =h0+1 !(3) Consider then the cohomology of gr h0 !,χv (Ψ ϱ,ξ ) computed through its filtration of stratification with graded parts, up to Galois shifts, and more particularly the induced filtration of the free quotient of H 0 (Sh I w,v (∞),sv , gr h0 !,χv (Ψ 1v,ξ )) m as before.As the level of non degeneracy of T [ℓ] is higher than that of the ℓ-torsion of H 0 (Sh I w,v (∞),sv , Fil h0 !,χv (Ψ 1v,ξ )) m , computed by means of the spectral sequence associated to (3.2.2), they must be graded parts of this filtration.We then have a filtration of the free quotient of H 0 (Sh I w,v (∞),sv , gr h0 !,χv (Ψ 1v,ξ )) m for which, among the graded parts, appear -torsion modules such as T , -and the free graded parts which are lattices Γ ξ,χv,m (I v , h 0 +i) of the free quotient of the localized cohomology of P ξ (χ v , h 0 + i) for 0 ⩽ i ⩽ d − h 0 .
We now go back to the level I w (h 0 ) = I w (∞) Iw v (d − h 0 , 1, . . ., 1): as q v modulo ℓ is of order strictly greater than d, the functor of Iw v (d − h 0 , 1, . . ., 1)-invariants is exact.As only the cohomology of P ξ (χ v , h 0 + i) for i = 0, 1 contributes, the result follows from the fact that T has a nontrivial invariant vector under Iw gr h0 !(Ψ ξ,χv ) so that we can find a filtration of gr h0 !(Ψ ξ,1v ) whose graded parts are free and isomorphic, after tensoring with Q ℓ , to gr h0 !(Ψ ξ,χv ).Arguing as in the proof of Lemma 3.1.10,using (3.2.3), we have the following result.Lemma 3.3.12.-For every t ⩾ 1, let j(t) be the minimal integer j such that the torsion of H j (Sh I w,v (∞),sv , gr t !(Ψ ξ,1v )) m has nontrivial invariant vectors under Iw v (d − h 0 , 1, . . ., 1).Then ) through the spectral sequence (2.3.2).Recall that for every p+q ̸ = 0, the free quotient of E p,q !,ϱ,1 are zero.By definition of the filtration, these E p,q !,ϱ,1 are trivial for p ⩾ 0 while, thanks to the previous lemma, for any p ⩽ −1 they are zero for p + q < j(p) := p − h 0 . (4)or those of H 0 (Sh I w (h 0 ),sv , P ξ (h 0 , χv))m as explained above , which is torsion and non zero, according to the previous lemma, is equal to Combining the result of Lemma 3.3.6 with the previous proposition, we then deduce that the cokernel T of Lemma 3.3.6verifies the following property.As an Then applying Lemma 3.2.5 at the place w, which satisfies the same hypothesis as v, we then deduce the following result.We can now repeat the arguments with gr k !,χv (Ψ ξ,1v ) for any 1 ⩽ k ⩽ h 0 .More precisely, cf. the last remark of Section 2.3, consider Fil i (gr k !,χc (Ψ ξ,1v )) for i = h 0 − k and i = h 0 − k + 2. As by hypothesis, the torsion of )) has trivial cohomology groups in level I w (h 0 ) because the irreducible constituents of Fil h0+2−k (gr k !,χv (Ψ ξ,1v )) ⊗ Z ℓ Q ℓ are, up to Galois shift, Harris-Taylor perverse sheaves P (t, χ v ) with t ⩾ h 0 + 2; -we can apply the previous argument relatively to gr h0 !,χv (Ψ ξ,1v ) to the quotient Q := Fil h0−k (gr k !,χv (Ψ ξ,1v ))/ Fil h0+2−k (gr h0 !,χv (Ψ ξ,1v )), so that, denoting by Γ ′ ξ,χv,!,m (I, h 0 + 1, k) the lattice of (3.3.4) given by the free quotient of H 0 (Sh and, as an F ℓ -representation of GL d (F w ), it does not contain any irreducible generic subquotient made of characters.
In addition of the previous arguments, we also have to deal with the torsion in the cohomology groups of )), which could modify the lattice Γ ′ ξ,χv,!,m (I, h 0 + 1, k) to give the good one denoted above by Γ ξ,χv,!,m (I, h 0 + 1, k).Note again that, as an F ℓ -representation of GL d (F w ), this ℓ-torsion does not contain any irreducible generic subquotient made of characters, so the cokernel of , is again such that, as an F ℓ -representation of GL d (F w ), its ℓ-torsion does not contain any irreducible generic subquotient made of characters.
We now compute H 0 (Sh I w (h0),sv , Ψ ξ,1v ) m by means of the spectral sequence of vanishing cycles using the filtration ) such that the graded parts are, after tensoring with Q ℓ and up to Galois shift, of the form P ξ (χ v , t) with χ v ∈ Cusp −1 (1 v ).As before, arguing by contradiction, we suppose that the torsion of H d−1 (Sh I,η v , V ξ,Z ℓ ) m is nontrivial, and we pay special attention to the lattices of (3.3.16)V ξ,χv,m (I, h 0 + 1)(δ) := H i (Sh I,sv , P ξ (h 0 + 1, for χ v ∈ Cusp −1 (1 v ) and various δ.
From Lemma 3.1.13,we can repeat the same argument with χ ′ v in place of χ v as it was announced.

Notation 1 . 1 .
11. -Associated to a partition d = (d 1 ⩾ d 2 ⩾ d s ) of d = s i=1 d i , we consider the following Iwahori type subgroup of GL d (O K ):

Notation 1 .
2.2.-Denote by I the set of open compact subgroups small enough of G(A ∞ ).For I ∈ I, write Sh I,η → Spec F for the associated Shimura variety of Kottwitz-Harris-Taylor type.Definition 1.2.3.-Denote by Spl the set of places v of F such that p where the χ v,i are unramified characters.Definition 1.3.3.-For a finite set S of places of Q containing the places where G is ramified, denote by T S abs := x̸ ∈S T x,abs the abstract unramified Hecke algebra, where T

Corollary 3 . 3 .
14.  -As an F ℓ -representation of GL d (F w ), the ℓ-torsion of the cokernel T of Lemma 3.3.6does not contain any irreducible generic subquotient with cuspidal support made of characters.
Lemma 1.1.12.-With the previous notations, St t1 (χ 1 )×• • •×St tr (χ r ) has nontrivial invariant vectors under Iw(d) if, and only if, d is smaller, for the Bruhat order, to the dual partition associated to (t 1 , . . ., t r ).Recall that one way to obtain the dual partition is to use Fejer's diagrams.To a partition (d 1 ⩾ d 2 ⩾ d r ) one can associate a Fejer diagram with rows of respective size d 1 , . . ., d r .Then one can read this Fejer diagram through its columns whose size gives the dual partition associated to (d 1

. Moreover it has no nontrivial invariant vectors under Iw(d ′ ) for any d ′ strictly greater than d. 1.2. Shimura varieties of KHT type.
then the dual partition of (1, . . ., 1) is (d) and χ 1 × • • • × χ r has nontrivial invariant vectors under Iw(d) = GL d (O K ) and so under all the Iwahori type subgroup Iw(d).
sv is geometrically induced under the action of the parabolic subgroup P h,d (O v /M n v ), defined as the stabilizer of the first h vectors of the canonical basis of F d v .Concretely this means there exists a closed subscheme Sh =h I v (n),sv,1 h stabilized by the Hecke action of P h,d (F v ) and such that Proposition 3.3.13.-Up to multiplicities, the set of irreducible F ℓ [GL d (F w )]subquotients of the ℓ-torsion of (4) H 0 (Sh I w (h0),sv , gr h0 !(Ψ ξ,1v )) m , are the same as those of H tor (Sh I w (h0),sv , gr t !(Ψ ξ,1v )) m are independent of t.