Galois irreducibility implies cohomology freeness for KHT Shimura varieties
Pascal Boyer



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Galois irreducibility implies cohomology freeness]Galois irreducibility implies 
cohomology freeness for KHT Shimura varieties






boyer@math.univ-paris13.fr
Université Sorbonne Paris Nord 

LAGA, CNRS, UMR 7539
 
F-93430, Villetaneuse (France) 

CoLoSS: ANR-19-PRC







Given a KHT Shimura variety with an action of its unramified Hecke algebra ,
we proved in <cit.>, see also <cit.> for other PEL Shimura 
varieties, that its localized cohomology groups at a generic maximal ideal 𝔪 of 
, appear to be free.
In this work, we obtain the same result for 𝔪 such that its associated
galoisian _l-representation ρ_𝔪 is irreducible,
under the hypothesis  that [F(exp(2iπ/l):F]>d where F is the reflex field, d
the dimension of the KHT Shimura variety and l the residual characteristic.






11F70, 11F80, 11F85, 11G18, 20C08









[
    Boyer Pascal
    December 30, 2023
=====================



headings arabic













§ 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 you consider 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 _l-coefficients and Shimura varieties of Kottwitz-Harris-Taylor types,
we proved in <cit.>, whatever is the weight of 
the coefficients, when the level is large enough at l, there are always non trivial torsion 
cohomology classes, so that the _l-cohomology can not be concentrated in middle
degre. Thus if you want a _l-analog of the previous _l-statement, 
you must cut off some part of the cohomology. 

In <cit.> for KHT Shimura varieties, and more generally
in <cit.> for any PEL proper Shimura variety, we obtain such a result
under some genericness hypothesis which can be stated as follows.
Let (_K)_K ⊂ G(^) be a tower, indexed by open compact subgroups
K of G(^), of compact Shimura varieties of Kottwitz type associated to some 
similitude group G: we denote by F=EF^+ its reflex field
where F^+ is totally real and E/ is an imaginary quadratic extension.
Let then 𝔪 be a system of Hecke eigenvalues appearing in 
H^n_0(_K ×_F F,_l).
By the main result of <cit.>, one can attach to such 𝔪,
a mod l Galois representation 

    ρ_𝔪: (F/F) ⟶ GL_d(_l).

From <cit.> definition 1.9,
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_n } of eigenvalues of 
ρ_𝔪(_v)  
satisfies λ_i/λ_j ∉{ q_v^± 1} for all i ≠ j 
(resp. and are pairwise distincts), where q_v is the cardinal
of the residue field at v. Then under the hypothesis[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.] that there exists such p with
𝔪 generic at p, the integer n_0 above is necessary equals to
the relative dimension of _K. In particular the 
H^i(_K ×_F F,_l)_𝔪
are all torsion free.























In this work we consider the particular case of Kottwitz-Harris-Taylor Shimura varieties _K 
of <cit.> 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 appears to be useful at least for our approach of
Ihara's lemma, cf. <cit.>.










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 _K with coefficients in any
_l-local system V_ξ, are all free.

Note that Koshikawa, cf. <cit.>, starting from <cit.> and using techniques from group theory, proved a similar result in low dimension.

Thanks to the hypothesis [F(exp(2iπ/l):F]>d and by Cebotarev theorem,
we can pick places v of F such that the order q_v of the residue field of F at v,
is of order strictly greater than d  in (/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 modulo l reduction has a supercuspidal
support made of characters, cf. the remark after <ref>.
This simplification is completely harmless and if one wants to take care about
these cuspidal representations, it suffices to used the proposition 2.4.2 
of <cit.>.


  * With this hypothesis we also note that the pro-order of GL_d(_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 the lemma <ref>.


  * Finally in the last section, we are able to construct a
sequence of intervals contained in the set of eigenvalues of 
ρ_𝔪(_v) so that at the end we obtain a full set
{λ q_v^n: n ∈} 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 we analyse the torsion in the
cohomology of Harris-Taylor perverse sheaves at some place
v with infinite level at another place w. 
The main observation which rests on the symmetric role of both v and w, 
cf. corollary <ref>, 
is that, considering
l-torsion as a _l-representation of GL_d(F_w), then
we never encounter any irreducible generic sub-quotient with cuspidal
support made of characters.

(2) As one can compute the cohomology of the Shimura variety through the
spectral sequence associated to the exhaustive filtration of stratification
^∙(Ψ_v) of the vanishing cycles perverse sheaf, where the
E_1 terms are given by the cohomology groups of the
Harris-Taylor perverse sheaves, we then observe that the various
lattices of H^d-1_free(_K×_F F,V_ξ,_l)_𝔪 given by the integral cohomology, 
are only slightly modified from the ones given by the cohomology of the
Harris-Taylor perverses 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 sub-quotient with cuspidal
support made of characters, cf. proposition <ref>.


(3) However if the torsion sub-module of 
H^d-1(_K×_F F,V_ξ,_l)_𝔪
were not trivial, we prove, using the geometrical induced structure of the 
 Newton strata , that it must exists 𝔪
such that the lattices of the previous point, are not isomorphic, cf. 
lemma <ref>

(4) Finally in <ref>, 
for any 𝔪⊂𝔪, then 
H^d-1(_K ×_F F,V_ξ,_l)_𝔪
induces a quotient stable lattice Γ_𝔪 of (Π_𝔪^)^K ⊗ρ_𝔪. As
ρ_𝔪 is supposed to be irreducible, then this
lattice is isomorphic to a tensorial 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 H^d-1(_K ×_F F,
V_ξ,_l)_𝔪 given
by the filtration of the nearby perverse sheaf, so that, using
diagrams as <ref>, we arrive at Γ_𝔪.
In the process we are able to construct an increasing sequence of interval
contained in the set of eigenvalues of ρ_𝔪( v)
so that at the end we obtain a full set {λ q_v^n : n ∈}
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 <ref> for more details.























































































































§ RECALLS FROM <CIT.>




 §.§ Representations of GL_d(K)



We fix a finite extension K/_p with residue field _q. We denote by |-| its absolute
value.

For a representation π of GL_d(K) and n ∈1/2, set 

    π{ n }:= π⊗ q^-n ∘.


 
For π_1 and π_2 representations of respectively GL_n_1(K) and
GL_n_2(K), we will denote by

    π_1 ×π_2:=_P_n_1,n_1+n_2(K)^GL_n_1+n_2(K)π_1 {n_2/2}⊗π_2 {-n_1/2},

the normalized parabolic induced representation where for any sequence
r=(0< r_1 < r_2 < ⋯ < r_k=d), we write P_r for 
the standard parabolic subgroup of GL_d with Levi

    GL_r_1× GL_r_2-r_1×⋯× GL_r_k-r_k-1.
 



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 _l.

  (see <cit.> 9 and <cit.> 1.4)
Let g be a divisor of d=sg and π an irreducible cuspidal 
_l-representation of GL_g(K). 
The induced representation

    π{1-s/2}×π{3-s/2}×⋯×π{s-1/2}
 
holds a unique irreducible quotient (resp. subspace) denoted _s(π) (resp.
_s(π)); it is a generalized Steinberg (resp. Speh) representation.

Moreover the induced representation _t(π{-r/2}) ×_r(π{t/2} )
(resp. of _t-1(π{-r-1/2}) ×_r+1(π{t-1/2} ))
owns a unique irreducible subspace (resp. quotient), denoted
LT_π(t-1,r).



These representations LT_π(t-1,r) appear in the cohomology of the Lubin-Tate
spaces, cf. <cit.>.

  (cf. <cit.> III.5.10)
Let π be an irreducible cuspidal representation of GL_g(K) with a stable
-lattice[We say that π is integral.], then its modulo l reduction
is irreducible and cuspidal but not necessary supercuspidal.



The supercuspidal support of the modulo l reduction of a cuspidal representation, 
is a segment associated to some irreducible _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.


 
We denote by m(ϱ) the order of the Zelevinsky line 
{ϱ(δ): δ∈} of
ϱ if it is not equal to 1, otherwise m(ϱ)=l.


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 π_v is of ϱ-type u with u ≥ -1.


For ϱ an irreducible _l-supercuspidal representation, we denote by
_ϱ (resp. _ϱ(u) for some u ≥ -1) 
the set of equivalence classes of irreducible
_l-cuspidal representations whose modulo l reduction has 
for supercuspidal
support a segment associated to ϱ (resp. of ϱ-type u).


Let u ≥ 0, π_v,u∈_ϱ(u) and τ=π_v,u[s]_D. Let then denote
by ι the image of _s(ϱ) by the modulo l Jacquet-Langlands 
correspondence
defined at 1.2.4 de <cit.>. Then the modulo l reduction of τ is isomorphic to
thm1

    ι{-m(τ)-1/2}⊕ι{-m(τ)-3/2}⊕⋯⊕ι{m(τ)-1/2}

where ι{ n }:=ι⊗ q^-n ∘.

We want now to recall the notion of level of non degeneracy from <cit.> 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

    V_d(K)={ (m_i,j∈ P_d(K): m_i,j= δ_i,j for  j < n }.
 
its unipotent radical. We fix a non trivial character ψ of K and let θ be the
character of V_d(K) defined by θ( (m_i,j))=ψ(m_d-1,d).
For G=GL_r(K) or M_r(K), we denote by (G) the abelian category of algebraic 
representations of G and, following <cit.>, we introduce

    Ψ^-: (M_d(K)) ⟶(GL_d-1(K),     Φ^-:  (M_d) ⟶ (M_d-1(K))

defined by Ψ^-=r_V_d,1 (resp. Φ^-=r_V_d,θ) the functor of V_d-1
coinvariants (resp. (V_d-1,θ)-coinvariants), cf. <cit.> 1.8.
We also introduce the normalized compact induced functor

    Ψ^+:=i_V,1: (GL_d-1(K)) ⟶ (M_d(K)),
 

    Φ^+:=i_V,θ: (M_d-1(K)) ⟶(M_d(K)).


 (<cit.> p451)


  * The functors Ψ^-, Ψ^+, Φ^- and Φ^+ are exact.


  * Φ^- ∘Ψ^+=Ψ^- ∘Φ^+=0.


  * Ψ^- (resp. Φ^+) is left adjoint to Ψ^+ (resp. Φ^-) and the
following adjunction maps 

    ⟶Φ^- Φ^+,     Ψ^+ Ψ^- ⟶,

are isomorphisms meanwhile

    0 →Φ^+ Φ^- ⟶⟶Ψ^+ Ψ^- 
    → 0.





For τ∈(M_d(K)), the representation 

    τ^(k):=Ψ^- ∘ (Φ^-)^k-1(τ)

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.


 (cf. <cit.> 4.3) 
Let π∈(GL_d(K)) (or π∈(M_d(K)). The maximal number k such that 
(π_|M_d(K))^(k)≠ (0) 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.



A representation π of GL_d(K), over _l or _l, is then said generic
if its level of non degeneracy λ(π) is equal to d.


Over _l, an irreducible generic representation of
GL_d(K) looks like _t_1(π_1) ×⋯×_t_r(π_r)
where π_1,⋯, π_r are irreducible cuspidal representations. Note
moreover that the modulo l reduction of any irreducible generic representation
owns a unique generic irreducible sub-quotient.






 §.§ Shimura varieties of KHT type




Let F=F^+ E be a CM field where E/ is quadratic imaginary and 
F^+/ is
totally real with a fixed real embedding τ:F^+ ↪. For a place v of F,
we will denote by


  * F_v the completion of F at v,


  * _v the ring of integers of F_v,


  * ϖ_v a uniformizer,


  * q_v the cardinal of the residual field κ(v)=_v/(ϖ_v).

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=β x^*
β^-1 and let G/ be the group of similitudes, denoted by 
G_τ in <cit.>, defined for every -algebra R by 

    G(R)  ≃{ (λ,g) ∈ R^×× (B^op⊗_ R)^× such that  
    gg^♯_β=λ}

with B^op=B ⊗_F,c F. 
If x is a place of  split x=yy^c in E then 
thm1

    G(_x) ≃ (B_y^op)^××_x^×≃_x^××∏_z_i (B_z_i^op)^×,

where, identifying places of F^+ over x with places of F over y,
x=∏_i z_i in F^+.

Convention: for x=yy^c a place of  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_z^op)^× in the formula (<ref>).


In <cit.>, the authors justify the existence of some G like before such that moreover


  * if x is a place of  non split in E then G(_x) is quasi split;


  * the invariants of G() are (1,d-1) for the embedding τ and (0,d) for the others.


As in  <cit.> bottom of page 90, a compact open subgroup U of G(^) is said 
small enough
if there exists a place x such that the projection from U^v to G(_x) does not contain any 
element of finite order except identity.


Denote by  the set of open compact subgroups small enough of G(^).
For I ∈, write _I,η⟶ F for the associated
Shimura variety of Kottwitz-Harris-Taylor type.



 
Denote by  the set of  places v of F such that p_v:=v_|≠ l is split in E and let
B_v^×≃ GL_d(F_v).  For each I ∈, we write
(I) for the subset of  of places which does not divide I.




In the sequel, v and w will denote places of F in . 
For such a place v,
the scheme _I,η has a projective model _I,v over _v
with special fiber _I,s_v. For I going through , the projective system (_I,v)_I∈ 
is naturally equipped with an action of G(^) × such that any
w_v in the Weil group W_v of F_v acts by - (w_v) ∈,
where =∘^-1 and ^-1:W_v^ab≃ F_v^× is the isomorphism of Artin
sending the geometric Frobenius to uniformizers.

 
For I ∈, the Newton stratification of the geometric special fiber _I,s̅_v is denoted by

    _I,s̅_v=:^≥ 1_I,s̅_v⊃^≥ 2_I,s̅_v⊃⋯⊃^≥ d_I,s̅_v

where ^=h_I,s̅_v:=^≥ h_I,s̅_v - ^≥ h+1_I,s̅_v is an affine 
scheme, smooth of pure dimension d-h built up by the geometric 
points whose connected part of its Barsotti-Tate group is of rank h.
For each 1 ≤ h <d, write

    i_h:^≥ h_I,s̅_v↪^≥ 1_I,s̅_v,   
    j^≥ h: ^=h_I,s̅_v↪^≥ h_I,s̅_v,

and j^=h=i_h ∘ j^≥ h.



Let σ_0:E ↪_l be a fixed embedding and write Φ for the set of embeddings 
σ:F ↪_l whose restriction to E equals σ_0.
There exists then, cf. <cit.> p.97, an explicit bijection between irreducible algebraic representations 
ξ of G over _l and (d+1)-uple
( a_0, (a_σ)_σ∈Φ )
where a_0 ∈ and for all σ∈Φ, we have a_σ=
(a_σ,1≤⋯≤ a_σ,d ). 
We then denote by 

    V_ξ,_l
 
the associated _l-local system on _.
Recall that an irreducible automorphic representation Π is said ξ-cohomological if there exists
an integer i such that

    H^i  ( (  G()) ⊗_,U,Π_⊗ξ^∨ ) ≠ (0),

where U is a maximal open compact subgroup modulo the center of G().
Let d_ξ^i(Π_) be the dimension of this cohomology group.





 §.§ Cohomology of the Newton strata




 
For 1 ≤ h ≤ d, let _v(h) be the set of open compact subgroups 

    U_v(m,h):= 
    U_v(m^v) × ( [      I_h        0;        0 K_v(m_1) ] ),

where K_v(m_1)= ( GL_d-h(_v) ⟶ GL_d-h(_v/ (ϖ_v^m_1))  ).
We then denote by [H^i(h,ξ)] (resp. [H^i_!(h,ξ)]) the image of

    lim_⟶I ∈_v(h) H^i(_I,s̅_v,1^≥ h, V_ξ,_l[d-h]) 
        resp. lim_⟶I ∈_v(h) H^i(_I,s̅_v,1^≥ h, j^≥ h_1,! V_ξ,_l[d-h])
 
inside the Grothendieck (v,h) of admissible representations of 
G(^) × GL_d-h(F_v) ×.


An element σ∈ W_v acts through -σ∈ and Π_p_v,0(^-1 (σ)).
We moreover consider the action of GL_h(F_v) through  
∘: GL_h(F_v) ⟶ and finally
P_h,d(F_v) through its Levi factor GL_h(F_v) × GL_d-h(F_v), i.e.
its unipotent radical acts trivially.


From <cit.> proposition 3.6, for any irreducible tempered automorphic representation 
Π of G() 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 representation ξ-cohomological, its local component at v is generic and so looks like

    Π_v ≃_t_1(π_v,1) ×⋯×_t_u(π_v,u),

where for i=1,⋯,u, π_v,i is an irreducible cuspidal representation of est une 
GL_g_i(F_v).



  (cf. <cit.> proposition 3.9)
With the previous notations, we order the π_v,i such that the first r-ones are unramified
characters. Then the Π^,v-isotypic component of [H^0(h,ξ)] is then equals to

    ( ♯^1(,G)/d∑_Π' ∈_G(Π^,v) 
    m(Π') d_ξ(Π'_)  )   ( ∑_1 ≤ k ≤ r:  t_k=hΠ_v^(k)⊗χ_v,kχ^d-h/2 )

where






  * ^1(,G) is the subset of elements of H^1(,G) which become trivial
in H^1(_p',G) for every prime p';


  * Π_v^(k):= _t_1(χ_v,1) ×⋯×_t_k-1(χ_v,k-1) ×_t_k+1(χ_v,k+1) ×⋯×_t_u(χ_v,u) and
 

  * Ξ:1/2⟶_l^× 
is defined by Ξ(1/2)=q_v^1/2. 


  * _G(Π^,v) is the set of equivalence classes of irreducible automorphic 
representations Π' of G() such that (Π')^,v≃Π^,v.



In particular if [H^0(h,ξ)] has non trivial invariant vectors under some open
compact subgroup I ∈_v(h) which is maximal at v, then the local component
of Π at v is of the following shape _h(χ_v,1) ×χ_v,2×⋯χ_v,d-h
where the χ_v,i are unramified characters.


For a finite set S of places of  containing the places where G is ramified, 
denote by ^S_abs:=∏_x ∉S_x,abs the 
abstract unramified  Hecke algebra


where
_x,abs≃_l[X^un(T_x)]^W_x for T_x a split torus,
W_x the spherical Weyl group and X^un(T_x) the set of _l-unramified 
characters of T_x. 


Example.
For w ∈, we have

    _w,abs=_l  [T_w,i:  i=1,⋯,d  ],

where T_w,i is the characteristic function of

    GL_d(_w) (ϖ_w,⋯,ϖ_w^i, 1,⋯,1^d-i ) 
    GL_d(_w) ⊂  GL_d(F_w).



 
Let ^S_ξ be the image of ^S_abs inside 

    ⊕_i =0^2d-2lim_→I H^i(_I,η̅,V_ξ,_l)

where the limit is taken over 
the ideals I which are maximal at each places outside S. For I
an open compact subgroup maximal at each places outside S,
we will also denote by ^S_I,ξ the image of ^S_abs inside
H^d-1(_I,η̅,V_ξ,_l).


Let state some remarks about these Hecke algebras.


  * In <cit.>, we proved that if 𝔪 is a maximal ideal of 
^S_ξ such that there exists i with 
H^i(_I,η̅,V_ξ,_l)_𝔪≠ (0) for I maximal at each places outside S, then
(^S_ξ)_𝔪≠ (0), i.e. torsion cohomology classes raise in 
characteristic zero. In particular to define ^S_I,ξ there is no difference
taking cohomology with _l or _l coefficients,
and consider torsion classes or only the free quotients.


  * As explained in the introduction, we will consider maximal ideals
𝔪 such that ρ_𝔪 is irreducible so that
the Qm_l-cohomology groups are all concentrated in middle degree,
i.e. in degree 0 if we deal with perverse sheaves.



  * With the notations of <ref> about the Harris-Taylor local systems,
in <cit.>, we proved that, except for the GL_d(F_v)-action,
the irreducible sub-quotients of the
_l-cohomology groups of j^≥ tg_! HT(π_v,Π_t) or 
p j^≥ tg_!* HT(π_v,Π_t) are also sub-quotients of
the cohomology of _I,η̅. Moreover their torsion classes also
raises in characteristic zero. In particular if 𝔪 is such that
ρ_𝔪 is irreducible, then the image of ^S_ξ
inside H^0(_I,s̅_v, p j^≥ tg_!* HT(π_v,Π_t))_𝔪
factors through ^S_I',χ for I' such that (I')^v=I^v. The same
is also true for j^≥ tg_! HT(π_v,Π_t). In the case where π_v
is a unramified character, then you can take I'_v for the Iwahori subgroup
of GL_d(_v).





The minimal prime ideals of ^S_ξ are the prime ideals above the zero ideal of 
_l and are then in bijection with the prime ideals of 
^S_ξ⊗__l_l. 
To such an ideal, which corresponds
to give a collection of Satake parameters, is then associated a unique near equivalence class in
the sense of <cit.>, denoted by Π_𝔪, which is the finite set of
irreducible automorphic cohomological representations whose multi-set of Satake parameters
at each place x ∈(I), is given by S_𝔪(x) the multi-set of roots of
the Hecke polynomial

    P_𝔪,w(X):=∑_i=0^d(-1)^i q_w^i(i-1)/2 
    T_w,i,𝔪 X^d-i∈_l[X]

i.e.

    S_𝔪(w) := {λ∈^S_ξ⊗__l_l / 𝔪≃_l  such that 
    P_𝔪,w(λ)=0 }.

Thanks to <cit.> and <cit.>, we denote by

    ρ_𝔪:(F/F) ⟶ GL_d(_l)
 
the Galois representation associated to any Π∈Π_𝔪.
Recall that the modulo l reduction of ρ_𝔪 depends only
of 𝔪, and was denoted above ρ_𝔪.
For every w ∈(I), we also denote by S_𝔪(w) the multi-set of modulo l
Satake parameters at w given as the multi-set of roots of

    P_𝔪,w(X):=∑_i=0^d(-1)^i q_w^i(i-1)/2T_w,i X^d-i∈_l[X]

i.e.

    S_𝔪(w) := {λ∈^S_ξ/𝔪≃_l 
     such that  P_𝔪,w(λ)=0 }.

Using the arguments of <cit.> and following <cit.> V.4.4, 
we then deduce the existence, for _F,S the Galois group of the maximal
extension of F unramified outside S, of
thm1

    ρ_ξ,𝔪: _F,S⟶ GL_d((^S_ξ)_𝔪)

interpolating the ρ_𝔪, so that in particular for all u ∉S,
(1-X _u|ρ_𝔪) is equal to the 
Hecke polynomial.

In  <cit.>, the author constructs 
ρ_𝔪: _F,S⟶ GL_d((^S_ξ)_𝔪/J)
where J is a nilpotent ideal but it seems from incoming work that on can arrange
J to be zero.



























§ ABOUT THE NEARBY CYCLE PERVERSE SHEAF



Our strategy to compute the cohomology of the KHT-Shimura variety _I,η̅ with
coefficients in V_ξ,_l, is to realize it as the outcome of the nearby cycles 
spectral sequence at some place v ∈.

Note that the role of the local system V_ξ,_l
associated to ξ is completely harmless when
dealing with sheaves: one just have 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,_l is crucial as it selects the
automorphic representations which contribute to the cohomology.




 §.§ The case where the level at v is maximal



By the smooth base change theorem, we have
H^i(_I,η̅_v, V_ξ) ≃ H^i(_I,s̅_v,V_ξ).
As for each 1 ≤ h ≤ d-1, the open Newton stratum ^=h_I,s̅_v is affine
then H^i_c(^=h_I,s̅_v,V_ξ,_l[d-h]) is zero for i<0 and free for i=0.
Using this property and the following short exact sequence
of free perverse sheaves

    0 → i_h+1,* V_ξ,_l,|_I,s̅_v^≥ h+1[d-h-1] ⟶ 
    j^≥ h_! j^≥ h,* V_ξ,_l,|^≥ h_I,s̅_v[d-h] 
    ⟶ 
    V_ξ,_l,|^≥ h_I,s̅_v[d-h] → 0,

we then obtain for every i>0
thm1

    0 → H^-i-1(^≥ h_I,s̅_v,V_ξ,_l[d-h]) ⟶
    H^-i(^≥ h+1_I,s̅_v,V_ξ,_l[d-h-1]) → 0,

and for i=0,
thm1

    0 → H^-1(^≥ h_I,s̅_v,V_ξ,_l[d-h]) ⟶
    H^0(^≥ h+1_I,s̅_v,V_ξ,_l[d-h-1]) ⟶
    
    H^0(^≥ h_I,s̅_v,j^≥ h_! j^≥ h,* V_ξ,_l[d-h] ) ⟶
    H^0(^≥ h_I,s̅_v,V_ξ,_l[d-h]) →⋯

In <cit.>, arguing by induction from h=d to h=1, we prove that for a maximal
ideal 𝔪 of ^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(_I,s̅_v^≥ h,V_ξ,_l)_𝔪 are free: note that in order
to deal with i≥ 0, one has to use the Grothendieck-Verdier duality.

Without this hypothesis, arguing similarly, we conclude that any torsion cohomology class
comes from a non strict map 
thm1

    H^0_free(^≥ h+1_I,s̅_v,V_ξ,_l[d-h-1])_𝔪⟶
    H^0(^≥ h_I,s̅_v,j^≥ h_! j^≥ h,* V_ξ,_l[d-h] )_𝔪.

In particular it raises in characteristic zero to some free subquotient of
H^0(^≥ h_I,s̅_v,j^≥ h_! j^≥ h,* V_ξ,_l[d-h] )_𝔪.






We argue by absurdity and we suppose there exists I ∈ such that there
exists non trivial torsion cohomology classes in the 𝔪-localized cohomology of 
_I,η̅_v with coefficients in V_ξ,_l. Fix such finite level I.

  (cf. <cit.> lemme 4.13)
Consider h_0(I) maximal such that there exists i ∈ with
H^d-h_0(I)+i(_I,s̅_v^≥ h_0(I), V_ξ,_l)_𝔪,tor≠ (0). Then we have the following properties:


  * i=0,1;


  * for all 1 ≤ h ≤ h_0(I) and i<h-h_0(I),

    H^d-h+i(_I,s̅_v^≥ h V_ξ,_l)_𝔪,tor =(0)
 
while for i=h-h_0(I) it is non trivial.




Note that any system of Hecke eigenvalues
𝔪 of ^S_ξ inside the torsion of some 
H^i(_I,η̅_v,V_ξ,_l) raises in characteristic zero, i.e. 
is associated to a minimal prime ideal 𝔪 of ^S ∪{ v }_ξ. 
More precisely, using the remark following the proposition <ref>,
there exists 𝔪⊂𝔪 such that
the local component at v of π_𝔪 is isomorphic to
_h_0(I)+1(χ_v) ×χ_v,1×⋯×χ_v,d-h_0(I)-1
where χ_v, χ_v,1,⋯ , χ_v,d-h_0(I)-1 are characters of F_v^×.




 §.§ Harris-Taylor perverse sheaves over _l



Consider now the ideals I^v(n):=I^vK_v(n) where
K_v(n):=(GL_d(_v) ↠ GL_d(_v/_v^n)).
Recall then that _I^v(n),s̅_v^=h is geometrically induced
under the action of the parabolic subgroup P_h,d(_v/_v^n), defined as the
stabilizer of the first h vectors of the canonical basis of F_v^d. Concretely this means there
exists a closed subscheme _I^v(n),s̅_v,1_h^=h stabilized by the Hecke 
action of P_h,d(F_v) and such that

    _I^v(n),s̅_v^=h = _I^v(n),s̅_v,1_h^=h×_P_h,d(_v/_v^n) GL_d(_v/_v^n),

meaning that _I^v(n),s̅_v^=h is the disjoint union of copies of
_I^v(n),s̅_v,1_h^=h indexed by 
GL_d(_v/_v^n)/P_h,d(_v/_v^n) and 
exchanged by the action of
GL_d(_v/_v^n).

 
For any g ∈ GL_d(_v/_v^n)/P_h,d(_v/_v^n), we denote by
^=h_I^v(n),s̅_v,g the pure Newton stratum defined as the image of
_I^v(n),s̅_v,1_h^=h by g.
Its closure in _I^v(n),s̅_v is then denoted by
^≥ h_I^v(n),s̅_v,g.


Let then denote by  𝔪^v the multiset of
Hecke eigenvalues given by 𝔪 but outside v and 
introduce for Π_h any representation of GL_h(F_v)

    H^i(_I^v(),s̅_v,1_h^≥ h, V_ξ,_l)_𝔪^v⊗Π_h:=
    lim_⟶n H^i(_I^v(n),s̅_v,1_h^≥ h,
    V_ξ,_l)_𝔪^v⊗Π_h,

as a representation of GL_h(F_v) × GL_d-h(F_v), where g ∈ GL_h(F_v)
acts on Π_h as well as on H^i(_I^v(n),s̅_v,1_h^≥ h,
V_ξ,_l)_𝔪^v through the determinant map : GL_h(F_v)
↠ F_v^×. Note moreover that the unipotent radical of
P_h,d(F_v) acts trivially on these cohomology groups. 
We then introduce their induced version

    H^i(^≥ h_I^v(),s̅_v,Π_h ⊗ V_ξ,_l)_𝔪^v≃_P_h,d(F_v)^GL_d(F_v) H^i(_I^v(),s̅_v,1_h^≥ h, V_ξ,_l)_𝔪^v⊗Π_h.


More generally, with the notations of <cit.>, replace now the trivial representation by
 an irreducible cuspidal representation π_v of GL_g(F_v) for some 1 ≤ g ≤ d.
 
  
Let 1 ≤ t ≤ s:=⌊ d/g ⌋ and Π_t any representation of 
 GL_d-tg(F_v). We then denote by

    HT_1(π_v,Π_t):=(π_v[t]_D)_1_tg⊗Π_t ⊗Ξ^tg-d/2
 
the Harris-Taylor local system on the Newton stratum ^=tg_I,s̅_v,1_tg where 


  * (π_v[t]_D)_1_tg is defined thanks to
Igusa varieties attached to the representation π_v[t]_D of the division algebra of dimension
(tg)^2 over F_v associated to _t(π_v) by the Jacquet-Langlands correspondence,


  * Ξ:1/2⟶_l^× defined by Ξ(1/2)=q^1/2.

We also introduce the induced version

    HT(π_v,Π_t):= ( (π_v[t]_D)_1_tg⊗Π_t ⊗Ξ^tg-d/2) ×_P_tg,d(F_v) GL_d(F_v),

where the unipotent radical of P_tg,d(F_v) acts trivially and the action of

    (g^,v, ( [  g_v^c      *;      0 g_v^et ] ),σ_v) 
    ∈ G(^,v) × P_tg,d(F_v) × W_v
 
is given


  * by the action of g_v^c on Π_t and 
(σ_v) ∈ on Ξ^tg-d/2, and


  * the action of (g^,v,g_v^et,( g_v^c)-σ_v)
∈ G(^,v) × GL_d-tg(F_v) × on __l
(π_v[t]_D)_1_h⊗Ξ^tg-d/2.

We also introduce

    HT(π_v,Π_t)_1_tg:=HT(π_v,Π_t)_1_tg[d-tg],

and the perverse sheaf

    P(t,π_v)_1_tg:=j^=tg_1,!* HT(π_v,_t(π_v))_1_tg⊗(π_v),

and their induced version, HT(π_v,Π_t) and P(t,π_v), where 

    j^=h=i^h ∘ j^≥ h:^=h_I,s̅_v↪^≥ h_I,s̅_v↪_I,s̅_v
 
and ^∨, the dual of , is the local Langlands correspondence.
Finally we will also use the indice ξ in the notations, for example HT_ξ(π_v,Π_t), 
when we twist the sheaf with V_ξ,_l.


With the previous notations, from (<ref>), we deduce the following equality in the
Grothendieck group of Hecke-equivariant local systems
thm1

    m(ϱ)l^u [ _ξ,_l(π_v,u[t]_D)  ] =  
     [ _ξ,_l(π_v,-1[tm(ϱ)l^u]_D)  ].


We want now to focus on the perverse Harris-Taylor sheaves. Note first that over 
_l, there are two notions of intermediate extension associated to 
the two classical t-structures p and p+.
So for every π_v ∈_ϱ of GL_g(F_v) and 1 ≤ t ≤ d/g, we can define:
thm1

    p j^=tg_!* HT(π_v,,Π_t) _+ p+ j^=tg_!* HT(π_v,Π_t),

the symbol _+ meaning bimorphism, i.e. both a monomorphism and epimorphism, so that
the cokernel for the t-structure p (resp. the kernel for p+) has support in
^≥ tg+1_I,s̅_v. When π_v is a character, i.e. when g=1, the associated
bimorphims are isomorphisms, as explained in the following lemma, 
but in general there are not.



  
With the previous notations, we have an isomorphism

    p j^≥ h_1_h,!*  HT(χ_v,Π_h) 
    ≃p+ j^≥ h_1_h,!* HT(χ_v,Π_h).




Recall that ^≥ h_I,s̅_v,1_h is smooth over 
_p. As, up to a modification of the action of the fondamental
group through the character χ_v, we have

    HT(χ_v,Π_h)_1_h[h-d]= (_l)_|^≥ h_I,s̅_v,1_h⊗Π_h.

Then HT(χ_v,Π_h)_1_h is perverse for the two t-structures with

    i^h ≤ +1,*_1_h HT(χ_v,Π_h)_1_h∈ p ^< 0 and 
    i^h ≤ +1,!_1_h HT(χ_v,Π_h)_1_h∈p+^≥ 1.



One of the main result of <cit.>, is the fact that
the previous lemma holds for any π_v ∈_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 the trivial representation, 
we do not need to bother about the representations 
π_v ∈_ϱ(u) for u ≥ 0,
 cf. the remark after <ref>. 
 











































 §.§ Filtrations of the nearby cycles perverse sheaf




 Let denote by[We decide not to add I in the list of indexes.] 

    Ψ_v:=RΨ_η_v(_l[d-1])(d-1/2)

the nearby cycles autodual free perverse sheaf on the geometric special fiber _I,s̅_v
of _I. We also denote by Ψ_ξ,v:=Ψ_v⊗ V_ξ,_l.

Using the Newton stratification and following the constructions of <cit.>,
we can define a _l-filtration ^∙(Ψ_v) whose graded parts are free, 
isomorphic to some free perverse Harris-Taylor sheaf. Moreover,
by denoting __l(g) for the set of inertial equivalence 
classes of irreducible _l-supercuspidal representations of 
GL_g(F_v),  in <cit.> 
proposition 3.1.3, we proved the following splitting
thm1

    Ψ_v≃⊕_g=1^d 
    ⊕_ϱ∈__l(g)Ψ_ϱ,

with the property that

the irreducible sub-quotients of 

    Ψ_ϱ⊗__l_l ≃⊕_π_v ∈_ϱΨ_π_v

are exactly the perverse Harris-Taylor sheaves, of level I, 
associated to an irreducible cuspidal _l-representation
of some GL_g(F_v) such that the supercuspidal support of the modulo l reduction of
π_v is a segment associated to the inertial class ϱ.


In <cit.>, we proved that if you always use the adjunction maps j^=h_! j^=h,*→ 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 a character in which case, cf. lemma <ref>, the p and p+ intermediate extensions 
associated to character χ_v,-1∈_ϱ(-1), coincide. 
Note that in the following
we will not use the results of <cit.>.



Denoting by ^k(Ψ_ξ,v):=^k(Ψ_ξ,v)/^k-1(Ψ_ξ,v),
we then have a spectral sequence
thm1

    E_1^p,q=H^p+q(_I,s̅_v,^-p(Ψ_ξ,v)) ⇒ 
    H^p+q(_I,η̅_v,V_ξ,_l),

where we recall that
thm1

    p j^=tg_!* HT_ξ(π_v,_t(π_v))(1-t+2i/2) _+  
    ^k(Ψ_ξ,v) _+ 
    p+ j^=tg_!* HT_ξ(π_v,_t(π_v))(1-t+2i/2),

for some irreducible cuspidal representation π_v of GL_g(F_v) with 1 ≤ t ≤ d/g and
0 ≤ i ≤⌊ d/g ⌋ -1.





Let consider now the filtration of stratification of Ψ_ξ,ϱ constructed using the
adjunction morphisms j^=t_! j^=t,* as in <cit.>

    ^0_!(Ψ_ξ,ϱ) ^1_!(Ψ_ξ,ϱ) ^2_!(Ψ_ξ,ϱ)
    ⋯^d_!(Ψ_ξ,ϱ)

where the symbol  means a monomorphism such that the cokernel
is torsion free which here means that
^t_!(Ψ_ξ,ϱ) is the saturated image of 
j^=t_!j^=t,*Ψ_ξ,ϱ⟶Ψ_ξ,ϱ.  
We then denote by ^k_!(Ψ_ξ,ϱ) the graded parts and
thm1

    E_!,ϱ,1^p,q=H^p+q(_I,s̅_v,^-p_!(Ψ_ξ,ϱ)) ⇒ 
    H^p+q(_I,η̅_v,Ψ_ξ,ϱ).



Over _l, in <cit.> we prove that
^k_!(Ψ_ξ,ϱ) ⊗__l_l
is N^k where N is the monodromy operator at v. Moreover
there is only one filtration of stratification of ^k_!(Ψ_ξ,ϱ)
which we denote simply by ^∙ (^k_!(Ψ_ξ,ϱ)) in
the following. Over _l, 
^∙ (^k_!(Ψ_ξ,π_v))
coincides with the filtration through the iterated image of N, i.e.
^r(^k_!(Ψ_ξ,π_v))= N^r ∩ N^k so that
we recover the usual bi-filtration of monodromy of <cit.>.

















































































































































§ IRREDUCIBILITY IMPLIES FREENESS




Recall that we argue by absurdity assuming there exists non trivial 
cohomology classes in some of the 
H^i(_I,η̅,V_ξ,_l)_𝔪.
The strategy is then to choose a place v ∈_I such that the order of q_v modulo
l is strictly greater than d, and to compute the middle cohomology group
 H^d-1(_I,η̅,V_ξ,_l)_𝔪 through the spectral
 sequence of vanishing cycles. This spectral sequence gives us in particular a filtration of the
 free quotient of H^d-1(_I,η̅,V_ξ,_l)_𝔪 

We will in fact consider different sorts of level I relatively to the place v and another
one denoted w verifying the same hypothesis as v:


  * either with infinite level at v or with I_v some particular Iwahori subgroup;


  * either maximal or infinite at w.


From now on, the localization at 𝔪, means that we prescribe
the modulo l Satake's parameters as usual, but outside { v,w }.


By hypothesis, there exists non trivial torsion cohomology classes in level I
with I_v and I_w maximal. As moreover we supposed the order of both q_v and
q_w is strictly greater than d, then the functors of invariants by any open compact
subgroups either at v or w, is exact. Thus this allows us to argue similarly with
all the mentioned level, cf. lemma <ref>.
Let now explain the main steps of the following sections.

(a) Following the arguments of the previous section, 
we first analyse the torsion cohomology
classes of Harris-Taylor perverse sheaves with infinite level at v, and we deduce, 
cf. lemma <ref>, that, as _l-representations
of GL_d(F_v), 
irreducible sub-quotients of the l-torsion
of their cohomology in infinite level at v, with highest non degeneracy level, appears in degree 0,1.

(b) In <ref>, considering always infinite level at v, 
we analyse the torsion
cohomology classes of the graded parts ^t_!(Ψ_ϱ) of the filtration of
stratification constructed using the adjunction property j^=t_! j^=t,*→.
We then deduce, cf. lemma <ref>, that the l-torsion of 
H^i(_I^v(),s̅_v,V_ξ,_l)_𝔪 
does not have, as a _l-representation of GL_d(F_v),
any irreducible generic subquotient whose supercuspidal support is made of
characters.

(c) In section <ref>, we obtain two fondamental results.


  * First, cf. lemma <ref>,  
under the hypothesis that there exists non trivial torsion cohomology classes, we show that
the graded pieces Γ_k of the filtration of the free quotient of
H^0(_I^v(),s̅_v,V_ξ,_l)_𝔪
are not always given by the lattice of
H^0(_I,s̅_v,P_ξ(t,χ_v)(1-t+2δ/2))_𝔪⊗__l_l given by the integral cohomology of
P_ξ(t,χ_v). Roughly there exists some k and a short exact
sequence Γ_0 ↪Γ_k ↠ T where Γ_0
is the lattice given by the integral cohomology of the associated Harris-Taylor perverse sheaf, 
and T is non trivial 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 <ref>,
is that as a _l-representation
of GL_d(F_w), all the irreducible sub-quotients of the l-torsion of T, up to multiplicities,
are also sub-quotients of the l-torsion of the global cohomology. In particular,
as v and w are playing symmetric roles, these sub-quotients are not generic,
cf. corollary <ref>.

 
(d) In <ref>, the last step is to prove, under the absurd hypothesis that
there exists non trivial torsion cohomology classes while
ρ_𝔪 being irreducible, then S_𝔪(v)
contains a full set {λ q_v^n: n ∈} which is of order the order
of q_v modulo l. As this order is supposed to be strictly greater than d,
this is absurd. For more insight on the strategy to prove this fact using the previous
properties about lattices, we refer to
the introduction of <ref>.




 §.§ 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) when the level at v  
is infinite. 

 
We will denote by I^v() ∈ a finite[and morally infinite at v] 
level outside v, and we also denote by
𝔪 the maximal ideal of ^S ∪{ v }_ξ 
associated to 𝔪, i.e. we do not prescribe the modulo l Satake's at v.
Let also denote

    H^i(_I^v(),s̅_v,_l)_𝔪:= 
    lim_→I_v 
    H^i(_I,s̅_v,_l)_𝔪,

which can be viewed as a _l[GL_d(F_v)]-module.


 We have
the following resolution of p j_!*^=t HT(χ_v,Π_t)
thm1

    0 → j_!^=d HT(χ_v,Π_t {t-s/2} ) ×_d-t(χ_v{ t/2 } ))
     ⊗Ξ^s-t/2⟶⋯
    ⟶ j_!^=t+1 HT(χ_v,Π_t { -1/2 }×χ_v{ t/2 } ) 
    ⊗Ξ^1/2⟶
     j_!^=t HT(χ_v,Π_t) 
    ⟶ p j_!*^=t HT(χ_v,Π_t) → 0.




Note that


  * as this resolution is equivalent to the computation of the sheaves 
cohomology groups
of p j_!*^=h HT(χ_v,_h(χ_v)) as explained for example in
<cit.> proposition B.1.5 of appendice B, then, 
over _l, it follows from the main results of <cit.>.


  * Over _l, as every terms are free perverse sheaves, then
all the maps are necessary strict. 


  * This resolution, for a a general supercuspidal representation
with supercuspidal modulo l reduction, is one of the main result 
of <cit.> 2.3.



For the case of a character χ_v as above, the argument is almost 
obvious. Indeed as the strata
^≥ h_I^v,s̅_v,1 are smooth, then, cf. the proof of the lemma
<ref>, the constant sheaf, up to shift, is perverse and
so equals to the intermediate extension of the constant sheaf, shifted by d-h, 
on ^=h_I^v,s̅_v,1. In particular its sheaves cohomology groups
are well known so that the resolution is completely obvious for 
p j_1_h,!*^=h HT_1_h(χ_v,_h(χ_v))
if one remember that _i(χ_v) is just the character χ_v ∘
of GL_i(F_v).

The stated resolution is then simply the induced version of the resolution
of p j^=h_1_h,!* HT_1_h(χ_v,_h(χ_v)):
recall that a direct sum of intermediate extensions is still an intermediate 
extension.











By the adjunction property, the map
thm1

    j_!^=t+δ HT(χ_v,Π_t {-δ/2} ) ×_δ
    (χ_v{ t/2 } )) ⊗Ξ^δ/2
    ⟶ j_!^=t+δ-1 HT(χ_v,Π_t {1-δ/2} ) ×_δ-1(χ_v{ t/2 } )) ⊗Ξ^δ-1/2

is given by 
thm1

    HT(χ_v,Π_t {-δ/2}×_δ(χ_v{ t/2 } )) 
    ⊗Ξ^δ/2⟶
     p i^t+δ,!  j_!^=t+δ-1 HT(χ_v,Π_t {1-δ/2} ) ×_δ-1(χ_v{ t/2 } )) ⊗Ξ^δ-1/2

 We have then
thm1

    p i^t+δ,!  j_!^=t+δ-1 HT(χ_v,Π_t {1-δ/2} ) ×_δ-1(χ_v{ t/2 } )) ⊗Ξ^δ-1/2
    ≃ 
    HT  ( χ_v, Π_t {1-δ/2} )  × ( _δ-1(χ_v{ -1/2 })
     ×χ_v{δ-1/2} ) { t/2 } ) ⊗Ξ^δ/2.

Indeed one can compute p i^h+1,! j_!^=h HT(χ_v,Π_h) through
the spectral sequence associated to the exhaustive filtration of stratification of
j_!^=h HT(χ_v,Π_h) 
thm1

    (0)=^0(χ_v ,h) ↪^-d(χ_v ,h) ↪⋯
    ↪^-h(χ_v ,h)=j^=h_! HT(χ_v ,Π_h)

with graded parts, using <ref> and <cit.>,

    ^-k(χ_v,h) ≃ p j^=k_!* 
    HT(χ_v,Π_h {h-k/2}⊗_k-h(χ_v{h/2 } ))(h-k/2).
 
As remarked before the sheaf cohomology groups of 

    i^h+1,* p j_!*^h+k  HT(χ_v,Π_h { -k/2 }×_k(χ_v)(h/2))

are torsion free, so, by the Grothendieck-Verdier duality, the same is true for

    i^h+1,! p j_!*^h+k  HT(χ_v,Π_h { -k/2 }×_k(χ_v)(h/2)).

The statement follows then from the fact that, over
_l, the previous spectral sequence degenerates at E_1.

This property is also true when we replace the character χ_v by
any irreducible cuspidal representation π_v, cf. <cit.>.

Fact.
In particular, up to homothety, the map (<ref>), and so those of (<ref>), is 
unique. Finally as the map of (<ref>) are strict, the given maps (<ref>) 
are uniquely determined, that is if we forget the infinitesimal parts, these maps are independent 
of the chosen t in (<ref>).

We want now to copy the arguments of <ref>.

 
For every 1 ≤ h ≤ d, let denote by i_I^v(h) the smaller index i such that 
H^i(_I^v(),s̅_v, p j^=h_!* HT_ξ(χ_v ,Π_h ))_𝔪
has non trivial torsion: if it doesn't exists then set i_I^v(h)=+. 


By duality, as 
p j_!*^=h=p+ j^=h_!* for Harris-Taylor local systems associated to a
character, note that when i_I(h) is finite then i_I^v(h) ≤ 0. 
Note also that, using the classical determinant map, it is in fact independent of the character
χ_v. However we do not need this fact in the following.

 
Suppose there exists I ∈ such that
there exists 1 ≤ h ≤ d with i_I^v(h) finite and denote by h_0(I^v) 
the bigger such h.


 
For 1 ≤ h ≤ h_0(I^v) then i_I^v(h)=h-h_0(I^v). Moreover
_v acts by χ_v(_v) q_v^h_0(I^v)+1-h/2.




Note first that for every h_0(I^v) ≤ h ≤ s, then the cohomology groups of 
j^=h_! HT_ξ(χ_v,Π_h) are torsion free. 






The ξ-associated spectral sequence associated to the filtration
(<ref>), localized at 𝔪, is then concentrated in 
middle degree and torsion free.


Consider then the spectral sequence associated to the resolution (<ref>): its 
E_1 terms are torsion free and it degenerates at E_2. 
As by hypothesis the aims of this spectral sequence is free
and equals to only one E_2 terms, we deduce that all the maps
thm1

    H^0  (_I^v(),s̅_v,j_!^=h+δ HT_ξ(χ_v,Π_h {-δ/2} ) ×_δ (χ_v{ t/2 } )) ⊗Ξ^δ/2 )_𝔪⟶
     H^0  (_I^v(),s̅_v,
    j_!^=h+δ-1 HT_ξ(χ_v,Π_h {1-δ/2} ) 
    ×_δ-1(χ_v{ t/2 } )) ⊗Ξ^δ-1/2 )_𝔪

are strict. Then from the previous fact stressed after (<ref>), this property
remains true when we consider the associated spectral sequence for 1 ≤ h' ≤ h_0.

Consider now  h=h_0(I^v) 
where we know the torsion to be non trivial. From what was observed above
we then deduce that the map
thm1

    H^0  (_I^v(),s̅_v,j_!^=h_0(I^v)+1 HT_ξ(χ_v,Π_h_0(I^v){-1/2} ) 
    ×χ_v{ h_0(I^v)/2 } )) ⊗Ξ^1/2 )_𝔪
    ⟶  H^0  (_I^v(),s̅_v,
    j_!^=h_0(I^v) HT_ξ(χ_v,Π_h_0(I^v))  )_𝔪

has a non trivial torsion cokernel so that i_I(h_0(I^v))=0.

Finally for any 1 ≤ h ≤ h_0, the map like (<ref>) for h+δ-1 < h_0
are strict so that the H^i(_I^v(),s̅_v, p j^=h_!* 
HT_ξ(χ_v,Π_h))_𝔪 are zero for i < h-h_0 while when 
h+δ-1=h_0 its cokernel has non trivial torsion which gives
then a non trivial torsion class in H^h-h_0(_I^v(),s̅_v, p j^=h_!* 
HT_ξ(χ_v,Π_h))_𝔪.



 
With the notation of <ref>, we have h_0(I^v) ≥ h_0(I).



Consider the previous map (<ref>) 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 (<ref>) when
replacing h_0(I^v) by h_0(I), has a cokernel which is not free, then the cokernel
of (<ref>), for h_0(I), is also not free.




















The main reason to go to infinite level at v is to be able to use
the notion of level of non degeneracy which will be convenient but we will
have to deal with the possibility that h_0(I^v) might be strictly greater
than h_0(I) so that appear extra torsion classes with no interest for us.

From the previous proof, we also deduce that all cohomology classes of any of 
the H^i(_I^v(),s̅_v,P_ξ(t,χ_v))_𝔪 comes 
from the non strictness of some of
the map (<ref>) where Π_v:=_t(χ_v). In the following we will focus
on H^i(_I^v(),s̅_v,P_ξ(t,χ_v))_𝔪[l] as a 
_l-representation of GL_d(F_v). 
More precisely we are interested in irreducible such sub-quotients which have
maximal non-degeneracy level at v.


Let first fix such non degeneracy level λ for GL_d(F_v)
in the sense of notation <ref>, which is maximal for torsion classes
in H^0(_I^v(),s̅_v,P_ξ(t,χ_v))_𝔪[l] for various 1 ≤ t ≤ d and χ_v ∈_ϱ(-1).


As mentioned after <ref>, for the definition of λ, you could also
consider any fixed χ_v ∈_ϱ(-1). 

 
Let ϱ be a _l-character of F_v^× and χ_v ∈_-1(ϱ.  
Then all _l[GL_d(F_v)]-irreducible sub-quotients of
H^i(_I^v(),s̅_v,P_ξ(t,π_v))_𝔪[l], for i≠ 0,1, have a level of 
non degeneracy strictly less than λ.


Recall that we only know, a priori, that the P(t,π_v) only verify

    p j^=tg_!* HT(π_v,_t(π_v)) _+ P(t,π_v) _+
    p+ j^=tg_!* HT(π_v,_t(π_v)).



It easily follows from the observation that the level of non degeneracy of 
the modulo l reduction of _h(χ_v) ≃χ_v is strictly less than
those of the modulo l reduction of _h(χ_v) which is irreducible as the order
of q_v modulo l is strictly greater than d ≥ h.


The general case where π_v ∈_u(ϱ) for u ≥ 0
is also true. One first have to deal with
p j^=t_!* HT_ξ,_l(ϱ,Π_t)[d-t], i.e.
if H^i(_I^v(),s̅_v, p j^=t_!* HT_ξ,_l(ϱ,Π_t)
[d-t])_𝔪
and then use the equality of proposition 2.4.2 of <cit.>.


































 §.§ Global torsion and genericity



Recall that v ∈ is such that the order of q_v modulo l is strictly
greater than d. Let 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 I^v I_v  ∈
for varying I_v.




















For π_v∈_ϱ, let denote by

    ^1_!(Ψ_ϱ) ↠^1_!,π_v(Ψ_ϱ)
 
such that
^1_!,π_v(Ψ_v) ⊗__l_l ≃^1_!(Ψ_π_v)
where Ψ_π_v is the direct factor of Ψ_v ⊗__l_l
associated to π_v, cf. <cit.>. 

In the following, we will mainly be concerned with the case where π_v
is a character χ_v. We will then write the main statement in this case.

Recall the following resolution of ^1_!,χ_v(Ψ_ϱ)
thm1

    0 → j^=d_! HT(χ_v,_d(χ_v)) ⊗ (χ_v(d-1/2))
    ⟶
    
     j^=d-1_!HT(χ_v,_d-1(χ_v)) ⊗ (χ_v(d-2/2)) 
    ⟶
    ⋯⟶ j^=1_! HT(χ_v,χ_v) ⊗(χ_v)
    ⟶^1_!,χ_v(Ψ_ϱ) → 0

which is proved in <cit.> over _l. I claim it is
also true over _l. Indeed, using <ref>,
it is equivalent to the fact the sheaf cohomology of ^1_!,χ_v(Ψ_ϱ)
are torsion free which follows then from <cit.>, the comparison 
theorem of Faltings-Fargues cf. <cit.> and the main theorem
of <cit.>.
 
In <cit.>, we prove the same resolution for any irreducible
cuspidal representation π_v in place of χ_v.


As in notation <ref>, let
𝔪 be the maximal ideal of ^S ∪{ v }_ξ 
associated to 𝔪.
We can then apply the arguments of the previous section so that 
H^i(_I^v(),s̅_v,^1_!,χ_v(Ψ_ϱ,ξ))_𝔪 
has non trivial torsion for i=1-t_0
and with free quotient zero for i ≠ 0. Clearly we can also repeat the same 
arguments for the other ^t_!(Ψ_ϱ) ↠^t_!,χ_v(Ψ_ϱ) with
thm1

    0 → j^=d_! HT(χ_v,LT_χ_v(t-1,d-t)) ⊗(χ_v(d-2t+1/2))
    ⟶
    
     j^=d-1_!HT(χ_v,LT_χ_v(t-1,d-t-1)) ⊗(χ_v(d-2t/2)) 
    ⟶
    ⋯⟶ j^=t_! HT(χ_v,_t(χ_v)) ⊗(χ_v)
    ⟶^t_!,χ_v(Ψ_ϱ) → 0.

Finally all the torsion cohomology classes of the 
H^i(_I^v(),s̅_v,^t_!,χ_v(Ψ_ϱ))_𝔪 
come from the non strictness of the maps
thm1

    H^0(_I^v(),s̅_v,j^=h+1_! HT(χ_v,Π_h+1))_𝔪⟶
    H^0(_I^v(),s̅_v,j^=h_! HT(χ_v,Π_h))_𝔪

where (Π_h,Π_h+1) is of the shape ( LT_χ_v(t-1,h-t),LT_χ_v(t-1,h+1-t)  ).

We can then copy the proof of lemma <ref> which gives us the following
statement.


For every 1 ≤ h ≤ h_0, the number 
i_I(h)=h-h_0 of notation <ref>, is also the lowest integer i so that the torsion
of H^i(_I^v(),s̅_v,^h_!,χ_v(Ψ_ϱ,ξ) )_𝔪 is non zero.


 
As a _l[GL_d(F_v)]-module, for every i, the l-torsion of
H^i(_I^v(),s̅_v,V_ξ,_l)_𝔪,
does not have an irreducible generic sub-quotient
whose cuspidal support is made of characters.


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.


Recall first that, as by hypothesis ρ_𝔪 is irreducible, the
_l-version of the spectral sequence (<ref>) 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) so that the result
follows from the previous maps (<ref>) and the fact that for
any r>0, the modulo l reduction of LT_χ_v(t-1,r) does not admit any irreducible
generic sub-quotient.


We could also prove the same result without restriction on the cuspidal support but
then we would have to deal with the problem mentioned in the remark after lemma
<ref> which is the main subject of <cit.>.




 §.§ Torsion and modified lattices



Recall that we argue by absurdity, assuming there exists I_0 ∈
unramified at the place v, such that the torsion of some of
the H^i(_I_0,s̅_v,V_ξ,_l)_𝔪 is non
zero. We then denote h_0:=h_0(I_0), cf. proposition <ref>.

We want to study the free quotient of 
H^0(_I,s̅_v,V_ξ,_l)_𝔪, for some particular
level I, through the spectral sequence of vanishing cycles. We first focus 
on the cohomology of ^h_0_!,χ_v(Ψ_ϱ,ξ) for a character
χ_v ∈_-1(ϱ) which is by definition a quotient of
^h_0_!(Ψ_ϱ,ξ).
To do so, consider first the filtration constructed in <cit.>

    ^d-h_0 ( ^h_0_!,χ_v(Ψ_ϱ,ξ)) ⊂⋯⊂^0(^h_0_!,χ_v(Ψ_ϱ,ξ)),

with successive free graded parts
^i(^h_0_!,χ_v(Ψ_ϱ)) which, for i ≥ 0, 
is a _l-structure of
the _l-perverse sheaf P(h_0+i,χ_v)  (1-h_0+i/2).




For any finite level I ∈,
we then now introduce two _l-lattices of
thm1

    H^0(_I,s̅_v,P_ξ(h_0+i,χ_v))_𝔪⊗__l_l.



  * The first one denoted by Γ_ξ,χ_v,𝔪(I,h_0+i) is given by the free integral cohomology: recall that as the order of q_v modulo l
is supposed to be strictly greater than d then the modulo l reduction of
_h_0+i(χ_v) and that of χ_v[t]_D, 
remains irreducible so that, up to homothety, there is an unique
stable lattice of P_ξ(h_0+i,χ_v).


  * The spectral sequence associated to the previous filtration of 
^h_0_!,χ_v(Ψ_ξ,ϱ), provides a filtration of H^0_free
(_I,s̅_v,^h_0_!,χ_v(Ψ_ξ,ϱ))_𝔪
and Γ_ξ,χ_v,!,𝔪(I,h_0+i,h_0) is then the lattice of the
sub-quotient in this filtration corresponding to (<ref>). 


By construction we have Γ_ξ,χ_v,𝔪(I,h_0+1) 

↪Γ_ξ,χ_v,!,𝔪(I,h_0+1,h_0) 
but the cokernel of torsion might be non trivial due to torsion
in the remaining of the E_ terms.

 
Let denote by

    _v(h_0):= 
    { g ∈ GL_d(_v)  such that 
     (g ϖ_v) ∈
    P_1,2,⋯,h_0,d(κ(v)); }.



For h ≥ h_0+1, then LT_χ_v(h,d-h-1) does not have non trivial
vector invariant by _v(h_0). Moreover for π_v an irreducible
representation of GL_d-h(F_v) with h ≥ h_0+1, 
then LT_χ_v(h_0-1,h-h_0) ×π_v
admits non trivial invariants vectors by _v(h_0) if and only if
π_v is unramified.

  
With the previous notations, there exists a finite level I ∈ with
I_v ≃_v(h_0), and a short exact sequence

    0 →Γ_ξ,χ_v,𝔪(I,h_0+1) 
    
    ⟶Γ_ξ,χ_v,!,𝔪
    (I,h_0+1,h_0) ⟶   T → 0

where T ≠ (0) and every irreducible sub-quotient of its l-torsion as a 
_ξ,𝔪^S ⊗__l_l-module, 
can be obtained as a sub-quotient of the torsion submodule of 
the cokernel of 
thm1

    H^0(_I,s̅_v,j^=h_0+1_! 
    HT_ξ(χ_v,_h_0+1(χ_v)))_𝔪
    ⟶ H^0(_I,s̅_v,j^=h_0_! 
    HT_ξ(χ_v,_h_0(χ_v)))_𝔪.




The idea is to compute the cohomology of ^h_0_!(Ψ_ξ,χ_v)
in two different ways, first through the spectral sequence associated to
(<ref>) and secondly through 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^v()-level: as q_v modulo l is of order >d taking
invariant under GL_d(_v) is then an exact functor. 
First note that the I^v()-version of 
(<ref>) is non strict 
if and only if the same is true for its non induced version in the next formula,
whatever is Π_h_0 a representation of GL_h_0(F_v)
thm1

    H^0(_I^v(),s̅_v,1_h_0,j^=h_0+1_1_h_0,! HT_1_h_0,ξ
    (χ_v,Π_h_0⊗χ_v ))_𝔪
    ⟶ H^0(_I^v(),s̅_v,1_h_0,j^=h_0_1_h_0,! 
    HT_1_h_0,ξ(χ_v,Π_h_0))_𝔪,

where we denote by ^=h_0+1_I^v(),s̅_v,1_h_0 the
disjoint union of the pure strata, cf. notation <ref>, 
^=h_0+1_I^v(),s̅_v,g
contained in ^≥ h_0_I^v(),s̅_v,1_h_0. As usual
the notation j^=h_0+1_1_h_0 designates the closed embedding
of ^=h_0+1_I^v(),s̅_v,1_h_0 in ^≥ 1_I^v(),s̅_v.
Recall that h_0 in <ref>, is chosen so that, as a _l[P_h_0,d(F_v)]-module, for Π_h_0 unramified, using also 
the fact that q_v  modulo l is of order >d so that the functor of P_h_0,d(_v)-invariants is exact, then the cokernel of (<ref>)
has non trivial vectors invariant under P_h_0,d(_v). We then deduce
the following facts, cf. also the remark after <ref>.

 


  * With Π_h_0=_h_0(χ_v), the cokernel of the induced version of
(<ref>) has non trivial vectors invariant 
under _v(h_0).


  * 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
thm1

    H^0(_I^v(),s̅_v,j^=h+1_! 
    HT_ξ(χ_v,Π_h+1 ))_𝔪
    ⟶ H^0(_I^v(),s̅_v,j^=h_! 
    HT_ξ(χ_v,Π_h))_𝔪,

does not have non zero invariant vector under GL_d(_v).


  * For Π_h=LT_χ_v(h_0-1,h-h_0) and 
Π_h+1=LT_χ_v(h_0-1,h-h_0+1), as in (<ref>),
the cokernel of the previous point does not have non zero invariant vector under
_v(h_0).



We then compute the 𝔪-localized cohomology of 
^h_0_!,χ_v(Ψ_ϱ,ξ)
in level I^v() having non trivial invariant under I with 
I_v ≃_v(h_0). 
By maximality of h_0, note that for h_0<t ≤ d, the cohomology groups of
P_ξ(t,χ_v) and j^=t_! HT_ξ(π_v,LT_χ_v(h_0-1,t-h_0)),
after localization by 𝔪, do no
have non trivial torsion vector invariant under _v(h_0) as explained in
the previous lemma.

(1) Following the proof of <ref> with the spectral sequence 
associated to 
(<ref>) and neglecting torsion classes
which do not have non trivial vectors invariants by _v(h_0), 
we then deduce that the
H^i_tor(_I^v(),s̅_v,
^h_0_!(Ψ_ξ,χ_v))_𝔪 do not have
non trivial vector invariant under _v(h_0) if i ≠ 0,1
while for i=0 the torsion is non trivial and the
vectors invariant by _v(h_0) are given by the non strictness of 
thm1

    H^0(_I^v(),s̅_v,j^=h_0+1_! HT_ξ(χ_v,LT_χ_v(h_0-1,1)))_𝔪
    ⟶ H^0(_I^v(),s̅_v,j^=h_0_! 
    HT_ξ(χ_v,_h_0(χ_v)))_𝔪.


(2) Concerning H^0(_I^v(),s̅_v,
P_ξ(χ_v,h_0))_𝔪, 
its torsion submodule is
parabolicaly induced, so that beside those coming from the non strictness of (<ref>),
there is also the contribution given by the non strictness of (<ref>), which 
contains in particular a subquotient, denoted T, 
such that T[l] is of level of non degeneracy strictly greater than those appearing 
in (<ref>). Note moreover that T[l]
has non trivial vectors under _v(h_0).

(3) Consider then the cohomology of ^h_0_!,χ_v(Ψ_ϱ,ξ)  
computed through
its filtration of stratification with graded parts, up to Galois shifts, the 
H^0(_I^v(),s̅_v,P_ξ(h_0+k,χ_v))_𝔪, 
for 0 ≤ k ≤ d-h_0,
and more particularly the induced filtration of the free quotient of 
H^0(_I^v(),s̅_v,^h_0_!,χ_v(Ψ_ϱ,ξ))_𝔪  as before.
As the level of non degeneracy of T[l] is higher than those of the l-torsion
of H^0(_I^v(),s̅_v,^h_0_!,χ_v(Ψ_ϱ,ξ))_𝔪, 
computed through the spectral sequence associated to (<ref>),
we then have a filtration of this free quotient where both appears


  * torsion modules such as T,


  * and the free sub-quotients are given by the lattices Γ_ξ,χ_v,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 know go back to the level I=I^v_v(h_0): as q_v modulo l is of order strictly
greater than d, the functor of _v(h_0)-invariants is exact. As only contributes the
cohomology of P_ξ( χ_v,h_0+i) for i=0,1 the result follows from
the fact that T has non trivial invariant under _v(h_0).


Recall that 

    ^h_0_!(Ψ_ϱ) ⊗__l_l ≃⊕_χ_v ∈_ϱ^h_0_!(Ψ_χ_v)

so that we can find a filtration of ^h_0_!(Ψ_ϱ) whose graded parts are
free and isomorphic, after tensoring with _l, to 
^h_0_!(Ψ_χ_v).



Arguing as in the proof of lemma <ref>, using (<ref>), 
we have the following result.

 
For every 1 ≤ t, let j(t) be the minimal integer j such that the torsion of
H^j(_I^v(),s̅_v,^t_!(Ψ_ξ,ϱ))_𝔪 has 
non trivial invariant vectors under _v(h_0). Then

    j(t)={[                 +    if  t ≥ h_0+1,;             t-h_0 for  1 ≤ t ≤ h_0. ].

Moreover as a _ξ,𝔪⊗__l_l-module,
up to multiplicities, the irreducible sub-quotients of 
H_tor^j(t)(_I^v_v(h_0),s̅_v,^t_!(Ψ_ξ,ϱ))_𝔪 are
independent of t.
 


Important fact:
Note that, up to multiplicities, the irreducible _ξ,𝔪⊗__l_l-sub-quotients[i.e. if one forget
the action of GL_d(F_v)]
of the l-torsion of the cohomology of ^h_0_!,χ_v(Ψ_ϱ,ξ) and
P_ξ(h_0,χ_v) in level I^v_v(h_0) 
are the same, given by the non strictness of the 
maps (<ref>).

The idea is now to increase the level at another place w ∈(I) verifying the same
hypothesis than v, i.e. q_w modulo l is of order strictly greater than d.
When the level at w is infinite, i.e. for I^w() with I_v ≃_v(h_0), using
again the exactness of invariant with GL_d(_w), from the previous observation we
deduce that considering irreducible _l-representations of
GL_d(F_w), the sets, i.e. without multiplicities, of irreducible sub-quotients of the l-torsion
of the cohomology of respectively P_ξ(h_0,χ_v) and 
^h_0_!,χ_v(Ψ_ϱ,ξ), are the same.


 
Up to multiplicities, the set of irreducible
_l[GL_d(F_w)]-sub-quotients of the l-torsion of[or
those of H^0(_I^w(),s̅_v,P_ξ(h_0,χ_v))_𝔪
as explained above]
H^0(_I^w(),s̅_v,^h_0_!(Ψ_ξ,ϱ))_𝔪,
are the same as those of
H^d-h_0(_I^w(),s̅_v,V_ξ,_l)_𝔪.



We compute H^d-h_0(I^v)(_I^w(),s̅_v,V_ξ,_l)_𝔪
using the filtration ^∙_!(Ψ_ξ,ϱ) through the spectral sequence 
(<ref>). 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 there are zero for
p+q < j(p):=p-h_0. Note then that
E_!,ϱ,1^-1,j(1)+1 which is torsion and non zero, according to the previous lemma,
is equal to E_!,ϱ,^j(1)≃
H^d-h_0(_I^w(),s̅_v,V_ξ,_l)_𝔪.


Consider as before I^w() such that its local component at v is 
_v(h_0).
Then combining the result of lemma <ref> in level I^w(), with the previous
proposition, we then deduce that the cokernel T of <ref> verifies
the following property. As a _l-representation of GL_d(F_w), every irreducible
sub-quotient of T[l] is also a sub-quotient of
H^d-h_0(_I^w(),s̅_v,V_ξ,_l)_𝔪.
Then applying lemma <ref> at the place w playing a symmetric role
as v, we then deduce the following result.

 
As a _l-representation of GL_d(F_w), the l-torsion of 
 the cokernel T of lemma <ref> in level I^w() as above,
does not contain any irreducible generic sub-quotient with cuspidal support made of characters. 


- We can now repeat the arguments with ^k_!,χ_v(Ψ_ϱ,ξ) for any
1 ≤ k ≤ h_0. More precisely, cf. the last remark of <ref>, 
consider ^i(^k_!,χ_c(Ψ_ϱ,ξ))
for i=h_0-k and i=h_0-k+2. Take a level 

    I=I^v,wI_w() _v(h_0),
 
infinite at w and Iwahori at v,
such that, arguing by absurdity, the torsion of 
H^d-1(_I^v,w GL_d(_w ⊗_v),η̅,V_ξ,_l)_𝔪
is non trivial[For the definition of h_0:=h_0(I^v), cf. notation <ref>.].

There exists 𝔪⊂𝔪 such that

    Π_𝔪≃_h_0+1(χ_v) ×χ_v,1×⋯×χ_v,d-h_0-1,

with χ_v ≡ϱ l. 


















Moreover as before


  * ^h_0+2-k(^k_!,χ_v(Ψ_ϱ,ξ))
has trivial cohomology groups in level I because I_v=_v(h_0) and
the irreducible constituents of 
^h_0+2-k(^k_!,χ_v(Ψ_ϱ,ξ)) 
⊗__l_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 
^h_0_!,χ_v(Ψ_ξ,ϱ) to the 
quotient Q:=^h_0-k(^k_!,χ_v(Ψ_ϱ,ξ)) 
/^h_0+2-k(^h_0_!,χ_v(Ψ_ϱ,ξ)), so that, denoting by
Γ'_ξ,χ_v,!,𝔪 (I,h_0+1,k)
the lattice of (<ref>) given by the free quotient of 
H^0(_I^w(),s̅_v,Q)_𝔪, the cokernel T'_k of

    0 →Γ_ξ,χ_v,𝔪(I,h_0+1) 
    ⟶Γ'_ξ,χ_v,!,𝔪
    (I,h_0+1,k) ⟶   T'_k → 0

is such that T'_k[l] ≠ (0) and, as a _l-representation of 
GL_d(F_w), it does not contain any irreducible generic sub-quotient made of 
characters.


  * In addition of the previous arguments, we also have to deal with the
torsion in the cohomology groups of 

    ^k_!,χ_v(Ψ_ϱ,ξ) /^h_0-k(^k_!,χ_v(Ψ_ϱ,ξ)),
 
which could modify the lattice Γ'_ξ,χ_v,!,𝔪
(I,h_0+1,k) to give the good one denoted above by 
Γ_ξ,χ_v,!,𝔪(I,h_0+1,k) . Note
again that,
as a _l-representation of GL_d(F_w), this l-torsion
does not contain any
irreducible generic sub-quotient made of characters, so the cokernel of 

    Γ'_ξ,χ_v,!,𝔪 (I,h_0+1,k)
    ↪Γ_ξ,χ_v,!,𝔪(I,h_0+1,k),

is again such that,  as a _l-representation of 
GL_d(F_w), its l-torsion does not contain any irreducible generic 
sub-quotient made of characters.

Forgetting again Galois shifts, we then conclude that the l-torsion of
the cokernel T_k of
thm1

    0 →Γ_ξ,χ_v,𝔪(I,h_0+1) 
    ⟶Γ_ξ,χ_v,!,𝔪
    (I,h_0+1,k) ⟶   T_k → 0,

is non zero and,
as a _l-representation of GL_d(F_w), it
does not contain any
irreducible generic sub-quotient made of characters.



- We now compute
H^d-1(_I,η̅,V_ξ,_l)_𝔪 through the spectral
sequence of vanishing cycles using the filtration

    ^1_!(Ψ_ϱ) ⋯^d-1_!(Ψ_ϱ) ^d_!(Ψ_ϱ) Ψ_ϱ,

and with level I=I^v,wI_w() _v(h_0) 
infinite at w and Iwahori at v. Recall that we can filtrate each of the
^k_!(Ψ_ϱ) as follows:


  * Define first ^∙(^k_!(Ψ_ϱ)) such that

    ^r(^k_!(Ψ_ϱ)) ⊗__l_l
    ≃⊕_π_v ∈_r(ϱ) P(π_v,t),

where t is such that k=tg_r(ϱ).


  * Then each of the previous ^r(^k_!(Ψ_ϱ)) has a filtration
with graded parts such that over _l we recover, up to Galois shift, 
P(π_v,t).


As we choose the level Iwahori at v, when computing the cohomology
we are only concerned with characters χ_v ∈_-1(ϱ).


As before, arguing by absurdity, 
we suppose that the torsion of 
H^d-1(_I^v,w GL_d(_w ⊗_v),η̅,V_ξ,_l)_𝔪
is non trivial[For the definition of h_0:=h_0(I^v), cf. notation <ref>.], and we pay special attention to the lattices of 
thm1

    V_ξ,χ_v,𝔪(I,h_0+1)(δ):=
    
    H^i(_I,s̅,P_ξ(h_0+1,χ_v))(δ)_𝔪⊗__l_l,

for χ_v ∈_-1(ϱ) and various δ.

We first start with δ=-h_0/2. Note that the 
H^i(_I,s̅_v,Ψ_ξ,ϱ/^h_0+1_!(Ψ_ξ,ϱ))_𝔪
are all zero. Indeed the torsion free graded parts ^k(Ψ_ϱ) 
of any exhaustive filtration of 
Ψ_ϱ / ^h_0+1_!(Ψ_ϱ), up to Galois torsion, are such that
^k(Ψ_ϱ) ⊗__l_l ≃ P(t,π_v) 
whith π_v ∈_ϱ
an irreducible cuspidal representation of some GL_g(F_v) with tg >h_0+1. Then
every irreducible constituant of 
H^i(_I^v(),s̅_v,^k(Ψ_ξ,ϱ))_𝔪⊗__l_l,
as a _l-representation of GL_d(F_v) is a sub-quotient of an induced 
representation
r_l(_t(π_v)){δ/2 }×τ for some irreducible _l-representation
τ of GL_d-tg(F_v). In particular such a representation does not have non trivial
invariants under _v(h_0), so that, as the functor of _v(h_0)-invariants is exact,
there is no cohomology in level I as stated.

We then deduce that
H^0(_I,s̅,P_ξ,_l(h_0+1,χ_v))(δ)_𝔪.

is a quotient of  
H^0(_I,s̅_v,Ψ_ξ,ϱ)_𝔪⊗__l_l
and we denote by Γ_ξ,χ_v,Ψ,𝔪(I,h_0+1,+) 

its stable lattice induced by 
H^0_free(_I,s̅_v,Ψ_ξ,ϱ)_𝔪.



 
With the previous notations, we have

    0 →Γ_ξ,χ_v,Ψ,𝔪(I,h_0+1,+)
    ⟶Γ_ξ,χ_v,𝔪(I,h_0+1) ⟶ T 
    → 0,

where T[l], as a 𝔽_l-representation of GL_d(F_w),
does not have any irreducible generic sub-quotient with cuspidal support
made of characters.





We compute H^0(_I,s̅_v,Ψ_ξ,ϱ)_𝔪 through 
the spectral sequence associated to the filtration

    ^1_!(Ψ_ξ,ϱ) ⋯^d-1_!(Ψ_ξ,ϱ)
    ^d_!(Ψ_ξ,ϱ) Ψ_ξ,ϱ.

Recall that H^0(_I,s̅_v,Ψ_ξ,ϱ)_𝔪=
H^0(_I,s̅_v,_!^h_0+1(Ψ_ξ,ϱ)_𝔪.
Let denote

    K_0 = (^h_0+1_!(Ψ_ξ,ϱ) ↠ 
    P_ξ(h_0+1,χ_v)(-h_0/2)).

Recall that over _l, all the cohomology groups are concentrated
in degree zero. We then have

    0 →Γ_ξ,χ_v,𝔪(I,h_0+1,+) ⟶Γ_ξ,χ_v,Ψ,𝔪(I,h_0+1)
    ⟶  T → 0,

where T ↪ H^1_tor(_I,s̅_v, K_0)_𝔪.
The statement about T[l] then follows from the previous section.












Consider now the case δ=h_0/2 and denote as before by
Γ_ξ,χ_v,Ψ,𝔪(I,h_0+1,-) the lattice of 
V_ξ,χ_v,𝔪(I,h_0+1)(h_0/2) induced by the free quotient
of H^0(_I,s̅_v,Ψ_ϱ,ξ)_𝔪.
Consider ^1_!,χ_v(Ψ_ϱ,ξ) ^1_!(Ψ_ϱ,ξ)
Ψ_ϱ,ξ.

Before ^1_!,χ_v(Ψ_ϱ,ξ) was defined as a quotient of
^1_!(Ψ_ϱ,ξ). To separate the χ_v ∈_-1(ϱ)
in ^1_!(Ψ_ϱ,ξ) we start with a filtration of
j^=1,*^1_!(Ψ_ϱ,ξ) with

    j^=1,*^1_!(Ψ_ϱ,ξ) ⊕_χ_v ∈_-1(ϱ)
    HT_ξ(χ_v,χ_v),

by considering an numbering of _-1(ϱ). In particular for a fixed
χ_v, when it appears in first or last position, we may to modify the lattice
of HT(χ_v,χ_v). But as, up to homothety, HT(χ_v,χ_v) has an unique
stable lattice, we then obtain the same ^1_!,χ_v(Ψ_ϱ,ξ).

We then has seen that the lattice Γ_ξ,χ_v,!,𝔪(I,h_0+1,1)
is such that
thm1

    0 →Γ_ξ,χ_v,𝔪(I,h_0+1) 
    ⟶Γ_ξ,χ_v,!,𝔪
    (I,h_0+1,1) ⟶   T → 0,

where T ≠ (0) is such that T[l], as a _l-representation of
GL_d(F_w), does not contain any
irreducible generic sub-quotient made of characters.
We then conclude by observing that

    Γ_ξ,χ_v,Ψ,𝔪(I,h_0+1,-)=
    Γ_ξ,χ_v,!,𝔪(I,h_0+1,1).


From proposition <ref> and (<ref>),
as the modulo l reduction of 
Γ_ξ,χ_v,Ψ,𝔪 (I,h_0+1,-)
has a generic sub-quotient, it is not isomorphic to 
Γ_ξ,χ_v,Ψ,𝔪 (I,h_0+1,+).




 §.§ Global lattices and generic representations








We argue on the set S_𝔪(v) of modulo l eigenvalues of 
ρ_𝔪(_v). 
By hypothesis there exists λ:=χ_v(ϖ_v) such that 

    {λ q_v^h_0/2,λ q_v^h_0/2-1,⋯,λ q_v^-h_0/2}⊂ S_𝔪(v),
 
where h_0 ≥ 1 is defined before under the hypothesis there exists non
trivial torsion. More precisely λ q_v^h_0/2 is the modulo l reduction
of the eigenvalue of _v acting on P_ξ(χ_v,h_0+1)(h_0/2)
such that the torsion of H^0(_I,s̅_v,P_ξ(χ_v,h_0)(h_0/2))_𝔪
is non zero.

We then start with any λ_0 q_v^h_0/2∈ S_𝔪(v) and
we want to prove that S_𝔪(v) contains a subset 
{λ_1 q_v^h_0/2,λ_0 q_v^h_0/2-1,⋯,λ_1 
q_v^-h_0/2} such that


  * there exists 0 < r < h_0+1 with 
λ_1 q_v^h_0/2=λ_0 q_v^h_0/2+r,


  * and λ_1 q_v^h_0/2 is the modulo l reduction of the eigenvalue 
of _v acting on P_ξ(χ'_v,h_0+1)(h_0/2) such that the torsion of 
H^0(_I,s̅_v,P_ξ(χ'_v,h_0)(h_0/2))_𝔪
is non zero.

We then obtain another interval inside S_𝔪(v) 
containing strictly the previous one and
we can then play again with λ_1 q_v^h_0 and repeat the above
property. At the end we then obtain the full set {λ_0 q_v^n: n ∈}
which is of order the order of q_v modulo l. But this order is by hypothesis
strictly greater than d although trivially the set S_𝔪(v) 
of eigenvalues of ρ(_v) is of order ≤ d.


We now explain how to increase the interval as stated above. To do so
start first with a classical fact concerning _l[G]-modules where
G is a group. Let then Γ, Γ_1 and Γ_2 three 
_l-free modules with an action of a group G such that
thm1

    0 →Γ_1 ⟶Γ⟶Γ_2 
    → 0,

which is G equivariant. We then suppose that this extension is split
over _l, i.e. 

    Γ⊗__l_l
    ≃ (Γ_1 ⊗__l_l ) ⊕
    (Γ_2 ⊗__l_l).

Let then denote by 
Γ'_2:=Γ∩ (Γ_2 ⊗__l_l)
and Γ'_1:=Γ/Γ_2' so that

    0 →Γ'_2 ⟶Γ⟶Γ'_1 
    → 0.

We then have the following commutative diagram
thm1

    Γ_1 @^(->[d] @=[r]    Γ_1 @^(->[d] 
    Γ'_2 @^(->[r] @=[d]    Γ@->>[r] @->>[d]    Γ'_1 @->>[d]  
    Γ'_2 @^(->[r]    Γ_2 @->>[r]     T,

where T is of torsion and zero if and only if (<ref>) is split, i.e.
Γ'_1=Γ_1 and Γ'_2=Γ_2.

Important remark:
in the following we will consider _I,ξ,𝔪[_F,S]-free modules
when Γ_i ⊗__l_l for i=1,2
are isotypic for the Galois action relatively to a character χ_i
such that χ_1 ≄χ_2. From the
previous diagram, then T=0 and Γ≃Γ_1 ⊕Γ_2.


The idea is to consider two distincts filtrations 
Γ:=H^0_free(_I,s̅_v,Ψ_ξ,ϱ)_𝔪,
and use the previous diagram to explain that, upon the hypothesis that the
torsion is non zero, we could not go through one to the other.
Recall first that

    Γ⊗__l_l ≃⊕_𝔪Π_𝔪^I ⊗ρ_𝔪,

so that, by fixing any numbering 

    {𝔪⊂𝔪 s.t. Π_𝔪^I  ≠ (0) }=
    {𝔪_1,⋯, 𝔪_r },

we define a filtration ^∙(Γ) with graded parts
^k(Γ) which is a stable lattice of
Π^I_𝔪_k⊗ρ_𝔪_k.
With previous notation, we suppose that Π_𝔪_r,v≃_h_0+1(χ_v) ×χ_v,1×⋯×χ_v,d-h_0-1
with χ_v ∈_-1(ϱ). As the modulo l reduction of
ρ_𝔪_r is irreducible, then 
^r(Γ) is typic, in the sense of <cit.> 5, i.e.

    ^r(Γ) ≃Γ_r ⊗Λ_r,

where Γ_r is a stable lattice of Π_𝔪_r on
which the Galois action is trivial and Λ_r is a stable lattice of
ρ_𝔪_r.

We now start from the filtration of Γ induced by the filtration
^∙_!(Ψ_ϱ) and we try, using diagrams like above, to end up
to the previous lattice ^r(Γ) ≃Γ_r ⊗Λ_r.
As explained in the previous section, Γ_ξ,χ_v,Ψ,𝔪
(I,h_0+1,+) is a quotient of Γ and so a quotient of ^r(Γ)
so that 

    Γ_r ≃Γ_ξ,χ_v,Ψ,𝔪 (I,h_0+1,+).

At the opposite, we see 
Γ_ξ,χ_v,Ψ,𝔪 (I,h_0+1,-)
as a quotient of 
H^0_free(_I,s̅_v,^1_!(Ψ_ξ,ϱ))_𝔪.
As noticed in the last remark of the previous section, 
Γ_ξ,χ_v,Ψ,𝔪 (I,h_0+1,-)
and Γ_ξ,χ_v,Ψ,𝔪 (I,h_0+1,+)
are not isomorphic.
We then deduced that there should exists χ'_v ∈_-1(ϱ)
and a diagram <ref> with 
Γ'_2=Γ_ξ,χ_v,Ψ,𝔪 (I,h_0+1,-)
and Γ'_1 associated to a sub-quotient of H^0(_I,s̅_v,
P_ξ(χ'_v,t)(t-1/2-δ))_𝔪. Note that
as this P_ξ(χ'_v,t)(t-1/2- δ) is a sub-quotient of some
^k_!(Ψ_ξ,ϱ) with k ≥ 2, then we must have 
2 ≤ t ≤ h_0+1 and δ>0.

In particular, from the previous important remark, 
we also deduce that χ'_v(t-1/2-δ) ≡χ_v(h_0/2)  l so that, by denoting λ_1=χ'_v(ϖ_v),
we can write λ_1 q_v^h_0/2=λ_0 q_v^h_0/2+r
with 0 < r <h_0+1.

To be able to play again with χ'_v in place of χ_v, we just need to prove
that the 𝔪-localized
cohomology groups of P_ξ(χ'_v,h_0) with level I, have torsion in degree
0 and 1. It is equivalent to look at the cohomology of 
P_ξ(χ'_v,h_0)(h_0+1-t+2δ/2)=P_x(χ_v,1,h_0)
with χ_v,1:=χ'_v (h_0+1-t+2δ/2) verifying 
χ_v,1≡χ_v  l.
Recall the well-known short exact sequence

    0 → H^i(X,P) ⊗__l_l ⟶
    H^i(X,P ⊗__l_l) ⟶
    H^i+1(X,P)[l] → 0.

We apply it to X=_I,s̅_v and P=P_ξ(χ_v,1,h_0), and we
recall that its _l-cohomology localized at 𝔪,
is concentrated in degree 0.
The same is true for P_ξ(χ_v,h_0) while it has torsion in degree 0 and 1
such that its _l-cohomology localized at 𝔪, is 
concentrated in degree -1,0,1.
The same is then true for the _l-cohomology, 
localized at 𝔪, of P_ξ(χ_v,1,h_0)
so that its _l-cohomology localized at 𝔪, must have 
torsion in degree 0 and 1.

 















































































































































































































































































































































































































































































































































































































































































































































































































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