# From local to global congruences for automorphic representations

Boyer Pascal<EMAIL_ADDRESS>Université Paris 13, Sorbonne Paris
Nord
LAGA, CNRS, UMR 7539
F-93430, Villetaneuse (France)
Coloss: ANR-19-PRC

###### Abstract

Given a irreducible automorphic representation $\Pi$ of a similitude group
$G/{\mathbb{Q}}$ giving rise to a KHT-Shimura variety, given a local
congruence of the local component of $\Pi$ at a fixed place $p$, we justify
the existence of a global automorphic representation $\Pi^{\prime}$ with the
same weight and the same level outside $p$ than $\Pi$, such that $\Pi$ and
$\Pi^{\prime}$ are weakly congruent. The arguments rest on the separation of
the various contributions coming either from torsion or on the distinct
families of automorphic representations, to the modulo $l$ reduction of the
cohomology of Harris-Taylor perverse sheaves.

###### Key words and phrases:

Shimura varieties, torsion in the cohomology, maximal ideal of the Hecke
algebra, localized cohomology, galois representation

###### 1991 Mathematics Subject Classification:

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

###### Contents

  1. 1 Introduction
  2. 2 Notations about representations of $GL_{n}(K)$
  3. 3 KHT-Shimura varieties and Harris-Taylor local systems
  4. 4 $\overline{\mathbb{Q}}_{l}$-cohomology groups
  5. 5 Integral Harris-Taylor perverse sheaves
  6. 6 Automorphic congruences

##  1\. Introduction

The first appearance of automorphic congruences can be trace back to
Ramanujan’s works on the $\tau$-functions. Now existence and construction of
higher dimension automorphic congruences play an essential role in particular
in the Langlands program.

One possible approach is to look at the cohomology groups of some
$\overline{\mathbb{Z}}_{l}$-local system on a Shimura variety associated to
some reductive group $G/{\mathbb{Q}}$, whose free quotients are expected to be
of automorphic nature. The idea is then to take two (or two families of) such
local systems those modulo $l$ reduction is related and be able to relate the
modulo $l$ reduction of their cohomology groups. We then face two main
problems.

  * (a)

The torsion may interfere and prevent us to relate the modulo $l$ reduction of
the free quotients of the cohomology groups of our two
$\overline{\mathbb{Z}}_{l}$-local systems.

  * (b)

Even if we can manage about the torsion, the
$\overline{\mathbb{Q}}_{l}$-cohomology of our local systems may involve
different sorts of automorphic representations and we would only be able to
construct imprecise automorphic congruences.

In this article we are able, playing with all the Harris-Taylor local systems
associated to one cuspidal representation, to deal with this two problems when
$G$ is a similitude group with signature $(1,d-1)$ and the associated Shimura
variety is of Kottwitz-Harris-Taylor type. More precisely we first prove the
conjecture 5.10 of [4] which says that the modulo $l$ reduction of the torsion
submodule of the cohomology groups of the Harris-Taylor perverse sheaves only
depends on the modulo $l$ reduction of the perverse sheaf. Secondly we are
able to separate the contribution of the automorphic representations according
to the shape of their local component at the place considered, in the
following sense.

Main result: We start with a irreducible automorphic representation $\Pi$ of
$G({\mathbb{A}})$ which locally at some place $v$ looks like
$\mathop{\mathrm{Speh}}\nolimits_{s}\bigl{(}{\mathop{\mathrm{St}}\nolimits}_{t_{1}}(\pi_{v,1})\times\cdots{\mathop{\mathrm{St}}\nolimits}_{t_{r}}(\pi_{v,r})\bigr{)}$
where $\pi_{v,1},\cdots,\pi_{v,r}$ are irreducible cuspidal representations.
We then consider some local congruence $\pi^{\prime}_{v,1}$ of $\pi_{v,1}$
which we suppose to be supercuspidal modulo $l$, and we then construct a
irreducible automorphic representation $\Pi^{\prime}$ of $G({\mathbb{A}})$
such that

  * •

locally $\Pi^{\prime}$ at the place $v$ is isomorphic to
$\mathop{\mathrm{Speh}}\nolimits_{s}({\mathop{\mathrm{St}}\nolimits}_{t^{\prime}_{1}}(\pi^{\prime}_{v,1})\times\cdots\times{\mathop{\mathrm{St}}\nolimits}_{t^{\prime}_{r^{\prime}}}(\pi^{\prime}_{v,r^{\prime}})$
with $t^{\prime}_{1}=t_{1}$ and $\pi^{\prime}_{v,i}$, for
$i=2,\cdots,r^{\prime}$ irreducible cuspidal representations,

  * •

globally $\Pi^{\prime}$ share with $\Pi$ the same weight and the same level
outside $v$.

More precisely theorem 6.2 gives a quantitative version in terms of a equality
between multiplicities in the space of automorphic forms.

###### Contents

  1. 1 Introduction
  2. 2 Notations about representations of $GL_{n}(K)$
  3. 3 KHT-Shimura varieties and Harris-Taylor local systems
  4. 4 $\overline{\mathbb{Q}}_{l}$-cohomology groups
  5. 5 Integral Harris-Taylor perverse sheaves
  6. 6 Automorphic congruences

##  2\. Notations about representations of $GL_{n}(K)$

We fix a finite extension $K/{\mathbb{Q}}_{p}$ with residue field
${\mathbb{F}}_{q}$. We denote by $|-|$ its absolute value.

######  Definition 2.1.

Two representations $\pi$ and $\pi^{\prime}$ of $GL_{n}(K)$ are said
inertially equivalent and we denote by $\pi\sim^{i}\pi^{\prime}$, if there
exists a character
$\chi:{\mathbb{Z}}\longrightarrow\bar{\mathbb{Q}}_{l}^{\times}$ such that

$\pi\simeq\pi^{\prime}\otimes(\chi\circ\mathop{\mathrm{val}}\nolimits\circ\mathop{\mathrm{det}}\nolimits).$

We then denote by $e_{\pi}$ the order of the set of characters
$\chi:{\mathbb{Z}}\longrightarrow\bar{\mathbb{Q}}_{l}^{\times}$, such that
$\pi\otimes\chi\circ\mathop{\mathrm{val}}\nolimits(\mathop{\mathrm{det}}\nolimits)\simeq\pi$.

For a representation $\pi$ of $GL_{d}(K)$ and $n\in\frac{1}{2}{\mathbb{Z}}$,
set

$\pi\\{n\\}:=\pi\otimes
q^{-n\mathop{\mathrm{val}}\nolimits\circ\mathop{\mathrm{det}}\nolimits}.$

######  Notations 2.2.

For $\pi_{1}$ and $\pi_{2}$ representations of respectively $GL_{n_{1}}(K)$
and $GL_{n_{2}}(K)$, we will denote by

$\pi_{1}\times\pi_{2}:=\mathop{\mathrm{ind}}\nolimits_{P_{n_{1},n_{1}+n_{2}}(K)}^{GL_{n_{1}+n_{2}}(K)}\pi_{1}\\{\frac{n_{2}}{2}\\}\otimes\pi_{2}\\{-\frac{n_{1}}{2}\\},$

the normalized parabolic induced representation where for any sequence
$\underline{r}=(0<r_{1}<r_{2}<\cdots<r_{k}=d)$, we write $P_{\underline{r}}$
for the standard parabolic subgroup of $GL_{d}$ with Levi

$GL_{r_{1}}\times GL_{r_{2}-r_{1}}\times\cdots\times GL_{r_{k}-r_{k-1}}.$

Recall that a representation $\varrho$ 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 $\overline{\mathbb{F}}_{l}$.

######  Definition 2.3.

(see [8] §9 and [2] §1.4) Let $g$ be a divisor of $d=sg$ and $\pi$ an
irreducible cuspidal $\overline{\mathbb{Q}}_{l}$-representation of
$GL_{g}(K)$. The induced representation

$\pi\\{\frac{1-s}{2}\\}\times\pi\\{\frac{3-s}{2}\\}\times\cdots\times\pi\\{\frac{s-1}{2}\\}$

holds a unique irreducible quotient (resp. subspace) denoted
${\mathop{\mathrm{St}}\nolimits}_{s}(\pi)$ (resp.
$\mathop{\mathrm{Speh}}\nolimits_{s}(\pi)$); it is a generalized Steinberg
(resp. Speh) representation.

Any generic irreducible representation $\Pi$ of $GL_{d}(K)$ is isomorphic to
${\mathop{\mathrm{St}}\nolimits}_{t_{1}}(\pi_{1})\times\cdots\times{\mathop{\mathrm{St}}\nolimits}_{t_{r}}(\pi_{r})$
where for $i=1,\cdots,r$, the $\pi_{i}$ are irreducible cuspidal
representations of $GL_{g_{i}}(K)$ and $t_{i}\geq 1$ are such that
$\sum_{i=1}^{r}t_{i}g_{i}=d$. For $\Pi$ a irreducible generic representation
and $s\geq 1$, we denote by

$\mathop{\mathrm{Speh}}\nolimits_{s}(\Pi)=\mathop{\mathrm{Speh}}\nolimits_{s}\bigl{(}{\mathop{\mathrm{St}}\nolimits}_{t_{1}}(\pi_{1})\bigr{)}\times\cdots\times\mathop{\mathrm{Speh}}\nolimits_{s}\bigl{(}{\mathop{\mathrm{St}}\nolimits}_{t_{r}}(\pi_{r})\bigr{)}$

the Langlands quotient of the parabolic induced representation
$\Pi\\{\frac{1-s}{2}\\}\times\Pi\\{\frac{3-s}{2}\\}\times\cdots\times\Pi\\{\frac{s-1}{2}\\}$.
In terms of the local Langlands correspondence, if $\sigma$ is the
representation of ${\mathop{\mathrm{Gal}}\nolimits}(\bar{F}/F)$ associated to
$\Pi$, then $\mathop{\mathrm{Speh}}\nolimits_{s}(\Pi)$ corresponds to
$\sigma\oplus\sigma(1)\oplus\cdots\oplus\sigma(s-1)$.

######  Definition 2.4.

Let $D_{K,d}$ be the central division algebra over $K$ with invariant $1/d$
and with maximal order denoted by ${\mathcal{D}}_{K,d}$.

The local Jacquet-Langlands correspondance is a bijection
$\mathop{\mathrm{JL}}\nolimits$ between the set of equivalence classes of
irreducible admissible representations of $D_{K,d}^{\times}$ and the one of
irreducible admissible essentially square integrable representations of
$GL_{d}(K)$.

######  Notation 2.5.

For $\pi$ a cuspidal irreducible $\bar{\mathbb{Q}}_{l}$-representation of
$GL_{g}(K)$ and for $t\geq 1$, we then denote by $\pi[t]_{D}$ the
representation
$\mathop{\mathrm{JL}}\nolimits^{-1}({\mathop{\mathrm{St}}\nolimits}_{t}(\pi))^{\vee}$
of $D_{K,tg}^{\times}$.

##  3\. KHT-Shimura varieties and Harris-Taylor local systems

Let $F=F^{+}E$ be a CM field where $E/{\mathbb{Q}}$ is quadratic imaginary and
$F^{+}/{\mathbb{Q}}$ totally real with a fixed real embedding
$\tau:F^{+}\hookrightarrow{\mathbb{R}}$. For a place $v$ of $F$, we will
denote by

  * •

$F_{v}$ the completion of $F$ at $v$,

  * •

${\mathcal{O}}_{v}$ the ring of integers of $F_{v}$,

  * •

$\varpi_{v}$ a uniformizer,

  * •

$q_{v}$ the order of the residual field
$\kappa(v)={\mathcal{O}}_{v}/(\varpi_{v})$.

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 $\beta\in B^{*=-1}$, denote by
$\sharp_{\beta}$ the involution $x\mapsto x^{\sharp_{\beta}}=\beta
x^{*}\beta^{-1}$ and $G/{\mathbb{Q}}$ the group of similitudes, denoted
$G_{\tau}$ in [6], defined for every ${\mathbb{Q}}$-algebra $R$ by

$G(R)\simeq\\{(\lambda,g)\in
R^{\times}\times(B^{op}\otimes_{\mathbb{Q}}R)^{\times}\hbox{ such that
}gg^{\sharp_{\beta}}=\lambda\\}$

with $B^{op}=B\otimes_{F,c}F$. If $x$ is a place of ${\mathbb{Q}}$ split
$x=yy^{c}$ in $E$ then

$G({\mathbb{Q}}_{x})\simeq(B_{y}^{op})^{\times}\times{\mathbb{Q}}_{x}^{\times}\simeq{\mathbb{Q}}_{x}^{\times}\times\prod_{z_{i}}(B_{z_{i}}^{op})^{\times},$
(3.1)

where, identifying places of $F^{+}$ over $x$ with places of $F$ over $y$,
$x=\prod_{i}z_{i}$ in $F^{+}$.

Convention: for $x=yy^{c}$ a place of ${\mathbb{Q}}$ split in $E$ and $z$ a
place of $F$ over $y$ as before, we shall make throughout the text, the
following abuse of notation by denoting $G(F_{z})$ in place of the factor
$(B_{z}^{op})^{\times}$ in the formula (3.1).

In [6], the author justify the existence of $G$ like before such that moreover

  * •

if $x$ is a place of ${\mathbb{Q}}$ non split in $E$ then
$G({\mathbb{Q}}_{x})$ is quasi split;

  * •

the invariants of $G({\mathbb{R}})$ are $(1,d-1)$ for the embedding $\tau$ and
$(0,d)$ for the others.

######  Definition 3.2.

Define ${\mathop{\mathrm{Spl}}\nolimits}$ as the set of places $v$ of $F$ such
that $p_{v}:=v_{|{\mathbb{Q}}}\neq l$ is split in $E$ and
$B_{v}^{\times}\simeq GL_{d}(F_{v})$. For each $I\in{\mathcal{I}}$, write
${\mathop{\mathrm{Spl}}\nolimits}(I)$ the subset of
${\mathop{\mathrm{Spl}}\nolimits}$ of places which do not divide the level
$I$.

As in [6] bottom of page 90, a compact open subgroup $U$ of
$G({\mathbb{A}}^{\infty})$ is said _small enough_ if there exists a place $x$
such that the projection from $U^{v}$ to $G({\mathbb{Q}}_{x})$ does not
contain any element of finite order except identity.

######  Notation 3.3.

We denote by ${\mathcal{I}}$ the set of open compact subgroups small enough of
$G({\mathbb{A}}^{\infty})$. For $I\in{\mathcal{I}}$, then
$\mathop{\mathrm{Sh}}\nolimits_{I,\eta}\longrightarrow\mathop{\mathrm{Spec}}\nolimits
F$ is the associated Shimura variety said of Kottwitz-Harris-Taylor type.

In the sequel, $v$ will denote a fixed place of $F$ in
${\mathop{\mathrm{Spl}}\nolimits}$. For such a place $v$ the scheme
$\mathop{\mathrm{Sh}}\nolimits_{I,\eta}$ has a projective model
$\mathop{\mathrm{Sh}}\nolimits_{I,v}$ over
$\mathop{\mathrm{Spec}}\nolimits{\mathcal{O}}_{v}$ with special fiber
$\mathop{\mathrm{Sh}}\nolimits_{I,s_{v}}$. For $I$ going through
${\mathcal{I}}$, the projective system
$(\mathop{\mathrm{Sh}}\nolimits_{I,v})_{I\in{\mathcal{I}}}$ is naturally
equipped with an action of $G({\mathbb{A}}^{\infty})\times{\mathbb{Z}}$ such
that $w_{v}$ in the Weil group $W_{v}$ of $F_{v}$ acts by
$-\mathop{\mathrm{deg}}\nolimits(w_{v})\in{\mathbb{Z}}$, where
$\mathop{\mathrm{deg}}\nolimits=\mathop{\mathrm{val}}\nolimits\circ\mathop{\mathrm{Art}}\nolimits^{-1}$
and $\mathop{\mathrm{Art}}\nolimits^{-1}:W_{v}^{ab}\simeq F_{v}^{\times}$ is
the Artin isomorphism which sends geometric Frobenius to uniformizers.

######  Notations 3.4.

(see [1] §1.3) For $I\in{\mathcal{I}}$, the Newton stratification of the
geometric special fiber $\mathop{\mathrm{Sh}}\nolimits_{I,\bar{s}_{v}}$ is
denoted by

$\mathop{\mathrm{Sh}}\nolimits_{I,\bar{s}_{v}}=:\mathop{\mathrm{Sh}}\nolimits^{\geq
1}_{I,\bar{s}_{v}}\supset\mathop{\mathrm{Sh}}\nolimits^{\geq
2}_{I,\bar{s}_{v}}\supset\cdots\supset\mathop{\mathrm{Sh}}\nolimits^{\geq
d}_{I,\bar{s}_{v}}$

where
$\mathop{\mathrm{Sh}}\nolimits^{=h}_{I,\bar{s}_{v}}:=\mathop{\mathrm{Sh}}\nolimits^{\geq
h}_{I,\bar{s}_{v}}-\mathop{\mathrm{Sh}}\nolimits^{\geq h+1}_{I,\bar{s}_{v}}$
is an affine scheme111see for example [7], 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 $1\leq h<d$, write

$i_{h}:\mathop{\mathrm{Sh}}\nolimits^{\geq
h}_{I,\bar{s}_{v}}\hookrightarrow\mathop{\mathrm{Sh}}\nolimits^{\geq
1}_{I,\bar{s}_{v}},\quad j^{\geq
h}:\mathop{\mathrm{Sh}}\nolimits^{=h}_{I,\bar{s}_{v}}\hookrightarrow\mathop{\mathrm{Sh}}\nolimits^{\geq
h}_{I,\bar{s}_{v}},$

and $j^{=h}=i_{h}\circ j^{\geq h}$.

Consider now the ideals $I^{v}(n):=I^{v}K_{v}(n)$ where

$K_{v}(n):=\mathop{\mathrm{Ker}}\nolimits(GL_{d}({\mathcal{O}}_{v})\twoheadrightarrow
GL_{d}({\mathcal{O}}_{v}/{\mathcal{M}}_{v}^{n})).$

Recall then that $\mathop{\mathrm{Sh}}\nolimits_{I^{v}(n),\bar{s}_{v}}^{=h}$
is geometrically induced under the action of the parabolic subgroup
$P_{h,d}({\mathcal{O}}_{v}/{\mathcal{M}}_{v}^{n})$. Concretely this means that
there exists a closed subscheme
$\mathop{\mathrm{Sh}}\nolimits_{I^{v}(n),\bar{s}_{v},\overline{1_{h}}}^{=h}$
stabilized by the Hecke action of $P_{h,d}(F_{v})$ and such that

$\mathop{\mathrm{Sh}}\nolimits_{I^{v}(n),\bar{s}_{v}}^{=h}=\mathop{\mathrm{Sh}}\nolimits_{I^{v}(n),\bar{s}_{v},\overline{1_{h}}}^{=h}\times_{P_{h,d}({\mathcal{O}}_{v}/{\mathcal{M}}_{v}^{n})}GL_{d}({\mathcal{O}}_{v}/{\mathcal{M}}_{v}^{n}),$

meaning that $\mathop{\mathrm{Sh}}\nolimits_{I^{v}(n),\bar{s}_{v}}^{=h}$ is
the disjoint union of copies of
$\mathop{\mathrm{Sh}}\nolimits_{I^{v}(n),\bar{s}_{v},\overline{1_{h}}}^{=h}$
indexed by
$GL_{d}({\mathcal{O}}_{v}/{\mathcal{M}}_{v}^{n})/P_{h,d}({\mathcal{O}}_{v}/{\mathcal{M}}_{v}^{n})$
and exchanged by the action of
$GL_{d}({\mathcal{O}}_{v}/{\mathcal{M}}_{v}^{n})$.

######  Notations 3.5.

For $1\leq t\leq s_{g}:=\lfloor d/g\rfloor$, let $\Pi_{t}$ be any
representation of $GL_{d-tg}(F_{v})$. We then denote by

$\widetilde{HT}_{1}(\pi_{v},\Pi_{t}):={\mathcal{L}}(\pi_{v}[t]_{D})_{\overline{1_{tg}}}\otimes\Pi_{t}\otimes\Xi^{\frac{tg-d}{2}}$

the Harris-Taylor local system on the Newton stratum
$\mathop{\mathrm{Sh}}\nolimits^{=tg}_{I,\bar{s}_{v},\overline{1_{tg}}}$ where

  * •

${\mathcal{L}}(\pi_{v}[t]_{D})_{\overline{1_{tg}}}$ is defined thanks to Igusa
varieties attached to the representation $\pi_{v}[t]_{D}$,

  * •

$\Xi:\frac{1}{2}{\mathbb{Z}}\longrightarrow\overline{\mathbb{Z}}_{l}^{\times}$
is defined by $\Xi(\frac{1}{2})=q^{1/2}$.

We also introduce the induced version

$\widetilde{HT}(\pi_{v},\Pi_{t}):=\Bigl{(}{\mathcal{L}}(\pi_{v}[t]_{D})_{\overline{1_{tg}}}\otimes\Pi_{t}\otimes\Xi^{\frac{tg-d}{2}}\Bigr{)}\times_{P_{tg,d}(F_{v})}GL_{d}(F_{v}),$

where the unipotent radical of $P_{tg,d}(F_{v})$ acts trivially and the action
of

$(g^{\infty,v},\left(\begin{array}[]{cc}g_{v}^{c}&*\\\
0&g_{v}^{et}\end{array}\right),\sigma_{v})\in G({\mathbb{A}}^{\infty,v})\times
P_{tg,d}(F_{v})\times W_{v}$

is given

  * •

by the action of $g_{v}^{c}$ on $\Pi_{t}$ and
$\mathop{\mathrm{deg}}\nolimits(\sigma_{v})\in{\mathbb{Z}}$ on
$\Xi^{\frac{tg-d}{2}}$, and

  * •

the action of
$(g^{\infty,v},g_{v}^{et},\mathop{\mathrm{val}}\nolimits(\mathop{\mathrm{det}}\nolimits
g_{v}^{c})-\mathop{\mathrm{deg}}\nolimits\sigma_{v})\in
G({\mathbb{A}}^{\infty,v})\times GL_{d-tg}(F_{v})\times{\mathbb{Z}}$ on
${\mathcal{L}}_{\overline{\mathbb{Q}}_{l}}(\pi_{v}[t]_{D})_{\overline{1_{tg}}}\otimes\Xi^{\frac{tg-d}{2}}$.

We also introduce

$HT(\pi_{v},\Pi_{t})_{\overline{1_{tg}}}:=\widetilde{HT}(\pi_{v},\Pi_{t})_{\overline{1_{tg}}}[d-tg],$

and the perverse sheaf

$P(t,\pi_{v})_{\overline{1_{tg}}}:=j^{=tg}_{1,!*}HT(\pi_{v},{\mathop{\mathrm{St}}\nolimits}_{t}(\pi_{v}))_{\overline{1_{tg}}}\otimes{\mathbb{L}}(\pi_{v}),$

and their induced version, $HT(\pi_{v},\Pi_{t})$ and $P(t,\pi_{v})$, where

$j^{=h}=i^{h}\circ j^{\geq
h}:\mathop{\mathrm{Sh}}\nolimits^{=h}_{I,\bar{s}_{v}}\hookrightarrow\mathop{\mathrm{Sh}}\nolimits^{\geq
h}_{I,\bar{s}_{v}}\hookrightarrow\mathop{\mathrm{Sh}}\nolimits_{I,\bar{s}_{v}}$

and ${\mathbb{L}}^{\vee}$, the dual of ${\mathbb{L}}$, is the local Langlands
correspondence.

For $\overline{\mathbb{Q}}_{l}$ or $\overline{\mathbb{F}}_{l}$ coefficients,
we will mention it in the index of the local system, as for example
$HT_{\overline{\mathbb{Q}}_{l}}(\pi_{v},\Pi_{t})$ or
$HT_{\overline{\mathbb{F}}_{l}}(\pi_{v},\Pi_{t})$.

##  4\. $\overline{\mathbb{Q}}_{l}$-cohomology groups

From now on, we fix a prime number $l$ unramified in $E$. Let us first recall
some known facts about irreducible algebraic representations of $G$, see for
example [6] p.97. Let $\sigma_{0}:E\hookrightarrow\overline{{\mathbb{Q}}}_{l}$
be a fixed embedding and et write $\Phi$ the set of embeddings
$\sigma:F\hookrightarrow\overline{\mathbb{Q}}_{l}$ whose restriction to $E$
equals $\sigma_{0}$. There exists then an explicit bijection between
irreducible algebraic representations $\xi$ of $G$ over
$\overline{\mathbb{Q}}_{l}$ and $(d+1)$-uple
$\bigl{(}a_{0},(\overrightarrow{a_{\sigma}})_{\sigma\in\Phi}\bigr{)}$ where
$a_{0}\in{\mathbb{Z}}$ and for all $\sigma\in\Phi$, we have
$\overrightarrow{a_{\sigma}}=(a_{\sigma,1}\leq\cdots\leq a_{\sigma,d})$.

For $K\subset\overline{\mathbb{Q}}_{l}$ a finite extension of
${\mathbb{Q}}_{l}$ such that the representation $\iota^{-1}\circ\xi$ of
highest weight
$\bigl{(}a_{0},(\overrightarrow{a_{\sigma}})_{\sigma\in\Phi}\bigr{)}$, is
defined over $K$, write $W_{\xi,K}$ the space of this representation and
$W_{\xi,{\mathcal{O}}}$ a stable lattice under the action of the maximal open
compact subgroup $G({\mathbb{Z}}_{l})$, where ${\mathcal{O}}$ is the ring of
integers of $K$ with uniformizer $\lambda$.

Remark. if $\xi$ is supposed to be $l$-small, in the sense that for all
$\sigma\in\Phi$ and all $1\leq i<j\leq n$ we have $0\leq
a_{\tau,j}-a_{\tau,i}<l$, then such a stable lattice is unique up to a
homothety.

######  Notation 4.1.

We will denote by $V_{\xi,{\mathcal{O}}/\lambda^{n}}$ the local system on
$\mathop{\mathrm{Sh}}\nolimits_{I,v}$ as well as

$V_{\xi,{\mathcal{O}}}=\lim_{\genfrac{}{}{0.0pt}{}{\longleftarrow}{n}}V_{\xi,{\mathcal{O}}/\lambda^{n}}\quad\hbox{and}\quad
V_{\xi,K}=V_{\xi,{\mathcal{O}}}\otimes_{{\mathcal{O}}}K.$

For $\overline{\mathbb{Z}}_{l}$ and $\overline{\mathbb{Q}}_{l}$ version, we
will write respectively $V_{\xi,\overline{\mathbb{Z}}_{l}}$ and
$V_{\xi,\overline{\mathbb{Q}}_{l}}$.

Remark. the representation $\xi$ is said _regular_ if its parameter
$\bigl{(}a_{0},(\overrightarrow{a_{\sigma}})_{\sigma\in\Phi}\bigr{)}$ verifies
for all $\sigma\in\Phi$ that $a_{\sigma,1}<\cdots<a_{\sigma,d}$.

######  Definition 4.2.

For a $\overline{\mathbb{Z}}_{l}$-sheaf ${\mathcal{F}}$ on
$\mathop{\mathrm{Sh}}\nolimits_{I,v}$, we will denote by ${\mathcal{F}}_{\xi}$
the sheaf ${\mathcal{F}}\otimes V_{\xi,\overline{\mathbb{Z}}_{l}}$.

######  Definition 4.3.

Let $\xi$ be a irreducible algebraic ${\mathbb{C}}$-representation with finite
dimension of $G$. Then a irreducible ${\mathbb{C}}$-representation
$\Pi_{\infty}$ of $G({\mathbb{A}}_{\infty})$ is said $\xi$-cohomological if
there exists an integer $i$ such that

$H^{i}(({\mathop{\mathrm{Lie\leavevmode\nobreak\
}}\nolimits}G({\mathbb{R}}))\otimes_{\mathbb{R}}{\mathbb{C}},U_{\tau},\Pi_{\infty}\otimes\xi^{\vee})\neq(0)$

where $U_{\tau}$ is a maximal open compact modulo center subgroup of
$G({\mathbb{R}})$, cf. [6] p.92. We then denote by $d_{\xi}^{i}(\Pi_{\infty})$
the dimension of this cohomology group.

Remark. If $\xi$ has $\bar{\mathbb{Q}}_{l}$-coefficients, then a irreducible
$\bar{\mathbb{Q}}_{l}$-representation $\Pi^{\infty}$ of
$G({\mathbb{A}}^{\infty})$ is said $\xi$-cohomolocal if there exists a
${\mathbb{C}}$-representation $\Pi_{\infty}$ of $G({\mathbb{A}}_{\infty})$
such that $\iota_{l}\Bigl{(}\Pi^{\infty}\Bigr{)}\otimes\Pi_{\infty}$ is an
automorphic ${\mathbb{C}}$-representation of $G({\mathbb{A}})$, where
$\iota_{l}:\overline{\mathbb{Q}}_{l}\simeq{\mathbb{C}}$.

Recall that $G({\mathbb{Q}}_{p})\simeq{\mathbb{Q}}_{p}^{\times}\times
GL_{d}(F_{v})\times\prod_{i=2}^{r}(B_{v_{i}}^{op})^{\times}$, for a fixed
place $v$ of $F$ above $p$. For $\Pi$ a irreducible representation of
$G({\mathbb{A}})$, its component for the similitude factor le facteur
${\mathbb{Q}}_{p}^{\times}$ is denoted as in [6], $\Pi_{p,0}$: as all the open
compact subgroup of ${\mathcal{I}}$ contain ${\mathbb{Z}}_{p}^{\times}$, the
representation $\Pi$ which appear in the cohomology groups will all verify
that $(\Pi_{p,0})_{|{\mathbb{Z}}_{p}^{\times}}=1$. We now consider

  * •

an admissible irreducible representation $\Pi$ of $G({\mathbb{A}})$ with
multiplicity $m(\Pi)$ in the space of automorphic forms,

  * •

and let $\pi_{v}$ be a cuspidal irreducible representation of $GL_{g}(F_{v})$.

  * •

We also fix a finite level $I$ and we denote by $S$ the set of places $x$ such
that $I_{x}$ is maximal and we moreover impose $v\in S$. We then denote by
${\mathbb{T}}^{S}$ the $\overline{\mathbb{Z}}_{l}$-unramified Hecke algebra
outside $S$.

From [2] and more precisely from [4], we now recall the main results about the
$\overline{\mathbb{Q}}_{l}$-cohomology groups of Harris-Taylor perverse
sheaves. We first introduce some notations.

  * •

For any $\overline{\mathbb{Q}}_{l}$-perverse sheaf $P$ on the projective
system of schemes $\mathop{\mathrm{Sh}}\nolimits_{I^{v}(n),\bar{s}_{v}}^{=h}$
with $I^{v}(n):=I^{v}K_{v}(n)$ where
$K_{v}(n):=\mathop{\mathrm{Ker}}\nolimits(GL_{d}({\mathcal{O}}_{v})\twoheadrightarrow
GL_{d}({\mathcal{O}}_{v}/{\mathcal{M}}_{v}^{n}))$, we consider

$H^{i}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},P):=\lim_{\genfrac{.}{.}{0.0pt}{1}{\longrightarrow}{n}}H^{i}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(n),\bar{s}_{v}},P).$

It is then equipped with an action of
${\mathbb{T}}^{S}_{\overline{\mathbb{Q}}_{l}}\times GL_{d}(F_{v})\times W_{v}$
and we denote by $[H^{*}(P_{\xi})]$ the image of
$\sum_{i}(-1)^{i}H^{i}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},P\otimes
V_{\xi,\overline{\mathbb{Q}}_{l}})$ in the Grothendieck group of admissible
representations of ${\mathbb{T}}^{S}\times GL_{d}(F_{v})\times W_{v}$.

  * •

For a fixed maximal ideal $\widetilde{\mathfrak{m}}$ of
${\mathbb{T}}^{S}_{\overline{\mathbb{Q}}_{l}}$, and a
$\overline{\mathbb{Q}}_{l}$-perverse sheaf $P$ as above, we then denote by
$H^{i}(P_{\xi})_{\widetilde{\mathfrak{m}}}$ the localization at
$\widetilde{\mathfrak{m}}$ of
$H^{i}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},P_{\xi})$ and
$[H^{*}(P_{\xi})]_{\widetilde{\mathfrak{m}}}$ will denote its image in the
Grothendieck group of admissible representations of $GL_{d}(F_{v})\times
W_{v}$. If $\widetilde{\mathfrak{m}}$ is associated to a irrreductible
representation $\Pi^{\infty,v}$ of $G({\mathbb{A}}^{\infty,v})$, we might also
denote it by $[Hj(P_{\xi})]\\{\Pi^{\infty,v}\\}$.

  * •

In particular for $HT(\pi_{v},\Pi_{t})$ a Harris-Taylor local system, we
denote by $[H^{i}_{c}(HT(\pi_{v},\Pi_{t}))]_{\widetilde{\mathfrak{m}}}$ (resp.
$[H^{i}_{!*}(HT(\pi_{v},\Pi_{t}))]_{\widetilde{\mathfrak{m}}}$), the image of
$H^{i}(\mathop{\mathrm{Sh}}\nolimits_{{\mathcal{I}}^{v},\bar{s}_{v}},j^{=h}_{!}HT(\pi_{v},\Pi_{t}))_{\widetilde{\mathfrak{m}}}$
(resp.
$H^{i}(\mathop{\mathrm{Sh}}\nolimits_{{\mathcal{I}}^{v},\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=h}_{!*}HT(\pi_{v},\Pi_{t}))_{\widetilde{\mathfrak{m}}}$)
in the Grothendieck group of $GL_{d}(F_{v})\times W_{v}$ admissible
representations. We also use a similar notation for
$HT_{1_{tg}}(\pi_{v},\Pi_{t})$ by replacing $GL_{d}(F_{v})$ by
$P_{tg,d}(F_{v})$.

It is also well known, cf. lemma 3.2 of [4] that if $\Pi$ is a irreducible
automorphic representation of $G({\mathbb{A}})$ then its local component
$\Pi_{v}$ is isomorphic to

$\mathop{\mathrm{Speh}}\nolimits_{s}({\mathop{\mathrm{St}}\nolimits}_{t_{1}}(\pi_{1,z}))\times\cdots\times\mathop{\mathrm{Speh}}\nolimits_{s}({\mathop{\mathrm{St}}\nolimits}_{t_{u}}(\pi_{u,z}))\simeq\mathop{\mathrm{Speh}}\nolimits_{s}\Bigl{(}{\mathop{\mathrm{St}}\nolimits}_{t_{1}}(\pi_{1,z})\times\cdots\times{\mathop{\mathrm{St}}\nolimits}_{t_{u}}(\pi_{u,z})\Bigr{)}$

where the $\pi_{i,z}$ are irreducible cuspidal representations.

######  Notation 4.4.

For $\pi_{v}$ a irreducible cuspidal representation of $GL_{g}(F_{v})$, we
denote by ${\mathcal{A}}_{\xi,\pi_{v}}(r,s)$ the set of equivalence classes of
automorphic irreducible representations of $G({\mathbb{A}})$ which are
$\xi$-cohomological and such that

  * •

$(\Pi_{p,0})_{|{\mathbb{Z}}_{p}^{\times}}=1$,

  * •

its local component at $v$ looks like
$\mathop{\mathrm{Speh}}\nolimits_{s}({\mathop{\mathrm{St}}\nolimits}_{t}(\pi^{\prime}_{v}))\times?$
with $r=s+t-1$, $\pi^{\prime}_{v}\sim^{i}\pi_{v}$ and $?$ is a irreducible
representation of $GL_{d-stg}(F_{v})$ which we do not want to precise.

Remark. An automorphic irreducible representation $\Pi$ of $G({\mathbb{A}})$
which is $\xi$-cohomological and such that

$\Pi_{v}\simeq\mathop{\mathrm{Speh}}\nolimits_{s}\Bigl{(}{\mathop{\mathrm{St}}\nolimits}_{t_{1}}(\pi_{1,v})\times\cdots\times{\mathop{\mathrm{St}}\nolimits}_{t_{u}}(\pi_{u,v})\Bigr{)}$

belongs to ${\mathcal{A}}_{\xi,\pi_{v,i}}(s+t_{i}-1,s)$ for $i=1,\cdots,u$.

######  Proposition 4.5.

(proposition 3.6 of [4])
Let $\pi_{v}$ be a irreducible cuspidal representation of $GL_{g}(F_{v})$ and
$1\leq r\leq d/g$. Then we have

$[H^{i}_{!*}(HT_{\overline{1_{rg}},\xi}(\pi_{v},\Pi_{r}))]\\{\Pi^{\infty,v}\\}=\frac{e_{\pi_{v}}\sharp\mathop{\mathrm{Ker}}\nolimits^{1}({\mathbb{Q}},G)}{d}\\\
\sum_{\genfrac{.}{.}{0.0pt}{1}{(s,t)}{\Pi\in{\mathcal{A}}_{\xi,\pi_{v}}(s+t-1,s)}}\sum_{\Pi^{\prime}\in{\mathcal{U}}_{G}(\Pi^{\infty,v})}\sum_{m_{s,t}(r,i)=1}m(\Pi^{\prime})d_{\xi}(\Pi^{\prime}_{\infty})\Bigl{(}\Pi_{r}\otimes
R_{\pi_{v}}(s,t)(r,i)(\Pi_{v})\Bigr{)}$

where

  * •

${\mathcal{U}}_{G}(\Pi^{\infty,v})$ is the set of equivalence classes of
automorphic irreducible representation $\Pi^{\prime}$ of $G({\mathbb{A}})$
such that $(\Pi^{\prime})^{\infty,v}\simeq\Pi^{\infty,v}$;

  * •

$\Pi_{v}$ is the local component of all the
$\Pi^{\prime}\in{\mathcal{U}}_{G}(\Pi^{\infty,v})$ such that
$d_{\xi}(\Pi^{\prime}_{\infty})\neq 0$, is the common value of the
$d_{\xi}^{i}(\Pi^{\prime}_{\infty})$ for $i\equiv s\mod 2$, cf. corollary
VI.2.2 of [6] and corollary 3.3 of [4]

Concerning $R_{\pi_{v}}(s,t)(r,i)(\Pi_{v})$ as a sum of representations of
$GL_{d-rg}(F_{v})\times{\mathbb{Z}}$, for
$\Pi_{v}\simeq\mathop{\mathrm{Speh}}\nolimits_{s}({\mathop{\mathrm{St}}\nolimits}_{t_{1}}(\pi_{1,v}))\times\cdots\times\mathop{\mathrm{Speh}}\nolimits_{s}({\mathop{\mathrm{St}}\nolimits}_{t_{u}}(\pi_{u,v}))$,
it is given by the formula

$R_{\pi_{v}}(r,i)(\Pi_{v})=\sum_{k:\leavevmode\nobreak\
\pi_{k,v}\sim_{i}\pi_{v}}m_{s,t_{k}}(r,i)R_{\pi_{v}}(s,t_{k})(r,i)(\Pi_{v},k)\otimes\Bigl{(}\xi_{k}\otimes\Xi^{i/2}\Bigr{)}$

where

  * •

the characters $\xi_{k}$ are such that
$\pi_{k,v}\simeq\pi_{v}\otimes\xi_{k}\circ\mathop{\mathrm{val}}\nolimits\circ\mathop{\mathrm{det}}\nolimits$;

  * •

$R_{\pi_{v}}(s,t_{k})(r,i)(\Pi_{v},k)$ can be written as

$R_{\pi_{v}}(s,t_{k})(r,i)(\Pi_{v},k):=\mathop{\mathrm{Speh}}\nolimits_{s}({\mathop{\mathrm{St}}\nolimits}_{t_{1}}(\pi_{1,v}))\times\cdots\times\mathop{\mathrm{Speh}}\nolimits_{s}({\mathop{\mathrm{St}}\nolimits}_{t_{k-1}}(\pi_{k-1,v}))\\\
\times R_{\pi_{k,v}}(s,t_{k})(r,i)\times\\\
\mathop{\mathrm{Speh}}\nolimits_{s}({\mathop{\mathrm{St}}\nolimits}_{t_{k+1}}(\pi_{k+1,v}))\times\cdots\times\mathop{\mathrm{Speh}}\nolimits_{s}({\mathop{\mathrm{St}}\nolimits}_{t_{u}}(\pi_{u,v})).$

  * •

$R_{\pi_{k,v}}(s,t_{k})(r,i)$ is a representation of $GL_{d-rg}(F_{v})$ which
can be computed combinatorially as explained below

  * •

and $m_{s,t}(r,i)\in\\{0,1\\}$ is given in the next definition.

The representation $R_{\pi_{k,v}}(s,t_{k})(r,i)$ is computed as follows: we
first apply the Jacquet functor $J_{P^{op}_{rg,d}}$ to
$\mathop{\mathrm{Speh}}\nolimits_{s}({\mathop{\mathrm{St}}\nolimits}_{t_{k}}(\pi_{k,v}))$
which can written as a sum

$\sum\langle a_{1}\rangle\otimes\langle a_{2}\rangle$

where $a_{1},a_{2}$ are multisegments in the Zelevinsky line of $\pi_{v}$; the
precise computation is given in corollary 1.5.6 of [2]. We then consider the
sum

$\sum R_{\pi_{v}[r]_{D}}\Bigl{(}\langle
a_{1}\rangle\Bigr{)}a_{2}=\sum_{\psi}\epsilon_{\psi}\psi\otimes\Pi_{\psi}\in\mathop{\mathrm{Groth}}\nolimits\Bigl{(}F_{v}^{\times}\times
GL_{(st-r)g}(F_{v})\Bigr{)}$

where $\psi$ describe the set of characters of $F_{v}^{\times}$ and
$\epsilon_{\psi}\in\\{-1,1\\}$.

Remark. In the previous formula when $\tau_{v}=\pi_{v}[r]_{D}$, the $\psi$
such that $\Pi_{\psi}$ are non zero, looks like $|-|^{k/2}$ with
$k\in{\mathbb{Z}}$, and

$R_{\pi_{k,v}}(s,t_{k})(r,i)=\Pi_{|-|^{-i/2}}.$

We do not need the precise computation of $R_{\pi_{v}}(s,t)(r,i)$, we just
want to use the fact that if $\pi^{\prime}_{v}$ is any irreducible cuspidal
representation, then $R_{\pi^{\prime}_{v}}(s,t)(r,i)$ is obtained from
$R_{\pi_{v}}(s,t)(r,i)$ by replacing $\pi_{v}$ by $\pi^{\prime}_{v}$ in its
combinatorial description. In particular if the modulo $l$ reduction of
$\pi^{\prime}_{v}$ is isomorphic to the modulo $l$ reduction of $\pi_{v}$,
then the modulo $l$ reduction of $R_{\pi_{v}}(s,t)(r,i)$ and those of
$R_{\pi^{\prime}_{v}}(s,t)(r,i)$, are isomorphic.

The assumptions on $i$ in proposition 3.6.1 of [2] are contained in the
following definition of $m_{s,t}(r,i)$.

######  Definition 4.6.

The point with coordinates $(r,i)$ such that $m_{s,t}(r,i)=1$, are contained
in the convex hull of the polygon with edges $(s+t-1,0)$, $(t,\pm(s-1))$ and
$(1,\pm(s-t))$ if $s\geq t$ (resp. $(t-s+1,0)$ if $t\geq s$); inside it for a
fixed $r$, the indexes $i$ start from the boundary en grows by $2$, i.e.
$m_{s,t}(r,i)=1$ if and only if

  * •

$\max\\{1,s+t-1-2(s-1)\\}\leq r\leq s+t-1$;

  * •

if $t\leq r\leq s+t-1$ then $0\leq|i|\leq s+t-1-r$ and $i\equiv s+t-1-r\mod
2$;

  * •

if $\max\\{1,s+t-1-2(s-1)\\}\leq r\leq t$ then $0\leq|i|\leq s-1-(t-r)$ and
$i\equiv s-t-1+r\mod 2$,

as represented in figures 1 and 2.

Figure 1.  The squares indicate the $(r,i)$ such that $m_{s,t}(r,i)=1$ for a
Speh ($t=1$) at left and a Steinberg ($s=1$) on the right Figure 2.
$m_{s,t}(r,i)=1$ when $s\geq t$ at left and $t\geq s$ on the right Figure 3.
Superposition to compute $m(r,i)$ for
$\Pi_{v}\simeq\mathop{\mathrm{Speh}}\nolimits_{4}(\pi_{v})\times\mathop{\mathrm{Speh}}\nolimits_{4}({\mathop{\mathrm{St}}\nolimits}_{3}(\pi_{v}))\times\mathop{\mathrm{Speh}}\nolimits_{4}({\mathop{\mathrm{St}}\nolimits}_{5}(\pi_{v}))$

Remark. For

$\Pi_{v}\simeq\mathop{\mathrm{Speh}}\nolimits_{s}({\mathop{\mathrm{St}}\nolimits}_{t_{1}}(\pi_{1,v}))\times\cdots\times\mathop{\mathrm{Speh}}\nolimits_{s}({\mathop{\mathrm{St}}\nolimits}_{t_{u}}(\pi_{u,v}))$

the set of $(r,i)$ such that $[H^{i}(\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{\geq
rg}_{!*}{\mathcal{F}}_{\bar{\mathbb{Q}}_{l},\xi}(\pi_{v},r)[d-rg])]\\{\Pi^{\infty,v}\\}\neq
0$ is obtained by superposition of the $u$ previous diagrams as in the figure
3. More precisely for a fixed $(r,i)$, the contribution of diagram of
$\mathop{\mathrm{Speh}}\nolimits_{s}({\mathop{\mathrm{St}}\nolimits}_{t_{k}}(\pi_{k,v}))$
is the same as in the starting point $(s+t_{k}-1,0)$ after replacing
$\mathop{\mathrm{Speh}}\nolimits_{s}({\mathop{\mathrm{St}}\nolimits}_{t_{k}}(\pi_{k,v}))$
by $R_{\pi_{k}}(s,t_{k})(r,i)$. We can then trace back any

$R_{\pi_{k}}(s,t_{k})(r,i)(\Pi^{\prime}_{v})\otimes\Bigl{(}\xi_{k}\otimes\Xi^{i/2}\Bigr{)}$

of $[H^{i}(\kern 0.5pt\vphantom{j}^{p}\kern-0.5ptj^{\geq
rg}_{!*}{\mathcal{F}}_{\bar{\mathbb{Q}}_{l},\xi}(\pi_{v},r)_{1}[d-rg])]\\{\Pi^{\infty,v}\\}$,
to

$R_{\pi_{k}}(s+t_{k},0)(r^{\prime},0)(\Pi^{\prime}_{v})\otimes\Bigl{(}\xi_{k}\otimes\Xi^{0}\Bigr{)}$

of $[H^{0}(\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{\geq(s+t_{k})g}_{!*}{\mathcal{F}}_{\bar{\mathbb{Q}}_{l},\xi}(\pi_{v},s+t_{k})_{1}[d-(s+t_{k})g])]\\{\Pi^{\infty,v}\\}$.
Note although that for $i=0$, some of the constituants of $[H^{0}(\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{\geq
rg}_{!*}{\mathcal{F}}_{\bar{\mathbb{Q}}_{l},\xi}(\pi_{v},r)_{1}[d-rg])]\\{\Pi^{\infty,v}\\}$
may or may not come from $r^{\prime}>r$.

Comments about the exemple of figure 3 for $r=4$:

  * •

$\mathop{\mathrm{Speh}}\nolimits_{4}(\pi_{v})\times\mathop{\mathrm{Speh}}\nolimits_{4}({\mathop{\mathrm{St}}\nolimits}_{3}(\pi_{v}))\times
R_{\pi_{v}}(4,5)(4,0)$ comes from $(8,0)$;

  * •

$\mathop{\mathrm{Speh}}\nolimits_{4}(\pi_{v})\times
R_{\pi_{v}}(4,3)(4,0)\times\mathop{\mathrm{Speh}}\nolimits_{4}({\mathop{\mathrm{St}}\nolimits}_{5}(\pi_{v}))$
comes from $(6,0)$;

  * •

$R_{\pi_{v}}(4,1)(4,0)\times\mathop{\mathrm{Speh}}\nolimits_{4}({\mathop{\mathrm{St}}\nolimits}_{3}(\pi_{v}))\times\mathop{\mathrm{Speh}}\nolimits_{4}({\mathop{\mathrm{St}}\nolimits}_{5}(\pi_{v}))$
does not come from any $(r^{\prime},0)$ for $r^{\prime}>4$.

##  5\. Integral Harris-Taylor perverse sheaves

We now consider a fixed irreducible cuspidal representation $\pi_{v}$ such
that its modulo $l$ reduction is still supercuspidal. Then for any $t\geq 1$,
the representation $\pi_{v}[t]_{D}$ remains irreducible modulo $l$ so that
${\mathcal{L}}(\pi_{v}[t]_{D})_{\overline{1_{tg}}}$ has an unique, up to
homothety, stable $\overline{\mathbb{Z}}_{l}$-lattice. Then for a
$\overline{\mathbb{Z}}_{l}$-representation $\Pi_{t}$ of $GL_{tg}(F_{v})$, we
have a well defined $\overline{\mathbb{Z}}_{l}$-local system
$HT(\pi_{v},\Pi_{t})$.

Remark. $\Pi_{t}$ is called the infinitesimal part of the Harris-Taylor local
system and it does not play any role here, we only mention it as it appears
everywhere in [4] or [1].

Over $\overline{\mathbb{Z}}_{l}$, we have two natural $t$-structures denoted
$p$ and $p+$ which are exchanged by Grothendieck-Verdier duality. We then have
two notions of intermediate extension, $\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj_{!*}$ and $\kern
0.5pt\vphantom{j}^{p+}\kern-0.5ptj_{!*}$. In [3] we explain, using the Newton
stratification, how to construct $\overline{\mathbb{Z}}_{l}$-filtrations of
perverse sheaves, with torsion free graded parts. In [5] we then prove the
following results.

  * •

When the modulo $l$ reduction of $\pi_{v}$ remains supercuspidal, the two
intermediate extensions $\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=tg}_{!*}HT(\pi_{v},\Pi_{t})$ and $\kern
0.5pt\vphantom{j}^{p+}\kern-0.5ptj^{=tg}_{!*}HT(\pi_{v},\Pi_{t})$ are
isomorphic, as it is, essentially formally, the case when $\pi_{v}$ is a
character.

  * •

The following resolution of $\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj_{!*}^{=t}HT(\pi_{v},\Pi_{t})$, proved over
$\overline{\mathbb{Q}}_{l}$ in [1], is still valid over
$\overline{\mathbb{Z}}_{l}$:

$0\rightarrow
j_{!}^{=s}HT(\pi_{v},\Pi_{t}\\{\frac{t-s}{2}\\})\times\mathop{\mathrm{Speh}}\nolimits_{s-t}(\pi_{v}\\{t/2\\}))\otimes\Xi^{\frac{s-t}{2}}\longrightarrow\cdots\\\
\longrightarrow
j_{!}^{=t+1}HT(\pi_{v},\Pi_{t}\\{-1/2\\}\times\pi_{v}\\{t/2\\})\otimes\Xi^{\frac{1}{2}}\longrightarrow\\\
j_{!}^{=t}HT(\pi_{v},\Pi_{t})\longrightarrow\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj_{!*}^{=t}HT(\pi_{v},\Pi_{t})\rightarrow 0.$
(5.1)

  * •

By the adjunction property, the map

$j_{!}^{=(t+\delta)g}HT(\pi_{v},\Pi_{t}\\{\frac{-\delta}{2}\\})\times\mathop{\mathrm{Speh}}\nolimits_{\delta}(\pi_{v}\\{t/2\\}))\otimes\Xi^{\delta/2}\\\
\longrightarrow
j_{!}^{=(t+\delta-1)g}HT(\pi_{v},\Pi_{t}\\{\frac{1-\delta}{2}\\})\times\mathop{\mathrm{Speh}}\nolimits_{\delta-1}(\pi_{v}\\{t/2\\}))\otimes\Xi^{\frac{\delta-1}{2}}$
(5.2)

is given by

$HT(\pi_{v},\Pi_{t}\\{\frac{-\delta}{2}\\}\times\mathop{\mathrm{Speh}}\nolimits_{\delta}(\pi_{v}\\{t/2\\}))\otimes\Xi^{\delta/2}\longrightarrow\\\
\kern
0.5pt\vphantom{i}^{p}\kern-0.5pti^{(t+\delta)g,!}j_{!}^{=(t+\delta-1)g}HT(\pi_{v},\Pi_{t}\\{\frac{1-\delta}{2}\\})\times\mathop{\mathrm{Speh}}\nolimits_{\delta-1}(\pi_{v}\\{t/2\\}))\otimes\Xi^{\frac{\delta-1}{2}}.$
(5.3)

We have then

$\kern
0.5pt\vphantom{i}^{p}\kern-0.5pti^{(t+\delta)g,!}j_{!}^{=(t+\delta-1)g}HT(\pi_{v},\Pi_{t}\\{\frac{1-\delta}{2}\\})\times\mathop{\mathrm{Speh}}\nolimits_{\delta-1}(\pi_{v}\\{t/2\\}))\otimes\Xi^{\frac{\delta-1}{2}}\\\
\simeq
HT\Bigl{(}\pi_{v},\Pi_{t}\\{\frac{1-\delta}{2}\\})\times\bigl{(}\mathop{\mathrm{Speh}}\nolimits_{\delta-1}(\pi_{v}\\{-1/2\\})\times\pi_{v}\\{\frac{\delta-1}{2}\\}\bigr{)}\\{t/2\\}\Bigr{)}\otimes\Xi^{\delta/2}.$
(5.4)

  * •

In particular, up to homothety, the map (5.4), and so those of (5.3), is
unique. Finally as the map of (5.1) are strict, the given maps (5.2) are
uniquely determined, that is if we forget the infinitesimal parts, these maps
are independent of the chosen $t$ in (5.1).

  * •

We also have a filtration

$\mathop{\mathrm{Fil}}\nolimits^{t-s)}_{!}(\pi_{v},\Pi_{t})\hookrightarrow\mathop{\mathrm{Fil}}\nolimits^{t-s+1}_{!}(\pi_{v},\Pi_{t})\hookrightarrow\cdots\\\
\hookrightarrow\mathop{\mathrm{Fil}}\nolimits^{0}_{!}(\pi_{v},\Pi_{t})=j^{=tg}_{!}HT(\pi_{v},\Pi_{t}),$
(5.5)

with graded parts
${\mathop{\mathrm{gr}}\nolimits}^{-\delta}_{!}(\pi_{v},\Pi_{t})\simeq\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{\geq(t+\delta)g}_{!*}HT(\pi_{v},\Pi_{t}\overrightarrow{\times}{\mathop{\mathrm{St}}\nolimits}_{\delta}(\pi_{v}))(\delta/2)$.

We fix a level $I^{v}$ outside $v$ and a maximal ideal $\mathfrak{m}$ of
${\mathbb{T}}^{S}$: recall that fixing $\mathfrak{m}$ means that we focus on
liftings $\widetilde{\mathfrak{m}}\subset\mathfrak{m}$, i.e. on
$\xi$-cohomological automorphic representations $\Pi$ of $G({\mathbb{A}})$
such that their modulo $l$ Satake parameters are prescribed by $\mathfrak{m}$.
As usual we then denote with an index $\mathfrak{m}$ for the localization
$\otimes_{{\mathbb{T}}^{S}}{\mathbb{T}}^{S}_{\mathfrak{m}}$.

######  Notation 5.6.

For $1\leq h\leq d$, we denote by $i(h)$ the smaller index $i$ such that
$H^{i}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=h}_{!*}HT_{\xi}(\pi_{v},\Pi_{h}))_{\mathfrak{m}}$
has non trivial torsion: if it does not exists then set $i(h)=+\infty$.

Remark. By duality, as $\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj_{!*}^{=h}=\kern
0.5pt\vphantom{j}^{p+}\kern-0.5ptj^{=h}_{!*}$ for Harris-Taylor local systems
associated to a irreducible cuspidal representation whose modulo $l$ reduction
remains supercuspidal, note that when $i_{I}(h)$ is finite then $i(h)\leq 0$.

We now suppose for the rest of this section, that $I^{v}$ and $\mathfrak{m}$
are chosen so that there exists $1\leq t\leq s_{g}$ with $i(t)$ finite and we
denote by $t_{0}$ the bigger such $t$.

######  Lemma 5.7.

For $1\leq t\leq t_{0}$, we have $i(t)=t-t_{0}$ and

$H^{i(t)}_{tor}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=tg}_{!*}HT(\pi_{v},\Pi_{t}))\otimes_{\overline{\mathbb{Z}}_{l}}\overline{\mathbb{F}}_{l}\simeq\\\
H^{0}_{tor}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=t_{0}g}_{!*}HT(\pi_{v},\Pi_{t}\\{\frac{t_{0}-t}{2}\\}\times\mathop{\mathrm{Speh}}\nolimits_{t-t_{0}}(\pi_{v})\\{\frac{t}{2}\\}\Xi^{\frac{t-t_{0}}{2}})\otimes_{\overline{\mathbb{Z}}_{l}}\overline{\mathbb{F}}_{l}.$
(5.8)

###### Proof.

Note first that for every $t_{0}<t\leq s_{g}$, then the cohomology groups of
$j^{=hg}_{!}HT_{\xi}(\pi_{v},\Pi_{h})$ are torsion free. Indeed consider the
spectral sequence associated to the filtration (5.5): by definition of
$t_{0}$, note that the $E_{1}^{p,q}$ are torsion free. Moreover as explained
in [2] over $\overline{\mathbb{Q}}_{l}$, we remark that all the $d_{r}^{p,q}$
for $p+q\geq 0$ are zero. So torsion can also appear in degree $\leq 0$ but as
the open Newton stratum $\mathop{\mathrm{Sh}}\nolimits^{=hg}_{I,\bar{s}_{v}}$
are affine, then the cohomology of $j^{=hg}_{!}HT_{\xi}(\pi_{v},\Pi_{h})$ is
zero in degree $<0$ and torsion free for $i=0$.

Consider then the spectral sequence associated to the resolution (5.1) for
$t>t_{0}$: 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

$H^{0}\bigl{(}\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},j_{!}^{=(t+\delta)g}HT_{\xi}(\pi_{v},\Pi_{t}\\{\frac{-\delta}{2}\\})\times\mathop{\mathrm{Speh}}\nolimits_{\delta}(\pi_{v}\\{t/2\\}))\otimes\Xi^{\delta/2}\bigr{)}_{\mathfrak{m}}\longrightarrow\\\
H^{0}\bigl{(}\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},j_{!}^{=(t+\delta-1)g}HT_{\xi}(\pi_{v},\Pi_{h}\\{\frac{1-\delta}{2}\\}\\\
\times\mathop{\mathrm{Speh}}\nolimits_{\delta-1}(\pi_{v}\\{t/2\\}))\otimes\Xi^{\frac{\delta-1}{2}}\bigr{)}_{\mathfrak{m}}$
(5.9)

are strict. Then from the previous fact stressed after (5.4), this property
remains true when we consider the associated spectral sequence for $1\leq
t^{\prime}\leq t_{0}$.

Consider now $t=t_{0}$ where we know the torsion to be non trivial. From what
was observed above we then deduce that the map

$H^{0}\bigl{(}\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},j_{!}^{=(t_{0}+1)g}HT_{\xi}(\pi_{v},\Pi_{t_{0}}\\{\frac{-1}{2}\\})\times\pi_{v}\\{t_{0}/2\\}))\otimes\Xi^{1/2}\bigr{)}_{\mathfrak{m}}\\\
\longrightarrow
H^{0}\bigl{(}\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},j_{!}^{=t_{0}g}HT_{\xi}(\pi_{v},\Pi_{t_{0}})\bigr{)}_{\mathfrak{m}}$
(5.10)

has a non trivial torsion cokernel so that $i(t_{0})=0$.

Finally for any $1\leq t\leq t_{0}$, the map like (5.10) for
$t+\delta-1<t_{0}$ are strict so that the
$H^{i}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=tg}_{!*}HT_{\xi}(\pi_{v},\Pi_{t}))_{\mathfrak{m}}$
are zero for $i<t-t_{0}$ while when $t+\delta-1=t_{0}$ its cokernel has non
trivial torsion which gives then a non trivial torsion class in
$H^{t-t_{0}}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=tg}_{!*}HT_{\xi}(\pi_{v},\Pi_{t}))_{\mathfrak{m}}$.
∎

##  6\. Automorphic congruences

From now on we consider

  * •

two irreducible cuspidal representations $\pi_{v}$ and $\pi^{\prime}_{v}$ such
that their modulo $l$ reduction are isomorphic and supercuspidal,

  * •

and as before, a maximal ideal $\mathfrak{m}$ of ${\mathbb{T}}^{S}$.

For $V$ a $\overline{\mathbb{Z}}_{l}$-free module, we denote by
$r_{l}(V)=V\otimes_{\overline{\mathbb{Z}}_{l}}\overline{\mathbb{F}}_{l}$ its
modulo $l$ reduction.

######  Notation 6.1.

For $\psi_{v}$ a $\overline{\mathbb{F}}_{l}$-representation of
$GL_{h}(F_{v})$, we denote by $\dim_{\overline{\mathbb{F}}_{l},n}\psi$ the set

$\Bigl{\\{}\dim_{\overline{\mathbb{F}}_{l}}\psi_{v}^{K_{v}(n)},\leavevmode\nobreak\
n\in{\mathbb{N}}\Bigr{\\}}.$

######  Theorem 6.2.

(cf. conjecture 5.2.1 of [4])
Let $r$ be maximal such that there exists $s$ and
$\widetilde{\mathfrak{m}}\subset\mathfrak{m}$ with
$\Pi_{\widetilde{\mathfrak{m}}}\in{\mathcal{A}}_{\xi,\pi_{v}}(r,s)$ and
$\Pi_{\widetilde{\mathfrak{m}},S}^{I^{v}}\neq(0)$. Then we have

$\sum_{\Pi\in{\mathcal{A}}_{\xi,\pi_{v}}(r,s)}m(\Pi)d_{\xi}(\Pi_{\infty})\dim_{\overline{\mathbb{Q}}_{l}}(\Pi^{\infty,v})^{I^{v}}\dim_{\overline{\mathbb{F}}_{l},n}r_{l}\bigl{(}R_{\pi_{v}}(r,r)(\Pi_{v})\bigr{)}\\\
=\\\
\sum_{\Pi^{\prime}\in{\mathcal{A}}_{\xi,\pi^{\prime}_{v}}(r,s)}m(\Pi^{\prime})d_{\xi}(\Pi^{\prime}_{\infty})\dim_{\overline{\mathbb{Q}}_{l}}(\Pi^{{}^{\prime},\infty,v})^{I^{v}}\dim_{\overline{\mathbb{F}}_{l},n}r_{l}\bigl{(}R_{\pi_{v}}(r,r)(\Pi^{\prime}_{v})\bigr{)}.$
(6.3)

Remark. The result is still valid without the maximality hypothesis on $r$ but
it is more tricky to expose, cf. the remark after the proof.

A qualitative version of the previous theorem could be formulated as follows.

  * •

Given a irreducible $\xi$-cohomological automorphic representation $\Pi$ of
$G({\mathbb{A}})$ such that
$\Pi_{v}\simeq\mathop{\mathrm{Speh}}\nolimits_{s}({\mathop{\mathrm{St}}\nolimits}_{t}(\pi_{v}))\times?$,
with $\pi_{v}$ cuspidal with modulo $l$ reduction still supercuspidal,

  * •

and $\pi^{\prime}_{v}$ such that its modulo $l$ reduction is isomorphic to
those of $\pi_{v}$,

then there exists a irreducible $\xi$-cohomological automorphic representation
$\Pi^{\prime}$ such that

  * •

outside $v$, $\Pi$ and $\Pi^{\prime}$ share the same level $I^{v}$,

  * •

their also share the same modulo $l$ Satake parameters at the places outside
$S$, i.e. they are weakly automorphic congruent,

  * •

at $v$, we have
$\Pi^{\prime}_{v}\simeq\mathop{\mathrm{Speh}}\nolimits_{s}({\mathop{\mathrm{St}}\nolimits}_{t}(\pi_{v}))\times?^{\prime}$,

where by convention, the symbols $?$ and $?^{\prime}$ mean any representation
we do not want to precise.

The strategy of the proof is the same as in [4]. The main point to notice is
that the left hand side of (6.3) is a part of the image of
$H^{0}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=rg}_{!*}HT_{\xi}(\pi_{v},\Pi_{r}))_{\mathfrak{m}}$.
In [4] §5.3, we look at the particular case where the localized cohomology
groups at $\mathfrak{m}$ are concentrated in degree $0$: this is for example
the case when $\xi$ is very regular, or if $\overline{\rho}_{\mathfrak{m}}$ is
irreducible. Then

$[H^{*}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=rg}_{!*}HT_{\xi}(\pi_{v},\Pi_{r}))_{\mathfrak{m}}]=[H^{0}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=rg}_{!*}HT_{\xi}(\pi_{v},\Pi_{r}))_{\mathfrak{m}}],$

and its modulo $l$ reduction is then equal to

$H^{*}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},{\mathbb{F}}\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=rg}_{!*}HT_{\xi}(\pi^{\prime}_{v},\Pi_{r}))_{\mathfrak{m}}=H^{*}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},j^{=rg}_{!*}HT_{\xi,\overline{\mathbb{F}}_{l}}(\pi^{\prime}_{v},\Pi_{r}))_{\mathfrak{m}}\\\
=r_{l}\Bigl{(}H^{0}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=rg}_{!*}HT_{\xi}(\pi^{\prime}_{v},\Pi_{r}))_{\mathfrak{m}}\Bigr{)},$

where
${\mathbb{F}}(\bullet)=\bullet\otimes^{\mathbb{L}}_{\overline{\mathbb{Z}}_{l}}\overline{\mathbb{F}}_{l}$
and by the main result of [5], as $\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=rg}_{!*}HT_{\xi}(\pi^{\prime}_{v},\Pi_{r})\simeq\kern
0.5pt\vphantom{j}^{p+}\kern-0.5ptj^{=rg}_{!*}HT_{\xi}(\pi^{\prime}_{v},\Pi_{r})$
for $\pi^{\prime}_{v}$ such that $r_{l}(\pi^{\prime}_{v})$ remains
supercuspidal, then

${\mathbb{F}}\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=rg}_{!*}HT_{\xi}(\pi^{\prime}_{v},\Pi_{r})\simeq
j^{=rg}_{!*}HT_{\xi,\overline{\mathbb{F}}_{l}}(\pi^{\prime}_{v},\Pi_{r}).$

To isolate the various contribution in these cohomology groups, we start from
$r=d$ towards $r=1$ so that for a fixed $r$, as for $r^{\prime}>r$, the modulo
$l$ reduction of elements of ${\mathcal{A}}_{\xi,\pi_{v}}(r^{\prime},1)$ was,
by an inductive argument, already identified with those of
${\mathcal{A}}_{\xi,\pi^{\prime}_{v}}(r^{\prime},1)$, we can then identify the
contribution of ${\mathcal{A}}_{\xi,\pi_{v}}(r,1)$ in the modulo $l$ reduction
of $[H^{*}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=rg}_{!*}HT_{\xi}(\pi_{v},\Pi_{r}))_{\mathfrak{m}}]$
to that of ${\mathcal{A}}_{\xi,\pi^{\prime}_{v}}(r,1)$.

In the general case where $s$ is not necessary equal to $1$ but its congruence
modulo $2$ is fixed, the first problem is that
${\mathcal{A}}_{\xi,\pi_{v}}(r,s)$ and
${\mathcal{A}}_{\xi,\pi_{v}}(r,s^{\prime})$ with $s>s^{\prime}$ first
contribute to
$[H^{0}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=rg}_{!*}HT_{\xi}(\pi_{v},\Pi_{r}))_{\mathfrak{m}}]$
at the same time. To separate them, the only solution seems to look at
$[H^{1-s}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=(r+1-s)g}_{!*}HT_{\xi}(\pi_{v},\Pi_{r}))_{\mathfrak{m}}]$
where only ${\mathcal{A}}_{\xi,\pi_{v}}(r,s)$ contributes. We are then led to
argue on individual cohomology groups where we recall the following short
exact sequence for a torsion free $\overline{\mathbb{Z}}_{l}$-perverse sheaf
$P$ over a $\overline{\mathbb{F}}_{p}$-scheme $X$:

$0\rightarrow{\mathbb{F}}H^{i}(X,P)\longrightarrow
H^{i}(X,{\mathbb{F}}P)\longrightarrow H^{i+1}(X,P)[l]\rightarrow 0,$

where in our situation $P=\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=tg}_{!*}HT(\pi_{v},\Pi_{t})$ and, as
recall before, as the $p$ and $p+$ intermediate extension of
$HT(\pi_{v},\Pi_{t})$ are isomorphic, then

${\mathbb{F}}\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=tg}_{!*}HT(\pi_{v},\Pi_{t})\simeq
j^{=tg}_{!*}HT_{\overline{\mathbb{F}}_{l}}(\pi_{v},\Pi_{t}).$

We then must face the problem of understanding the $l$-torsion and its
contribution in the modulo $l$ reduction of the cohomology. In [4] §5.4, we
formulate the conjecture that, for a fixed representation $\Pi_{t}$ of
$GL_{tg}(F_{v})$, the modulo $l$ reduction of the torsion in each cohomology
group of $\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=tg}_{!*}HT(\pi_{v},\Pi_{t})$ should depend
only on the modulo $l$ reduction of $\pi_{v}$ and we explain how this
conjecture implies the previous theorem.

###### Proof.

We first look at
$H^{0}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=rg}_{!*}HT_{\xi}(\pi_{v},\Pi_{t}))_{\mathfrak{m}}$
which, by maximality of $r$ and the spectral sequence associated to (5.1), is
isomorphic to
$H^{0}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},j^{=rg}_{!}HT_{\xi}(\pi_{v},\Pi_{t}))_{\mathfrak{m}}$
and so is torsion free. To this cohomology group contribute the irreducible
automorphic representations of ${\mathcal{A}}_{\xi,\pi_{v}}(r,s)$ with
$r=s+t-1$ for some set

${\mathcal{B}}(\pi_{v})=\Bigl{\\{}(s,t)\hbox{ s. t.
}\exists\widetilde{\mathfrak{m}}\subset\mathfrak{m},\Pi_{\widetilde{\mathfrak{m}}}\in{\mathcal{A}}_{\xi,\pi_{v}}(r,s)\hbox{
and }(\Pi_{\widetilde{\mathfrak{m}}}^{\infty,v})^{I_{S}}\neq(0)\Bigr{\\}}.$

The qualitative version of the theorem asks to prove that
${\mathcal{B}}(\pi_{v})={\mathcal{B}}(\pi^{\prime}_{v})$ and the quantitative
one then follows from the formula of the multiplicities in 4.5.

(a) About the quantitative version, consider first the easiest case when the
qualitative version tells us
${\mathcal{B}}(\pi_{v})={\mathcal{B}}(\pi^{\prime}_{v})=\\{(s,t)\\}$. Then the
modulo $l$ reduction of

$[H^{0}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=rg}_{!*}HT_{\xi}(\pi_{v},\Pi_{t}))_{\mathfrak{m}}]=[H^{*}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=rg}_{!*}HT_{\xi}(\pi_{v},\Pi_{t}))_{\mathfrak{m}}]$

is equal to that of
$H^{*}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=rg}_{!*}HT_{\xi}(\pi^{\prime}_{v},\Pi_{t}))_{\mathfrak{m}}$
which is also equal to that of
$H^{0}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(\infty),\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=rg}_{!*}HT_{\xi}(\pi^{\prime}_{v},\Pi_{t}))_{\mathfrak{m}}$.
We can then conclude, as these two cohomology groups are torsion free, from
the explicit computation of the multiplicities in 4.5.

(b) We now focus on the sequence of the dimension

$d_{k,n}=\dim_{\overline{\mathbb{F}}_{l}}H^{k}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(n),\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=(r-k)g}_{!*}HT_{\xi}(\pi_{v},\mathds{1}_{r-k}))_{\mathfrak{m}}\otimes_{\overline{\mathbb{Z}}_{l}}\overline{\mathbb{F}}_{l},$

where $\mathds{1}_{r_{k}}$ is the trivial representation of
$GL_{(r-k)g}(F_{v})$, for $k=0,\cdots,r-1$ and $n\in{\mathbb{N}}$. More
specifically we will focus on the $d_{k,n}-d_{k+1,n}$. Note the following
facts:

  * •

By maximality of $r$, we have
$H^{k+1}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(n),\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=(r-k)g}_{!*}HT_{\xi}(\pi_{v},\mathds{1}))_{\mathfrak{m}}=(0)$
so that

$H^{k}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(n),\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=(r-k)g}_{!*}HT_{\xi}(\pi_{v},\mathds{1}_{r-k}))_{\mathfrak{m}}\otimes_{\overline{\mathbb{Z}}_{l}}\overline{\mathbb{F}}_{l}\\\
\simeq
H^{k}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(n),\bar{s}_{v}},j^{=(r-k)g}_{!*}HT_{\xi,\overline{\mathbb{F}}_{l}}(\pi_{v},\mathds{1}_{r-k}))_{\mathfrak{m}}.$

  * •

By lemma 5.7, the dimension of the modulo $l$ reduction of the torsion of
$H^{k}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(n),\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=(r-k)g}_{!*}HT_{\xi}(\pi_{v},\mathds{1}))_{\mathfrak{m}}$
is prescribed by those of
$H^{1}(\mathop{\mathrm{Sh}}\nolimits_{I^{v}(n),\bar{s}_{v}},\kern
0.5pt\vphantom{j}^{p}\kern-0.5ptj^{=(r-1)g}_{!*}HT_{\xi}(\pi_{v},\mathds{1}))_{\mathfrak{m}}$.

  * •

From 4.5 and the description in 4.6 of the $m_{s,t}(r,i)$, the behavior of the
contribution to the sequence $d_{k,n}-d_{k+1,n}$ of the modulo $l$ reduction
of the free quotients of each cohomology groups is completely determined by
${\mathcal{B}}(\pi_{v})$. Indeed note that the contribution of some
automorphic representation $\Pi$ such that
$\Pi_{v}\simeq\mathop{\mathrm{Speh}}\nolimits_{s}({\mathop{\mathrm{St}}\nolimits}_{t}(\pi_{v}))\times?$
with $s+t-1=r$, only contributes to $d_{k,n}$ for $k=0,\cdots,s-1$ so that
$d_{s-1,n}-d_{s,n}$, after eliminating the contribution of the torsion part,
will detect such $\Pi$.

So as the contribution of the torsion to the sequence $(d_{k,n})_{k\geq
1,n\in{\mathbb{N}}}$ depends only on $(d_{1,n})_{n\in{\mathbb{N}}}$, we can
then infer the elements $(s,t)\in{\mathcal{B}}(\pi_{v})$ (resp.
${\mathcal{B}}(\pi^{\prime}_{v})$) with $s\geq 2$ and prove the qualitative
previous theorem for $s\geq 2$. To deduce the result for $s=1$, we then look
at $(d_{0,n})_{n\in{\mathbb{N}}}$ where the contribution of the torsion is
zero and where the contribution of the free quotient of each of the $s\geq 2$
are the same for $\pi_{v}$ and $\pi^{\prime}_{v}$. The remaining part for
$\pi_{v}$ and $\pi^{\prime}_{v}$ should then match which give us the case of
$s=1$.

∎

Remark. We could keep on tracing the contribution of the torsion sub-modules,
to the modulo $l$ reduction of the others cohomology groups. Once the
contribution coming from $r$ maximal is understood, we could consider the next
$r^{\prime}<r$ and repeat the previous argument taking into account what is
already known about the contribution of what is related to $r$.

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