# Cohomology of non-generic character stacks

Tommaso Scognamiglio
Université Paris Cité/IMJ-PRG
<EMAIL_ADDRESS>
###### Abstract

We study (compactly supported) cohomology of character stacks of punctured
Riemann surface with prescribed semisimple local monodromies at punctures. In
the case of generic local monodromies, the cohomology of these character
stacks has been studied in [27], [42].

In this paper we extend the results and conjectures of [27] to the non-generic
case. In particular we compute the E-series and give a conjectural formula for
the mixed Poincaré series. We prove our conjecture in the case of the
projective line with $4$ punctures.

In the non-punctured case, formulas of a similar kind have already appeared in
the recent work of Davison, Hennecart, Schlegel-Mejia [11]. Their approach
involves non-abelian Hodge theory and BPS sheaves, which is not yet available
in the more general context of this paper, i.e the cohomology of BPS sheaves
is not known in the punctured case and the use of non-abelian Hodge theory
makes difficult the study of the weight filtration on character stacks and in
particular the study of E-series.

###### Contents

  1. 1 Introduction
    1. 1.1 Review of generic character stacks
    2. 1.2 Overview of the paper
    3. 1.3 Main results
      1. 1.3.1 E-series of character stacks
      2. 1.3.2 Conjecture for mixed Poincaré series of character stacks
    4. 1.4 Final remark
  2. 2 Preliminaries and notations
    1. 2.1 Partitions and multipartitions
    2. 2.2  Multitypes
    3. 2.3 Lambda rings and plethystic operations
      1. 2.3.1 Multitypes and plethysm
    4. 2.4 Varieties over finite fields
  3. 3 Finite reductive groups, subtori and multitypes
    1. 3.0.1 Twisted Frobenius of maximal tori
    2. 3.0.2 The case of finite general linear groups
    3. 3.1 Levi subgroups and parabolic subgroups of general linear groups
    4. 3.2 Admissible subtori of general linear groups
    5. 3.3 Regular elements and Möbius function for admissible tori
      1. 3.3.1 Poset of $F$-stable admissible subtori
      2. 3.3.2 Möbius functions of locally finite posets
      3. 3.3.3 Möbius function for admissible subtori
    6. 3.4 Levi subgroups and graphs
      1. 3.4.1 Notations on graphs
      2. 3.4.2 Graph associated to an admissible torus
      3. 3.4.3 Admissible subtori and admissible graphs
    7. 3.5 Inclusion of admissible subtori
  4. 4 Log-compatible functions and plethystic identities
    1. 4.1 Plethysm and log compatibility: main result
  5. 5 Computation of convolution product for products of finite general linear groups
    1. 5.1 Convolution product of class functions of a finite group
    2. 5.2 Deligne-Lusztig induction
    3. 5.3 Unipotent characters
    4. 5.4 Characters of tori and graphs
    5. 5.5 Characters of tori and Levi subgroups of finite general linear group
      1. 5.5.1 Reduced characters and connected centralizers
      2. 5.5.2 Characters of Levi subgroups
    6. 5.6 Irreducible characters of finite general linear group
    7. 5.7 Type of an irreducible character
    8. 5.8 Multiplicative parameters
    9. 5.9 Convolution product for general linear group and dual Log compatiblity
  6. 6 Multiplicative quiver stacks for star-shaped quivers
    1. 6.1 Quiver representations
    2. 6.2 Star-shaped quivers and multiplicative quiver stacks
    3. 6.3 Character stacks for Riemann surfaces
      1. 6.3.1 Resolution of adjoint orbits
      2. 6.3.2 Multiplicative quiver stacks and character stacks
    4. 6.4 Dual Log compatibility for moment map
    5. 6.5 Proof of Theorem 6.4.1
    6. 6.6 Remarks
  7. 7 Cohomology of character stacks
    1. 7.1 Mixed Poincaré series of Artin stacks
    2. 7.2 E-series of multiplicative quiver stacks and character stacks
      1. 7.2.1 Cohomological results for generic $k$-tuples
      2. 7.2.2 Main result
      3. 7.2.3 E-series for character stacks with semisimple monodromies
  8. 8 Mixed Poincaré series of character stacks for $\mathbb{P}^{1}_{\mathbb{C}}$ with four punctures
    1. 8.1 Cohomology computations
      1. 8.1.1 Cohomology of the character variety in the non-generic case
      2. 8.1.2 Cohomology of the character stack in the non-generic case

##  1 Introduction

Consider a Riemann surface $\Sigma$ of genus $g\geq 0$, a subset
$D=\\{p_{1},\dots,p_{k}\\}\subseteq\Sigma$ of $k$ points and
$\mathcal{C}=(\mathcal{C}_{1},\dots,\mathcal{C}_{k})$ a $k$-tuple of
semisimple adjoint orbits of $\operatorname{GL}_{n}(\mathbb{C})$.

The associated character stack $\mathcal{M}_{\mathcal{C}}$ is defined as the
quotient stack

$\mathcal{M}_{\mathcal{C}}\coloneqq\left[\biggl{\\{}\rho\in\operatorname{Hom}(\pi_{1}(\Sigma\setminus
D),\operatorname{GL}_{n}(\mathbb{C}))\ |\ \rho(x_{i})\in\mathcal{C}_{i}\text{
for }i=1,\dots,k\biggr{\\}}\middle/\operatorname{GL}_{n}(\mathbb{C})\right]$
(1.0.1)

where each $x_{i}$ is a small loop around the point $p_{i}$. These stacks
classify local systems on $\Sigma\setminus D$ such that the monodromy around
the point $p_{i}$ lies in $\mathcal{C}_{i}$, for $i=1,\dots,k$ and are
naturally related to certain moduli spaces of (strongly) parabolic Higgs
bundles on $\Sigma$ via the non-abelian Hodge correspondence, see for example
the work of Simpson [49].

The stack $\mathcal{M}_{\mathcal{C}}$ has therefore the following explicit
form in terms of matrix equations:

$\mathcal{M}_{\mathcal{C}}=\left[\biggl{\\{}(A_{1},B_{1},\dots,B_{g},X_{1},\dots,X_{k})\in\operatorname{GL}^{2g}_{n}(\mathbb{C})\times\prod_{j=1}^{k}\mathcal{C}_{j}|\
\prod_{i=1}^{g}[A_{i},B_{i}]\prod_{j=1}^{k}X_{j}=1\biggr{\\}}\middle/\operatorname{GL}_{n}(\mathbb{C})\right].$
(1.0.2)

In what follows, for a complex stack of finite type $\mathcal{X}$, we will
denote by $H^{*}_{c}(\mathcal{X})\coloneqq H^{*}_{c}(\mathcal{X},\mathbb{C})$
its compactly supported cohomology with $\mathbb{C}$-coefficients (this is
well defined thanks to the work of Laszlo and Olsson [33]).

Recall that each vector space $H^{i}_{c}(\mathcal{X})$ is endowed with the
weight filtration $W^{i}_{\bullet}H^{i}_{c}(\mathcal{X})$, from which we
define the mixed Poincaré series $H_{c}(\mathcal{X},q,t)$

$H_{c}(\mathcal{X},q,t)\coloneqq\sum_{m,i}\dim(W^{i}_{m}/W^{i}_{m+1})q^{\frac{m}{2}}t^{i}.$

The E-series $E(\mathcal{X},q)$ is the specialization of
$H_{c}(\mathcal{X},q,t)$ obtained by plugging $t=-1$ and the Poincaré series
$P_{c}(\mathcal{X},t)$ is the specialization of $H_{c}(\mathcal{X},q,t)$
obtained by plugging $q=1$.

###  1.1 Review of generic character stacks

The cohomology of character stacks and its mixed Hodge structure have been
extensively studied from different perspectives. So far, most of the results
have been obtained in the case where the $k$-tuple $\mathcal{C}$ is generic
(see section 7.2.1 for a precise definition of genericity).

In the generic case, the stack $\mathcal{M}_{\mathcal{C}}$ is smooth and it is
a $\mathbb{G}_{m}$-gerbe over the associated GIT quotient, which we denote by
$M_{\mathcal{C}}$. Therefore the cohomology of $\mathcal{M}_{\mathcal{C}}$ can
be easily deduced from that of the character variety $M_{\mathcal{C}}$.

We give here a quick review of the results obtained about the cohomology of
generic character stacks and varieties, see section 7.2 for more details.

The first results concerning this subject were obtained in the case where
$k=1$ and $\mathcal{C}$ is a central conjugacy class. For $n\in\mathbb{N}$ and
$d\in\mathbb{Z}$, let $\mathcal{M}_{n,d}$ be the stack
$\mathcal{M}_{\mathcal{C}}$ for $k=1$ and $\mathcal{C}=\\{e^{\frac{2\pi
id}{n}}I_{n}\\}$ i.e

$\mathcal{M}_{n,d}=\left[\Biggl{\\{}(A_{1},B_{1},\dots,A_{g},B_{g})\in\prod_{i=1}^{2g}\operatorname{GL}_{n}(\mathbb{C})\
|\ \prod_{i=1}^{g}[A_{i},B_{i}]=e^{\frac{2\pi
id}{n}}I_{n}\Biggr{\\}}\middle/\operatorname{GL}_{n}(\mathbb{C})\right].$

The orbit $\mathcal{C}=\\{e^{\frac{2\pi id}{n}}\\}$ is generic if and only if
$(n,d)=1$.

Hitchin [29] computed the Poincaré polynomial $P_{c}(M_{n,d},t)$ in the
generic case for $n=2$, using non abelian Hodge correspondence and Morse
theory on the moduli space of Higgs bundles. Gothen [23] extended his result
for $n=3$.

Their approach was later extended to compute the Poincaré polynomial
$P_{c}(M_{\mathcal{C}},t)$ in the case where $n=2$, any $k$ and any generic
$k$-tuple $\mathcal{C}$ by Boden, Yogokawa [5] and where $n=3$, any $k$ and
any generic $k$-tuple $\mathcal{C}$ by García-Prada, Gothen and Munoz [20].

However, Morse theoretic techniques do not give information about weight
filtration and were hard to generalize to any $n$.

Hausel and Rodriguez-Villegas [26] were the first to obtain a general result
about cohomology of character stacks for any $n$. The authors computed the
E-series $E(\mathcal{M}_{n,d},q)$ of the stacks $\mathcal{M}_{n,d}$ for any
coprime $n,d$, by counting points over finite fields and proposed a
conjectural formula for the mixed Poincaré series
$H_{c}(\mathcal{M}_{n,d},q,t)$.

Schiffmann [47] found an expression for the Poincaré series
$P_{c}(\mathcal{M}_{n,d},t)$ in the generic case and Mellit [43] later checked
that Schiffmann’s formula agrees with the specialization of Hausel and
Rodriguez-Villegas’ conjecture at $q=1$.

Hausel, Letellier and Rodriguez-Villegas afterwards generalized the results of
[26] and computed [27, Theorem 1.2.3] the E-series
$E(\mathcal{M}_{\mathcal{C}},q)$ of the stacks $\mathcal{M}_{\mathcal{C}}$ for
any generic $k$-tuple $\mathcal{C}$. We quickly explain more precisely their
result since it is the starting point of this paper.

Let $\mathcal{P}$ be the set of partitions. In [27], the authors introduced,
for each multipartition $\bm{\mu}\in\mathcal{P}^{k}$, a rational function
$\mathbb{H}_{\bm{\mu}}(z,w)\in{\mathbb{Q}}(z,w)$, defined in terms of
Macdonald polynomials (for a precise definition see section 7.2.1). The
authors [27, Theorem 1.2.3] showed that there is an equality

$E(\mathcal{M}_{\mathcal{C}},q)=\dfrac{q^{\frac{d_{\bm{\mu}}}{2}}}{q-1}\mathbb{H}_{\bm{\mu}}\left(\sqrt{q},\dfrac{1}{\sqrt{q}}\right)$
(1.1.1)

where $2d_{\bm{\mu}}=\dim(\mathcal{M}_{\mathcal{C}})+1$ and
$\bm{\mu}=(\mu^{1},\dots,\mu^{k})$ is the multipartition given by the
multiplicities of the eigenvalues of $\mathcal{C}_{1},\dots,\mathcal{C}_{k}$
respectively.

In the same paper, the authors [27, Conjecture 1.2.1] proposed the following
conjectural formula for the mixed Poincaré series
$H_{c}(\mathcal{M}_{\mathcal{C}},q,t)$, which generalizes Hausel and
Rodriguez-Villegas’ conjecture stated in [26] and naturally deformes Identity
(1.1.1):

$H_{c}(\mathcal{M}_{\mathcal{C}},q,t)=\dfrac{(qt^{2})^{\frac{d_{\mu}}{2}}}{qt^{2}-1}\mathbb{H}_{\bm{\mu}}\left(t\sqrt{q},-\dfrac{1}{\sqrt{q}},\right).$
(1.1.2)

Mellit [42, Theorem 7.12] later computed the Poincaré series
$P_{c}(\mathcal{M}_{\mathcal{C}},t)$ using non-abelian Hodge correspondence.
His formula matches with the specialization at $q=1$ of the conjectural
formula (1.1.2) for the mixed Poincaré series.

###  1.2 Overview of the paper

One of the aim of this paper is the study of the cohomological invariants of
the stacks $\mathcal{M}_{\mathcal{C}}$ in the case where $\mathcal{C}$ is not
necessarily generic. The cohomology of character stacks
$\mathcal{M}_{\mathcal{C}}$ for non-generic $k$-tuples $\mathcal{C}$ has not
been much studied in the literature until recently.

The most explicit and general results in the non-generic case were obtained
mainly for the stacks $\mathcal{M}_{n,d}$.

Hausel and Rodriguez-Villegas [26, Theorem 3.8.1] expressed the E-series for
the stacks $\mathcal{M}_{n,0}$ in terms of the E-series for the generic
character stacks $\mathcal{M}_{n,1}$ by the following formula:

$\operatorname{Exp}\left(\sum_{n\in\mathbb{N}}\dfrac{E(\mathcal{M}_{n,1},q)}{q^{n^{2}(g-1)}}T^{n}\right)=\sum_{n\in\mathbb{N}}\dfrac{E(\mathcal{M}_{n,0},q)}{q^{n^{2}(g-1)}}T^{n}$
(1.2.1)

where $\operatorname{Exp}$ is the plethystic exponential in the ring of formal
power series ${\mathbb{Q}}(q)[[T]]$ (see section 2.3 for details about
plethystic operations). The authors’ result is obtained by counting points
over finite fields.

Fix now $r\in{\mathbb{Q}}$. Recently, Davison, Hennercart and Schelegel-Mejia
[11, Theorem 14.3, Corollary 14.7] proved the following formula expressing the
compactly supported Poincaré series of $\mathcal{M}_{n,d}$ for any $n,d$, in
terms of the Poincaré series for the generic character stacks
$\mathcal{M}_{n,1}$:

$\sum_{\begin{subarray}{c}(n,d)\in\mathbb{N}_{>0}\times\mathbb{Z}\\\
d=rn\end{subarray}}\dfrac{P_{c}(\mathcal{M}_{n,d},t)}{t^{n^{2}(2g-2)}}z^{n}w^{d}=\operatorname{Exp}\left(\sum_{\begin{subarray}{c}(n,d)\in\mathbb{N}_{>0}\times\mathbb{Z}\\\
d=rn\end{subarray}}\dfrac{P_{c}(\mathcal{M}_{n,1},t)}{t^{n^{2}(2g-2)}}z^{n}w^{d}\right)$
(1.2.2)

and formulated a similar conjecture for the mixed Poincaré series of
$H_{c}(\mathcal{M}_{n,d},q,t)$ for any $n,d$ (see the discussion after [11,
Theorem 14.10]).

They obtained this formula by relating the cohomology of a character stack
with the cohomology of the so-called BPS sheaves. The latter are certain
perverse sheaves defined on character varieties and are well-understood for
the stacks $\mathcal{M}_{n,d}$. More precisely the non-abelian Hodge
correspondence for stacks, proved in [11], and the recent work of Koseki and
Kinjo [32] about BPS sheaves for the moduli stack of Higgs bundles, give a way
to compute the cohomology of BPS sheaves for a stack $\mathcal{M}_{n,d}$.

However notice that, since the authors use non abelian Hodge correspondence
which does not preserve weight filtration on cohomology, their method does not
allow to prove an analogous formula for the E-series or the mixed Poincaré
series of $\mathcal{M}_{n,d}$.

Finally, the cohomology of BPS sheaves for character stacks
$\mathcal{M}_{\mathcal{C}}$ is not understood for an arbitrary $\mathcal{C}$
and so a generalisation of Formula (1.2.2) for an arbitrary $\mathcal{C}$ is
still unproved.

One of the main results of our paper is a generalization of (1.2.2) to
arbitrary $\mathcal{C}$ for the E-series $E(\mathcal{M}_{\mathcal{C}},q)$
instead of the Poincaré series $P_{c}(\mathcal{M}_{\mathcal{C}},t)$. As a
result we get an explicit formula for $E(\mathcal{M}_{\mathcal{C}},q)$ for any
$k$-tuple $\mathcal{C}$, see Theorem 1.3.3 below.

We also give a conjectural formula (see Conjecture 1.3.5) for the mixed
Poincaré series $H_{c}(\mathcal{M}_{\mathcal{C}},q,t)$, which we verify in the
case of $\Sigma=\mathbb{P}^{1}_{\mathbb{C}}$, $|D|=4$ and a certain family of
non-generic quadruples, for more details see section 8.

Conjecture 1.3.5 for the stacks $\mathcal{M}_{n,d}$ has already appeared in
[11], see the discussion there after Theorem $14.10$.

Finally let us notice that our approach is very different from that of [11] as
we do not use non-abelian Hodge theory nor BPS sheaves.

###  1.3 Main results

An important tool to formulate and prove the main results of this paper is the
construnction of character stacks as multiplicative quiver stacks, as first
introduced by Crawley-Boevey and Shaw [8],[9], which we quickly recall here
(see section 6.1 for more details).

Notice that this construction is not needed for studying generic character
stacks and does not appear for example in the articles [27], [42]. However, it
is a key point in our paper, as it allows to distinguish between different
levels of non-genericity for non-generic character stacks.

Let $s_{1},\dots,s_{k}\in\mathbb{N}$ be such that, for each $i=1,\dots,k$, the
conjugacy class $\mathcal{C}_{i}$ has $s_{i}+1$ distinct eigenvalues
$\gamma_{i,0},\dots,\gamma_{i,s_{i}}$ with multiplicities
$m_{i,0},\dots,m_{i,s_{i}}$ respectively. Let $Q=(I,\Omega)$ be the following
star-shaped quiver with $g$ loops on the central vertex

${\circ^{[1,1]}}$${\circ^{[1,2]}}$${\dots}$${\circ^{[1,s_{1}]}}$${\circ^{[2,1]}}$${\circ^{[2,2]}}$${\dots}$${\circ^{[2,s_{2}]}}$${\circ^{0}}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\circ^{[k,1]}}$${\circ^{[k,2]}}$${\dots}$${\circ^{[k,s_{k}]}}$$\scriptstyle{\cdots}$

Recall that for any $\beta\in\mathbb{N}^{I}$, there is a representation
variety $R(\overline{Q},\beta)^{\circ,\ast}$ and a multiplicative moment map

$\Phi^{\ast}_{\beta}:R(\overline{Q},\beta)^{\circ,\ast}\to\operatorname{GL}_{\beta}(\mathbb{C})\coloneqq\prod_{i\in
I}\operatorname{GL}_{\beta_{i}}(\mathbb{C}).$

For any $s\in(\mathbb{C}^{*})^{I}$, we denote by $s$ the central element
$s\coloneqq(s_{i}I_{\beta_{i}})_{i\in I}\in\operatorname{GL}_{\beta}$. The
multiplicative quiver stack with parameters $\beta,s$ is the quotient stack

$\mathcal{M}_{s,\beta}^{\ast}\coloneqq[(\Phi^{\ast}_{\beta})^{-1}(s)/\operatorname{GL}_{\beta}].$

Consider now the dimension vector $\alpha_{\mathcal{C}}\in\mathbb{N}^{I}$
defined as

$(\alpha_{\mathcal{C}})_{[i,j]}=\sum_{h=j}^{s_{i}}m_{i,h}$

for every $j=0,\dots,s_{i}$, where we are identifiying $[i,0]=0$ for each
$i=1,\dots,k$. Notice that $(\alpha_{\mathcal{C}})_{0}=n$. Let moreover
$\gamma_{\mathcal{C}}\in(\mathbb{C}^{*})^{I}$ be defined as follows

$\displaystyle(\gamma_{\mathcal{C}})_{[i,j]}=\begin{cases}\displaystyle\prod_{i=1}^{k}\gamma^{-1}_{i,0}\text{
if }j=0\\\ \gamma^{-1}_{i,j}\gamma_{i,j-1}\text{ otherwise }\par\end{cases}.$

The result of Crawley-Boevey [8, Proposition 2] implies that, for the elements
$\alpha_{\mathcal{C}},\gamma_{\mathcal{C}}$, there is an isomorphism of stacks

$\mathcal{M}_{\mathcal{C}}\cong\mathcal{M}^{\ast}_{\gamma_{\mathcal{C}},\alpha_{\mathcal{C}}}.$

Let now $(\mathbb{N}^{I})^{*}\subseteq\mathbb{N}^{I}$ be the subset of vectors
with non-increasing coordinates along the legs and denote by
$\mathcal{H}^{*}_{\gamma_{\mathcal{C}},\alpha_{\mathcal{C}}}\subseteq(\mathbb{N}^{I})^{*}$
the subset defined as

$\mathcal{H}_{\gamma_{\mathcal{C}},\alpha_{\mathcal{C}}}^{*}=\\{\delta\in(\mathbb{N}^{I})^{*}\
|\ \gamma_{\mathcal{C}}^{\delta}=1\ \text{ and
}\delta\leq\alpha_{\mathcal{C}}\\}$

where $\gamma_{\mathcal{C}}^{\delta}=\displaystyle\prod_{i\in
I}(\gamma_{\mathcal{C}})_{i}^{\delta_{i}}$.

######  Example 1.3.1.

It can be checked that if
$\mathcal{H}^{*}_{\gamma_{\mathcal{C}},\alpha_{\mathcal{C}}}=\\{\alpha_{\mathcal{C}}\\}$
then the $k$-tuple $\mathcal{C}$ is generic.

The subsets $\mathcal{H}^{*}_{\gamma_{\mathcal{C}},\alpha_{\mathcal{C}}}$
allow to define a natural stratification on the set of $k$-tuples
$\mathcal{C}$ and so of character stacks $\mathcal{M}_{\mathcal{C}}$.

The introduction of this stratification is one of the key ingredients to study
the cohomology of $\mathcal{M}_{\mathcal{C}}$ in the non-generic case.

Notice that altough not explicitly defined, the subsets
$\mathcal{H}^{*}_{\gamma_{\mathcal{C}},\alpha_{\mathcal{C}}}$ appear
implicitly in [11].

####  1.3.1 E-series of character stacks

For any $\beta\in(\mathbb{N}^{I})^{*}$ and for any $j=1,\dots,k$, the integers
$(\beta_{[j,0]}-\beta_{[j,1]},\dots,\beta_{[j,s_{j}-1]}-\beta_{[j,s_{j}]},\beta_{[j,s_{j}]})$
up to reordering form a partition $\mu_{\beta}^{j}\in\mathcal{P}$. Denote by
$\bm{\mu}_{\beta}\in\mathcal{P}^{k}$ the multipartition
$\bm{\mu}_{\beta}=(\mu_{\beta}^{1},\dots,\mu_{\beta}^{k})$ and by
$\mathbb{H}_{\beta}(z,w)$ the function $\mathbb{H}_{\bm{\mu}_{\beta}}(z,w)$.

######  Remark 1.3.2.

Notice that for a $k$-tuple $\mathcal{C}$, the multipartition
$\bm{\mu}_{\alpha_{\mathcal{C}}}\in\mathcal{P}^{k}$ is the multipartition
given by the multiplicities of the orbits
$\mathcal{C}_{1},\dots,\mathcal{C}_{k}$ respectively.

Moreover, it can be checked that
$\dim(\mathcal{M}_{\mathcal{C}})=2(\alpha_{\mathcal{C}},\alpha_{\mathcal{C}})+1$,
where $(,)$ is the Euler form of $Q$. The result [27, Theorem 1.2.3] of
Hausel, Letellier, Rodriguez-Villegas for a generic $k$-tuple $\mathcal{C}$
can then be rewritten as follows:

$\dfrac{E(\mathcal{M}_{\mathcal{C}},q)}{q^{(\alpha_{\mathcal{C}},\alpha_{\mathcal{C}})}}=\dfrac{q\mathbb{H}_{\alpha_{\mathcal{C}}}\left(\sqrt{q},\frac{1}{\sqrt{q}}\right)}{q-1}.$
(1.3.1)

The main result about character stacks of this paper (see Theorem 7.2.7) is
the following Theorem:

######  Theorem 1.3.3.

For any $k$-tuple of semisimple orbits $\mathcal{C}$, it holds:

$\operatorname{Coeff}_{\alpha_{\mathcal{C}}}\left(\operatorname{Exp}\left(\sum_{\begin{subarray}{c}\beta\in\mathcal{H}_{\gamma_{\mathcal{C}},\alpha_{\mathcal{C}}}^{*}\end{subarray}}\dfrac{q\mathbb{H}_{\beta}\left(\sqrt{q},\frac{1}{\sqrt{q}}\right)}{q-1}y^{\beta}\right)\right)=\dfrac{E(\mathcal{M}_{\mathcal{C}},q)}{q^{(\alpha_{\mathcal{C}},\alpha_{\mathcal{C}})}}.$
(1.3.2)

######  Remark 1.3.4.

One of the interesting aspects of Theorem 1.3.3 is that it expresses the
E-series $E(\mathcal{M}_{\mathcal{C}},q)$ for any $k$-tuple $\mathcal{C}$ in
terms of the functions
$\mathbb{H}_{\beta}\left(\sqrt{q},\frac{1}{\sqrt{q}}\right)$, i.e in terms of
the E-series for generic $k$-tuples.

Similar type of results, relating non-generic to generic, have already
appeared elsewhere, see for example the discussion in section 1.2 and also
Letellier’s paper [35].

In Chapter section 7, we compute the E-series of the complex character stacks
$\mathcal{M}_{\mathcal{C}}$ through the approach introduced in [26],[27],[38],
i.e by reduction over finite fields and point counting. Namely, recall that if
there exists a rational function $Q(t)\in{\mathbb{Q}}(t)$ such that, for any
$\mathbb{F}_{q}$-stack $\mathcal{M}_{\mathcal{C},\mathbb{F}_{q}}$ obtained
from $\mathcal{M}_{\mathcal{C}}$ by base change and any $m$, it holds

$\\#\mathcal{M}_{\mathcal{C},\mathbb{F}_{q}}(\mathbb{F}_{q^{m}})=Q(q^{m}),$

we have an equality

$E(\mathcal{M}_{\mathcal{C}},q)=Q(q),$

for more details see section 7.1.

However, the way we count rational points of character stacks in this paper is
quite different from that of [27]. The description of the rational functions
$Q(t)$ for non-generic character stacks is given through the results of
Chapter section 5 and section 6, where we show how to compute the rational
points of a multiplicative quiver stack for a star-shaped quiver over
$\mathbb{F}_{q}$.

Notice that in the articles [28],[27], the authors did not need to introduce
multiplicative quiver stacks to compute $\mathbb{F}_{q}$-points of generic
character stacks.

The results of section 5 and section 6 about the rational functions $Q(t)$
will be obtained as a consequence of the main technical result of this paper
which is Theorem 4.1.2. The latter theorem is very general and works for
certain families of rational functions called _Log compatible_.

Therefore, to prove Theorem 1.3.3 we will have to prove that the rational
functions involved in it satisfy this Log compatibility property.

In [48], we proved similar formulas in a different context : quiver
representations and multiplicities in tensor product of irreducible characters
of ${\rm GL}_{n}(\mathbb{F}_{q})$. Using categories of representations we
could avoid using the technical Theorem 4.1.2.

The proof of Theorem 4.1.2 will be one of the main technical results of the
first part of this paper. We will have to use combinatorial objects different
from the ones used for the generic case in [28],[27]. They we will be
introduced in Chapters section 3,section 4.

####  1.3.2 Conjecture for mixed Poincaré series of character stacks

Hausel, Letellier, Rodriguez Villegas conjectural formula (1.1.2) for the
mixed Poincaré series of character stacks for generic $k$-tuples and Theorem
1.3.3 suggest the following conjecture for the mixed Poincaré series of
character stacks:

######  Conjecture 1.3.5.

For any $k$-tuple of semisimple orbits $\mathcal{C}$, it holds:

$\operatorname{Coeff}_{\alpha_{\mathcal{C}}}\left(\operatorname{Exp}\left(\sum_{\begin{subarray}{c}\beta\in\mathcal{H}^{*}_{\gamma_{\mathcal{C}},\alpha_{\mathcal{C}}}\end{subarray}}\dfrac{(qt^{2})\mathbb{H}_{\beta}\left(t\sqrt{q},\frac{1}{\sqrt{q}}\right)}{qt^{2}-1}y^{\beta}\right)\right)=\dfrac{H_{c}(\mathcal{M}_{\mathcal{C}},q,t)}{(qt^{2})^{(\alpha_{\mathcal{C}},\alpha_{\mathcal{C}})}}.$
(1.3.3)

In Chapter 8, we verify that Conjecture 1.3.5 holds in the case of
$\Sigma=\mathbb{P}^{1}_{\mathbb{C}}$, $|D|=4$ and the following family of non-
generic quadruples.

Pick
$\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}\in\mathbb{C}^{*}\setminus\\{1,-1\\}$
and denote by $\mathcal{C}_{j}$ the conjugacy class of the diagonal matrix

$\begin{pmatrix}\lambda_{j}&0\\\ 0&\lambda_{j}^{-1}\end{pmatrix}.$

Assume moreover that $\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}$ have
the following property. Given $\epsilon_{1},\dots,\epsilon_{4}\in\\{1,-1\\}$
such that $\lambda_{1}^{\epsilon_{1}}\cdots\lambda_{4}^{\epsilon_{4}}=1$, then
either $\epsilon_{1}=\cdots=\epsilon_{4}=1$ or
$\epsilon_{1}=\cdots=\epsilon_{4}=-1$.

###  1.4 Final remark

In the generic case [27], the conjectural formula for the mixed Poincaré
series is written in the language of symmetric functions. More precisely, let
$\Lambda_{k}$ the ring of functions with values in the rational functions
${\mathbb{Q}}(z,w)$ and separately symmetric in $k$ sets of infinite variables
$\mathbf{x_{1}},\dots,\mathbf{x_{k}}$. Then the function
$\mathbb{H}_{\bm{\mu}}(z,w)$ mentioned in section 1.1 is defined as

$\mathbb{H}_{\bm{\mu}}(z,w)\coloneqq(z^{2}-1)(1-w^{2})\langle\operatorname{Coeff}_{T^{n}}(\operatorname{Log}(\Omega(z,w))),h_{\bm{\mu}}\rangle$

where $\Omega(z,w)\in\Lambda_{k}[[T]]$ (see [27, Paragraph 2.3.6] for a
definition), $h_{\bm{\mu}}=h_{\mu^{1}}(\mathbf{x_{1}})\cdots
h_{\mu^{k}}(\mathbf{x_{k}})$ is the complete symmetric functions and
$\langle,\rangle$ is the natural extension to $\Lambda_{k}$ of the bilinear
product on symmetric functions making Schur functions orthonormal.

One of the advantages of this approach is that the cohomology of every generic
character stack is encoded in the single object $\Omega(z,w)$. Moreover, this
allows to immediately generalize the results and the conjectures to the
cohomology of generic character stacks with non necessary semisimple local
monodromies by replacing the complete symmetric functions in the pairing by
other symmetric functions (for instance it would be Schur symmetric functions
for conjugacy classes which are the product of a unipotent conjugacy class
with a central element, see for example [35]).

It would be interesting to "symmetrize" our main formulas and conjectures.

##### Acknowledgements.

The author is very grateful to Ben Davison, Fernando Rodriguez-Villegas and
Olivier Schiffmann for many useful discussions about the topics dealt in this
paper.

The author would also like to thank Emmanuel Letellier for many useful
comments about a first draft of this paper. This article is a part of the
author’s PhD thesis.

##  2 Preliminaries and notations

In this Section, $I$ will be a fixed finite set. In the cases relevant to the
main result of this paper on non-generic character stacks, $I$ will be the set
of vertices of a star-shaped quiver.

In section 2.2 and section 2.3, we will introduce and recall the properties of
some combinatorial objects defined in terms of $I$, such as the multitypes and
an associated ring, which will be a key technical point of this paper and were
not needed in the works about the generic case [28],[27].

We endow the set $\mathbb{N}^{I}$ with the partial order $\leq$ such that, for
$\alpha,\beta\in\mathbb{N}^{I}$, we have $\beta\geq\alpha$ if and only if
$\beta_{i}\geq\alpha_{i}$ for each $i\in I$.

###  2.1 Partitions and multipartitions

Let $\mathcal{P}$ be the set of all partitions and
$\mathcal{P}^{*}\subseteq\mathcal{P}$ the subset of nonzero partitions. A
partition $\lambda$ will be denoted by
$\lambda=(\lambda_{1},\lambda_{2}\dots,\lambda_{h})$ with
$\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{h}$ or by
$\lambda=(1^{m_{1}},2^{m_{2}},\dots)$ where $m_{k}$ is the number of
occurrencies of the number $k$ in the partition $\lambda$. We will denote by
$\lambda^{\prime}$ the partition conjugate to $\lambda$.

The size of $\lambda$ is $\displaystyle|\lambda|=\sum_{i}\lambda_{i}$ and its
length $l(\lambda)$ is the largest $i$ such that $\lambda_{i}\neq 0$. For each
$n\in\mathbb{N}$, we denote by $\mathcal{P}_{n}$ the subset of partitions of
size $n$. For two partitions $\lambda,\mu$, we denote by
$\langle\lambda,\mu\rangle$ the quantity

$\langle\lambda,\mu\rangle=\sum_{i}\lambda_{i}^{\prime}\mu_{i}^{\prime}.$

Recall that the set $\mathcal{P}$ admits different possible orderings. In the
following, we will denote by $\lambda\leq\mu$ the ordering induced by the
lexicographic order.

Recall that the conjugacy classes of the symmetric group $S_{n}$ are indexed
by the partitions $\mathcal{P}_{n}$. For each $\lambda\in\mathcal{P}_{n}$,
denote by $z_{\lambda}$ the cardinality of the centralizer of an element of
the conjugacy class associated to $\lambda$. If
$\lambda=(1^{m_{1}},2^{m_{2}},\dots,)$, we have

$z_{\lambda}=\prod_{j}j!j^{m_{j}}.$

Moreover, recall that the set of irreducible characters of $S_{n}$ is in
bijection with $\mathcal{P}_{n}$. In our bijection we associate to the
partition $(n)$ the trivial character of $S_{n}$. We denote the irreducible
character of $S_{n}$ associated to $\lambda$ by $\chi^{\lambda}$.

For any three partitions
$\lambda\in\mathcal{P}_{n},\mu\in\mathcal{P}_{m},\nu\in\mathcal{P}_{n+m}$, we
denote by $c_{\lambda,\mu}^{\nu}$ the integer

$c_{\lambda,\mu}^{\nu}\coloneqq\langle\chi^{\nu},\operatorname{Ind}_{S_{n}\times
S_{m}}^{S_{n+m}}(\chi^{\lambda}\boxtimes\chi^{\mu})\rangle.$

Consider also the set of multipartitions $\mathcal{P}^{I}$. The elements of
$\mathcal{P}^{I}$ will be usually be denoted in bold letters
$\bm{\lambda}\in\mathcal{P}^{I}$. To avoid confusion with the notation used
for partition, we will use the notation $\bm{\lambda}=(\lambda^{i})_{i\in I}$.
For $\bm{\lambda}\in\mathcal{P}^{I}$, the size
$|\bm{\lambda}|\in\mathbb{N}^{I}$ of $\bm{\lambda}$ is defined as

$|\bm{\lambda}|_{i}\coloneqq|\lambda^{i}|.$

For an element $\alpha\in\mathbb{N}^{I}$, we will denote by
$(1^{\alpha})\in\mathcal{P}^{I}$ the multipartition $((1^{\alpha_{i}}))_{i\in
I}$ and by $(\alpha)\in\mathcal{P}^{I}$ the multipartition
$((\alpha_{i}))_{i\in I}$.

The order $\leq$ induces an ordering on $\mathcal{P}^{I}$ through the
lexicographical ordering, which we still denote by $\leq$.

###  2.2  Multitypes

A multitype is a function
$\omega:\mathbb{N}\times\mathcal{P}^{I}\to\mathbb{N}$ such that its support
(i.e the elements $(d,\bm{\mu})$ such that $\omega(d,\bm{\mu})\neq 0$) is
finite and $\omega(0,\bm{\lambda})=\omega(d,0)=0$ for any
$\bm{\lambda}\in\mathcal{P}^{I}$ and $d\in\mathbb{N}$.

On the set $\mathbb{N}\times\mathcal{P}^{I}$ put the total order defined as
follows. If $d>d^{\prime}$ then $(d,\bm{\lambda})>(d^{\prime},\bm{\mu})$, if
$|\bm{\lambda}|>|\bm{\mu}|$ then $(d,\bm{\lambda})>(d^{\prime},\bm{\mu})$ and
if $|\bm{\lambda}|=|\bm{\mu}|$ then $(d,\bm{\lambda})>(d^{\prime},\bm{\mu})$
if $\bm{\lambda}>\bm{\mu}$.

We can alternately think of a multitype $\omega$ as a non-decreasing sequence
$\omega=(d_{1},\bm{\lambda}_{1})\dots(d_{r},\bm{\lambda}_{r})$, where the
value $\omega(d,\bm{\lambda})$ corresponds to the number of times the element
$(d,\bm{\lambda})$ appear in the sequence
$(d_{1},\bm{\lambda}_{1})\dots(d_{r},\bm{\lambda}_{r})$.

We will denote by $\mathbb{T}_{I}$ the set of multitypes. If $|I|=1$, we call
multitypes simply types.

For $\omega\in\mathbb{T}_{I}$ where
$\omega=(d_{1},\bm{\lambda}_{1})\dots(d_{r},\bm{\lambda}_{r})$ and
$d\in\mathbb{N}_{>0}$, we denote by $\psi_{d}(\omega)$ the multitype

$\psi_{d}(\omega)\coloneqq(dd_{1},\bm{\lambda}_{1})\dots(dd_{r},\bm{\lambda}_{r}).$

Define the size $|\omega|$ of a multitype $\omega$ to be the following element
of $\mathbb{N}^{I}$

$\displaystyle|\omega|\coloneqq\sum_{(d,\bm{\mu})\in\mathbb{N}\times\mathcal{P}^{I}}d\omega(d,\bm{\mu})|\bm{\mu}|$

and $w(\omega)$ as the quantity

$w(\omega)\coloneqq\prod_{(d,\bm{\mu})\in\mathbb{N}\times\mathcal{P}^{I}}d^{\omega(d,\bm{\mu})}\omega(d,\bm{\mu})!.$

For $\alpha\in\mathbb{N}^{I}$, we denote by
$\mathbb{T}_{\alpha}\subseteq\mathbb{T}_{I}$ the subset of multitypes of size
$\alpha$.

The sum of multitypes endows the set $\mathbb{T}_{I}$ with an associative
operation $\ast:\mathbb{T}_{I}\times\mathbb{T}_{I}\to\mathbb{T}_{I}$. More
precisely, for $\omega_{1},\omega_{2}\in\mathbb{T}_{I}$, we define
$\omega_{1}\ast\omega_{2}$ as the multitype such that

$\omega_{1}\ast\omega_{2}(d,\bm{\lambda})\coloneqq\omega_{1}(d,\bm{\lambda})+\omega_{2}(d,\bm{\lambda}).$

The reason for this choice of notation will be clear later.

We view $\mathbb{N}_{>0}\times\mathbb{N}^{I}$ as a subset of
$\mathbb{N}_{>0}\times\mathcal{P}^{I}$ by associating to $(d,\alpha)$ the
element $(d,(1^{\alpha}))$. We call a multitype $\omega$ semisimple if its
support is contained in $\mathbb{N}_{>0}\times\mathbb{N}^{I}$. Given a
semisimple $\omega$ we will see it as a function
$\mathbb{N}_{>0}\times\mathbb{N}^{I}\to\mathbb{N}$ which we still denote by
$\omega$ where

$\omega(d,\alpha)\coloneqq\omega(d,(1^{\alpha})).$

Whenever the context is clear, we will frequently switch between the two
notations for semisimple multitypes.

For each $\alpha\in\mathbb{N}^{I}$, we denote by $\omega_{\alpha}$ the
semisimple multitype such that $\omega_{\alpha}(1,\alpha)=1$ and
$\omega_{\alpha}(d,\beta)=0$ for every other element
$(d,\beta)\in\mathbb{N}_{>0}\times\mathbb{N}^{I}$.

We denote by $\mathbb{T}_{I}^{ss}\subseteq\mathbb{T}_{I}$ the subset of
semisimple multitypes. Notice that for any semisimple multitype
$\omega\in\mathbb{T}_{I}^{ss}$, there exists
$d_{1},\dots,d_{r}\in\mathbb{N}_{>0}$ and
$\alpha_{1},\dots,\alpha_{r}\in\mathbb{N}^{I}$ such that

$\omega=\psi_{d_{1}}(\omega_{\alpha_{1}})\ast\cdots\ast\psi_{d_{r}}(\omega_{\alpha_{r}}).$

For a type
$\omega=(d_{1},\bm{\lambda}_{1})\dots(d_{r},\bm{\lambda}_{r})\in\mathbb{T}_{I}$,
we define its semisimplification $\omega^{ss}\in\mathbb{T}_{I}^{ss}$ as

$\omega^{ss}\coloneqq(d_{1},(1^{|\bm{\lambda}_{1}|}))\dots(d_{r},(1^{|\bm{\lambda}_{r}|})),$

i.e
$\omega^{ss}=\psi_{d_{1}}(\omega_{|\bm{\lambda}_{1}|})\ast\cdots\ast\psi_{d_{r}}(\omega_{|\bm{\lambda}_{r}|})$.

To a semisimple multitype $\omega=(d_{1},\alpha_{1})\dots(d_{r},\alpha_{r})$,
we associate the following polynomial $P_{\omega}(t)\in\mathbb{Z}[t]$

$P_{\omega}(t)\coloneqq\prod_{j=1}^{r}(t^{d_{j}}-1).$

Notice that for any $\alpha\in\mathbb{N}^{I}$, we have

$P_{\omega_{\alpha}}(t)=t-1.$

For a semisimple multitype
$\omega=(d_{1},\beta_{1})\cdots(d_{r},\beta_{r})\in\mathbb{T}^{ss}_{\alpha}$
put

$C_{\omega}^{o}:=\begin{cases}\mu(d)d^{r-1}(-1)^{r-1}(r-1)!&\text{ if
}d_{1}=d_{2}=\cdots=d_{r}=d\\\ 0&\text{ otherwise}.\end{cases}$

where $\mu$ denotes the ordinary Möbius function.

Lastly, we introduce the notion of levels for semisimple multitypes.

######  Definition 2.2.1.

For a subset $V\subseteq\mathbb{N}^{I}$ and a semisimple multitype $\omega$
with

$\omega=\psi_{d_{1}}(\omega_{\alpha_{1}})\ast\cdots\ast\psi_{d_{r}}(\omega_{\alpha_{r}}),$

we say that $\omega$ is of level $V$ if $\alpha_{j}\in V$ for each
$j=1,\dots,r$.

######  Example 2.2.2.

Notice that, for any $\alpha\in\mathbb{N}^{I}$, the multitype
$\omega_{\alpha}$ is of level $\\{\alpha\\}$. Conversely, the only multitype
$\omega\in\mathbb{T}_{\alpha}$ of level $\\{\alpha\\}$ is $\omega_{\alpha}$.

###  2.3 Lambda rings and plethystic operations

In this Paragraph we recall the definition and some properties of
$\lambda$-rings. We follow [45].

######  Definition 2.3.1.

A $\lambda$-ring $R$ is a commutative ${\mathbb{Q}}$-algebra with
homomorphisms $\psi_{d}:R\to R$ for any $d\geq 1$ such that
$\psi_{d^{\prime}}(\psi_{d}(r))=\psi_{dd^{\prime}}(r)$ for every
$d,d^{\prime}\in\mathbb{N}_{>0}$ and $r\in R$.

The morphisms $\psi_{d}$ are called Adams operations. For any partition
$\mu=(\mu_{1},\dots,\mu_{h})$, we denote by $\psi_{\mu}:R\to R$ the
homomorphism defined by
$\psi_{\mu}(r)=\psi_{\mu_{1}}(r)\cdots\psi_{\mu_{h}}(r)$.

For every integer $n\in\mathbb{N}$, denote by $\sigma_{n}(f)$ the element

$\sigma_{n}(f)=\sum_{\lambda\in\mathcal{P}_{n}}\dfrac{\psi_{\lambda}(f)}{z_{\lambda}}$
(2.3.1)

For a $\lambda$-ring $R$, consider now the ring $R[[y_{i}]]_{i\in I}$. For
$\alpha\in\mathbb{N}^{I}$ we will denote by $y^{\alpha}$ the monomial
$\displaystyle\prod_{i\in I}y_{i}^{\alpha_{i}}$. We endow the ring
$R[[y_{i}]]_{i\in I}$ with the $\lambda$-ring structure induced by the Adams
operations defined by

$\psi_{d}(ry^{\alpha})\coloneqq\psi_{d}(r)y^{d\alpha}$

for $r\in R$ and $\alpha\in\mathbb{N}^{I}$.

Denote by $R[[y_{i}]]_{i\in I}^{+}$ the ideal generated by the $y_{i}$’s. The
plethystic exponential is the following map
$\operatorname{Exp}:R[[y_{i}]]_{i\in I}^{+}\to 1+R[[y_{i}]]_{i\in I}^{+}$:

$\operatorname{Exp}(f)=\exp\left(\sum_{n\geq 1}\dfrac{\psi_{n}(f)}{n}\right).$
(2.3.2)

Notice that for $f,g\in R[[y_{i}]]_{i\in I}^{+},$ we have
$\operatorname{Exp}(f+g)=\operatorname{Exp}(f)\operatorname{Exp}(g)$.

######  Example 2.3.2.

Consider $R={\mathbb{Q}}$ and $|I|=1$. In the ring ${\mathbb{Q}}[[T]]$ we
have:

$\operatorname{Exp}(T)=\exp\left(\sum_{n\geq
1}\frac{T^{n}}{n}\right)=\exp\left(\log\left(\frac{1}{1-T}\right)\right)=\frac{1}{1-T}$
(2.3.3)

We have the following Lemma (see [45, Appendix A]):

######  Lemma 2.3.3.

For any $f\in R[[y_{i}]]^{+}_{i\in I}$, we have

$\operatorname{Exp}(f)=\sum_{n\geq 1}\sigma_{n}(f).$ (2.3.4)

The plethystic exponential admits an inverse operation
$\operatorname{Log}:1+R[[y_{i}]]_{i\in I}^{+}\to R[[y_{i}]]_{i\in I}^{+}$
known as plethystic logarithm. The plethystic logarithm can either be defined
by the property $\operatorname{Log}(\operatorname{Exp}(f))=f$ or by the
following explicit rule. For $\alpha\in\mathbb{N}^{I}$ we put

$\overline{\alpha}\coloneqq\gcd(\alpha_{i})_{i\in I}$

and we define $U_{\alpha}\in R$ by:

$\log(f)=\sum_{\alpha\in\mathbb{N}^{I}}\frac{U_{\alpha}}{\overline{\alpha}}y^{\alpha}.$
(2.3.5)

Then put

$\operatorname{Log}(f)\coloneqq\sum_{\alpha\in\mathbb{N}^{I}}V_{\alpha}y^{\alpha}$
(2.3.6)

where

$V_{\alpha}\coloneqq\dfrac{1}{\overline{\alpha}}\sum_{d|\overline{\alpha}}\mu(d)\psi_{d}(U_{\frac{\alpha}{d}})$
(2.3.7)

Notice that, for $A,B\in 1+R[[y_{i}]]^{+}_{i\in I}$, we have

$\operatorname{Log}(AB)=\operatorname{Log}(A)+\operatorname{Log}(B).$

####  2.3.1 Multitypes and plethysm

Let $\mathcal{K}^{ss}_{I}$ be the ${\mathbb{Q}}$-vector space having as a base
the semisimple multitypes $\mathbb{T}_{I}^{ss}$. The size of the types endows
$\mathcal{K}^{ss}_{I}$ with an $\mathbb{N}^{I}$-grading and the operation
$\ast$ endows $\mathcal{K}^{ss}_{I}$ with the structure of an
$\mathbb{N}^{I}$-graded ${\mathbb{Q}}$-algebra.

The operation $\psi_{d}$ on multitypes endows the ${\mathbb{Q}}$-algebra
$\mathcal{K}^{ss}_{I}$ with the structure of a $\lambda$-ring with Adams
operations defined as

$\psi_{d}(q_{1}\omega_{1}+\cdots+q_{r}\omega_{r})=q_{1}\psi_{d}(\omega_{1})+\cdots+\psi_{d}(\omega_{r})$

for any element
$q_{1}\omega_{1}+\cdots+q_{r}\omega_{r}\in\mathcal{K}^{ss}_{I}$ wth
$q_{1},\dots,q_{r}\in{\mathbb{Q}}$ and
$\omega_{1},\dots,\omega_{r}\in\mathbb{T}_{I}^{ss}$.

Notice that given a semisimple multitype
$\omega=(d_{1},(1^{\alpha_{1}}))\dots(d_{r},(1^{\alpha_{r}}))$, in the ring
$\mathcal{K}^{ss}_{I}$ we have an equality

$\omega=\psi_{d_{1}}(\omega_{\alpha_{1}})\ast\cdots\ast\psi_{d_{r}}(\omega_{\alpha_{r}}).$

We deduce therefore that $\mathcal{K}^{ss}_{I}$ is isomorphic to the ring of
polynomials in the variables $\psi_{d}(\omega_{\alpha})$ for
$(d,\alpha)\in\mathbb{N}_{>0}\times\mathbb{N}^{I}$.

Consider now the ring
$\hat{\mathcal{K}}^{ss}_{I}\coloneqq\mathcal{K}^{ss}_{I}[[y_{i}]]_{i\in I}$.
For semisimple multitypes of level $V$, we have the following lemma:

######  Lemma 2.3.4.

For any $V\subseteq\mathbb{N}^{I}$, we have the following identity in the ring
$\hat{\mathcal{K}}^{ss}_{I}$:

$\operatorname{Exp}\left(\sum_{\alpha\in
V}\omega_{\alpha}y^{\alpha}\right)=\sum_{\begin{subarray}{c}\omega\in\mathbb{T}^{ss}_{I}\\\
\text{ of level }V\end{subarray}}\dfrac{\omega}{w(\omega)}y^{|\omega|}$
(2.3.8)

###### Proof.

By eq.(2.3.1), there is an equality

$\operatorname{Exp}\left(\sum_{\alpha\in
V}\omega_{\alpha}y^{\alpha}\right)=\prod_{\alpha\in
V}\left(\sum_{n\in\mathbb{N}}\sigma_{n}(\omega_{\alpha})y^{n\alpha}\right)=\prod_{\alpha\in
V}\left(\sum_{\lambda\in\mathcal{P}}\dfrac{\psi_{\lambda}(\omega_{\alpha})}{z_{\lambda}}y^{|\lambda|\alpha}\right).$
(2.3.9)

For each semisimple multitype $\omega$ of level $V$, there exist unique
$\beta_{1}\neq\beta_{2}\neq\dots\neq\beta_{h}\in V$ and integers
$d_{1,1},\dots d_{1,l_{1}},d_{2,1},\dots,d_{h,l_{h}}$ such that
$\omega=(d_{1,1},(1^{\beta_{1}}))(d_{1,2},(1^{\beta_{1}}))\cdots(d_{h,l_{h}},(1^{\beta_{h}}))$
i.e

$\omega=\psi_{d_{1,1}}(\omega_{\beta_{1}})\ast\cdots\ast\psi_{d_{h,l_{h}}}(\omega_{\beta_{h}}).$

Up to reordering, we can assume that for each $j=1,\dots,h$, the integers
$(d_{j,1},\dots,d_{j,l_{j}})$ form a partition $\lambda_{j}$. Therefore we
have

$\displaystyle\omega=\prod_{j=1}^{h}\psi_{\lambda_{j}}(\omega_{\beta_{j}}).$

Notice moreover that $z_{\lambda_{1}}\cdots z_{\lambda_{h}}=w(\omega)$. This
implies that the RHS of eq.(2.3.9) is equal to the RHS of eq.(2.3.8).

∎

###  2.4 Varieties over finite fields

Let $q=p^{r}$ where $p$ is a prime number, $\mathbb{F}_{q}$ the field with $q$
elements and $\overline{\mathbb{F}}_{q}$ its algebraic closure. In the
following, a variety over $\mathbb{F}_{q}$ will be a pair $(X,F)$ where $X$ is
a reduced scheme of finite type over $\overline{\mathbb{F}}_{q}$ and $F$ is an
$\mathbb{F}_{q}$\- Frobenius morphism $F:X\to X$ (for more details see for
example Milne’s book [44]).

Whenever the $\mathbb{F}_{q}$-structure of $X$ is clear, we will often drop
the Frobenius morphism in the notation and we will simply use the terminology
"the $\mathbb{F}_{q}$-variety $X$".

A morphism of $\mathbb{F}_{q}$-varieties $f:(X,F)\to(Y,F)$ is a morphism
$f:X\to Y$ which commutes with the corresponding Frobenius maps.

Given an affine variety $(X,F)$ over $\mathbb{F}_{q}$ with Frobenius $F:X\to
X$, consider the variety $X^{d}$ equipped with the twisted Frobenius

$F_{d}:X^{d}\to X^{d}$

given by

$F_{d}(x_{1},\dots x_{d})=(F(x_{d}),F(x_{1}),\dots,F(x_{d-1})).$

In the following, we will denote the $\mathbb{F}_{q}$-variety $(X^{d},F_{d})$
simply by $(X)_{d}$. Notice that there is a bijection
$(X)_{d}(\mathbb{F}_{q})=X(\mathbb{F}_{q^{d}})$.

For $n\in\mathbb{N}$, we denote by $\operatorname{GL}_{n}$ the general linear
group over $\overline{\mathbb{F}}_{q}$. The group $\operatorname{GL}_{n}$ is
endowed with the canonical Frobenius morphism $F((a_{i,j}))=(a_{i,j}^{q})$ for
a matrix $(a_{i,j})\in\operatorname{GL}_{n}$.

For $\alpha\in\mathbb{N}^{I}$, we denote by $\operatorname{GL}_{\alpha}$ the
group

$\operatorname{GL}_{\alpha}\coloneqq\displaystyle\prod_{i\in
I}\operatorname{GL}_{\alpha_{i}}$

and by $\operatorname{GL}_{\alpha}(\mathbb{F}_{q})$ the finite group
$\operatorname{GL}_{\alpha}(\mathbb{F}_{q})=\operatorname{GL}_{\alpha}^{F}$.

######  Remark 2.4.1.

For each $n,d\geq 1$, we will define an embedding
$(\operatorname{GL}_{n})_{d}\subseteq\operatorname{GL}_{nd}$ defined over
$\mathbb{F}_{q}$ in the following way. Let
$\Delta:\operatorname{GL}_{n}^{d}\to\operatorname{GL}_{nd}$ be the block
diagonal embedding.

Notice that $\Delta$ is not defined over $\mathbb{F}_{q}$ when
$\operatorname{GL}_{n}^{d}$ si equipped with the Frobenius structure $F_{d}$.
Consider then the partition $\sigma\in S_{nd}$ given by $\sigma=(1\
(n+1)\cdots(n(d-1)+1))\cdots(n\ 2n\cdots dn)$ and the associated partition
matrix $J_{\sigma}\in\operatorname{GL}_{nd}$.

Fix an element $g_{\sigma}\in\operatorname{GL}_{nd}$ such that
$g_{\sigma}^{-1}F(g_{\sigma})=J_{\sigma}$ (such an element exists because of
the surjectivity of the Lang map, see for example [14, Theorem 4.29]). The
embedding

$g_{\sigma}\Delta
g_{\sigma}^{-1}:(\operatorname{GL}_{n}^{d},F_{d})\to(\operatorname{GL}_{nd},F)$

is defined over $\mathbb{F}_{q}$.

Similarly, we get an $\mathbb{F}_{q}$-embedding of
$(\operatorname{GL}_{\alpha})_{d}$ inside $\operatorname{GL}_{\alpha d}$ for
any $d\geq 1$ and any $\alpha\in\mathbb{N}^{I}$.

######  Example 2.4.2.

Let $n=2$. Fix $x\in\mathbb{F}_{q^{2}}^{*}\setminus\mathbb{F}^{*}_{q}$ and let
$T_{\epsilon}$ be the torus

$T_{\epsilon}\coloneqq\Biggl{\\{}\dfrac{1}{x^{q}-x}\begin{pmatrix}ax^{q}-bx&-a+b\\\
(a-b)xx^{q}&-ax+bx^{q}\end{pmatrix}\ |\
a,b\in\overline{\mathbb{F}_{q}}^{*}\Biggr{\\}}.$

The torus $T_{\epsilon}$ is $F$-stable and is
$\operatorname{GL}_{2}(\mathbb{F}_{q})$-conjugated to the torus
$(\mathbb{G}_{m})_{2}$ embedded inside $\operatorname{GL}_{2}$ as in Remark
2.4.1 above.

##  3 Finite reductive groups, subtori and multitypes

In this Section and in the rest of the article, $G$ is a reductive group
defined over $\mathbb{F}_{q}$ with a fixed Frobenius morphism $F:G\to G$. In
the cases that interest us in this article, $G$ will always be taken to be a
product of factors of type $(\operatorname{GL}_{n})_{d}$’s.

We denote by $\operatorname{rank}(G)$ the dimension of a maximal torus of $G$.
Recall that we always have an $F$-stable maximal subtorus $T\subseteq G$.

We denote by $\epsilon_{G}$ the rank of a maximal split $F$-stable subtorus of
$G$. Notice that in general $\epsilon_{G}\neq\operatorname{rank}(G)$. If
$\operatorname{rank}(G)=\epsilon_{G}$, we say that $G$ is split.

######  Example 3.0.1.

Consider the group $(\operatorname{GL}_{n})_{d}$. Let $T\subseteq G$ be the
maximal torus $T_{n}\times\cdots\times
T_{n}\subseteq(\operatorname{GL}_{n})d$, where we denote by
$T_{n}\subseteq\operatorname{GL}_{n}$ the torus of diagonal matrices. Notice
that $T$ is $F$-stable and $\dim(T)=\operatorname{rank}(G)=nd$. However, it is
possible to verify that $\epsilon_{G}=n$, i.e $(\operatorname{GL}_{n})_{d}$ is
split if and only if $d=1$.

For a maximal torus $T$, we denote by

$X_{*}(T)\coloneqq\operatorname{Hom}(T,\mathbb{G}_{m})$

and

$Y_{*}(T)\coloneq\operatorname{Hom}(\mathbb{G}_{m},T)$

the group of characters and cocharacters of $T$ respectively. Recall that
these are free abelian groups of rank $\operatorname{rank}(G)$ and that there
is a coupling

$\langle,\rangle:Y_{*}(T)\times X_{*}(T)\to\mathbb{Z},$

where, for $\beta\in Y_{*}(T),\alpha\in X_{*}(T)$, we have

$\alpha\circ\beta(z)=z^{\langle\beta,\alpha\rangle}$

for any $z\in\mathbb{G}_{m}$.

Denote by $W_{G}(T)$ the Weyl group of $T$, i.e $W_{G}(T)=N_{G}(T)/T$. Notice
that $W_{G}(T)$ acts on $X_{*}(T)$ as follows

$w\cdot\alpha(t)=\alpha(w\cdot t)$

for each $w\in W_{G}(T)$, $t\in T$ and $\alpha\in X_{*}(T)$. For each $w\in
W_{G}(T)$, we denote by $w:X_{*}(T)\to X_{*}(T)$ the corresponding
endomorphism.

Recall that inside $X_{*}(T)$ there is the root system $\Phi(T)\subseteq
X_{*}(T)$ given by the characters appearing in the weight space decomposition
of the adjoint action of $T$ on $\mathfrak{g}=\operatorname{Lie}(G)$. Recall
that for any $\epsilon\in\Phi(T)$, there is an injective homomorphism
$u_{\epsilon}:\mathbb{G}_{a}\to G$ such that for any
$x\in\overline{\mathbb{F}}_{q}$ and any $t\in T$, we have

$tu_{\epsilon}(x)t^{-1}=u_{\epsilon}(\epsilon(t)x).$

We denote by $U_{\epsilon}\subseteq G$ the subgroup
$U_{\epsilon}\coloneqq\operatorname{Imm}(u_{\epsilon})$.

Moreover, inside $Y_{*}(T)$ there is a dual root system $\Phi^{\vee}(T)$,
provided with a canonical bijection

$\Phi(T)\leftrightarrow\Phi^{\vee}(T)$
$\epsilon\leftrightarrow\epsilon^{\vee}$

such that $\langle\epsilon^{\vee},\epsilon\rangle=2$ for every
$\epsilon\in\Phi(T)$.

If $T$ is $F$-stable, the Frobenius acts on the groups $X_{*}(T),Y_{*}(T)$ as
follows

$F:X_{*}(T)\to X_{*}(T)$ $\alpha\to\alpha\circ F$

and

$F:Y_{*}(T)\to Y_{*}(T)$ $\beta\to F\circ\beta.$

####  3.0.1 Twisted Frobenius of maximal tori

Fix now a $F$-stable maximal torus $T\subseteq G$. As $T$ is $F$-stable, the
Frobenius acts on the Weyl group too. Given two elements $h_{1},h_{2}\in
W_{G}(T)$, we say that they are $F$-conjugated if there exists $w\in W_{G}(T)$
such that $h_{1}=wh_{2}F(w)^{-1}$.

The set of $F$-conjugacy classes of $W_{G}(T)$, usually denoted by
$H^{1}(F,W_{G}(T))$, parametrize the $G^{F}$-conjugacy classes of $F$-stable
maximal tori in the following way.

Given a $F$-stable maximal torus $T^{\prime}$ there exists $g\in G$ such that
$gTg^{-1}=T^{\prime}$. As $F(T^{\prime})=T^{\prime}$ we see that
$\dot{w}=g^{-1}F(g)$ belongs to $N_{G}(T)$ and so determines an associated
element $w\in W_{G}(T)$.

We can reformulate this correspondence in terms of the twisted
$\mathbb{F}_{q}$-structures of the torus $T$. While the conjugation by $g$
provides an isomorphism $T^{\prime}\cong T$ over $\overline{\mathbb{F}}_{q}$,
this isomorphism is not in general an $\mathbb{F}_{q}$-morphism
$(T^{\prime},F)\to(T,F)$.

However, when $T$ is equipped with the $\mathbb{F}_{q}$ structure coming from
the twisted Frobenius $\dot{w}F:T\to T$, the conjugation by $g$ is an
$\mathbb{F}_{q}$ -isomorphism

$(T^{\prime},F)\cong_{\mathbb{F}_{q}}(T,\dot{w}F).$

In the following, we assume to have fixed, for each $w\in W_{G}(T)$, a
corresponding $F$-stable maximal torus $T_{w}\subseteq G$.

######  Example 3.0.2.

Consider the case of $G=\operatorname{GL}_{n}$ and $T=T_{n}$ the torus of
diagonal matrices. In this case, we have $W_{G}(T_{n})=S_{n}$ and the
$F$-action on $S_{n}$ is trivial. In particular, the $F$-conjugacy classes of
$S_{n}$ are the conjugacy classes of $S_{n}$ and are therefore indexed by the
partitions $\mathcal{P}_{n}$ of size $n$.

For any $\lambda=(\lambda_{1},\dots,\lambda_{h})\in\mathcal{P}_{n}$, any
associated torus $F$-stable maximal torus $T^{\prime}$ is
$\operatorname{GL}_{n}(\mathbb{F}_{q})$-conjugated to

$(\mathbb{G}_{m})_{\lambda_{1}}\times\cdots\times(\mathbb{G}_{m})_{\lambda_{h}},$

i.e

$(T^{\prime},F)\cong(\mathbb{G}_{m})_{\lambda_{1}}\times\cdots\times(\mathbb{G}_{m})_{\lambda_{h}}.$

######  Example 3.0.3.

Consider the group $G=(\operatorname{GL}_{n})_{d}$ and the $F$-stable maximal
torus $T$ introduced above. The Weyl group $W_{G}(T)$ is isomorphic to
$S_{n}^{d}$ and the corresponding Frobenius action $F:S_{n}^{d}\to S_{n}^{d}$
is given by

$F(\sigma_{1},\dots,\sigma_{d})=(\sigma_{d},\sigma_{1},\dots,\sigma_{d-1}).$

The $F$-conjugacy classes of $S_{n}^{d}$ are in bijection with the conjugacy
classes of $S_{n}$ in the following way. Consider
$\tau=(\tau_{1},\dots,\tau_{d}),\sigma=(\sigma_{1},\dots,\sigma_{d})\in
S_{n}^{d}$. The element $\tau\sigma F(\tau)^{-1}$ is equal to
$(\tau_{1}\sigma_{1}\tau_{d}^{-1},\tau_{2}\sigma_{2}\tau_{1}^{-1},\dots,\tau_{d}\sigma_{d}\tau_{d-1}^{-1})$.
Notice that we have

$\prod_{i=0}^{d-1}(\tau\sigma
F(\tau)^{-1})_{d-i}=\tau_{d}(\sigma_{d}\sigma_{d-1}\cdots\sigma_{1})\tau_{d}^{-1}=\tau_{d}\left(\prod_{i=0}^{d-1}\sigma_{d-i}\right)\tau_{d}^{-1}.$

We deduce therefore that $\sigma,\sigma^{\prime}\in S_{n}^{d}$ are
$F$-conjugated if and only if
$\displaystyle\prod_{i=0}^{d-1}\sigma_{d-i},\prod_{i=0}^{d}\sigma^{\prime}_{d-i}$
are conjugated in $S_{n}$.

Consider a pair of $F$-stable maximal tori $T,T^{\prime}$ with
$gT^{\prime}g^{-1}=T$ and $\dot{w}=g^{-1}F(g)\in N_{G}(T^{\prime})$ and $w\in
W_{G}(T^{\prime})$ as above. There is an isomorphism of abelian groups

$\Psi_{g}:X_{*}(T^{\prime})\to X_{*}(T)$ $\alpha\to\alpha(g^{-1}-g)$

such that $\Psi_{g}(\Phi(T^{\prime}))=\Phi(T)$. Notice that in general,
$\Psi_{g}$ does not commute with the respective Frobenius morphisms on
$X_{*}(T),X_{*}(T^{\prime})$ and indeed we have

$\Psi_{g}^{-1}F\Psi_{g}=w\circ F:X_{*}(T^{\prime})\to X_{*}(T^{\prime}).$
(3.0.1)

####  3.0.2 The case of finite general linear groups

For $m\in\mathbb{N}$, let $\operatorname{GL}_{m}$ be the general linear group
over $\mathbb{F}_{q}$, with the canonical $\mathbb{F}_{q}$-structure
$F:\operatorname{GL}_{m}\to\operatorname{GL}_{m}$. Consider the maximal torus
of diagonal matrices $T_{m}\subseteq\operatorname{GL}_{m}$. As mentioned
before, in this case $W_{G}(T_{m})=S_{m}$ and the $F$-action on $W_{G}(T_{m})$
is trivial.

Let $\epsilon_{i}\in X_{*}(T_{m})$ be the homomorphism

$\epsilon_{i}\left(\begin{pmatrix}z_{1}&0&0&0&0&\dots&0\\\
0&z_{2}&0&0&0&\dots&0\\\ 0&0&z_{3}&0&0&\dots&0\\\
\vdots&\vdots&\vdots&\ddots&\dots&\dots&0\\\
0&0&0&0&0&0&z_{n}\end{pmatrix}\right)=z_{i}.$

Notice that the subset $\\{\epsilon_{1},\dots,\epsilon_{m}\\}\subset
X_{*}(T_{m})$ is a basis of the free abelian group $X_{*}(T_{m})$, which we
denote by $\mathcal{B}(T_{m})$. Notice that, for each $i=1,\dots,m$, we have
that $F(\epsilon_{i})=q\epsilon_{i}$.

Moreover, for such a basis, we have that

$\Phi(T_{m})=\\{\pm\epsilon_{i}\mp\epsilon_{j}\ |\ i\neq
j\in\\{1,\dots,m\\}\\}.$

For $h,j\in\\{1,\dots,m\\}$, denote by
$\epsilon_{h,j}\coloneqq\epsilon_{h}-\epsilon_{h}$. We denote by
$\Phi^{+}(T_{m})$ the set of positive roots with respect to the Borel subgroup
of upper triangular matrices, i.e

$\Phi^{+}(T_{m})=\\{\epsilon_{i,j}\ |\ i<j\\}.$

For any other $F$-stable maximal torus $T\subseteq\operatorname{GL}_{m}$, fix
$g$ such that $gT_{m}g^{-1}=T$ and the corresponding permutation $w\in
W_{G}(T_{m})=S_{m}$ as the end of paragraph above. Denote by
$\mathcal{B}(T)=\Psi_{g}(\mathcal{B}(T_{m}))$.

Whenever the torus $T$ is fixed and the context is clear we will denote by
$\epsilon_{i}$ also the element $\Psi_{g}(\epsilon_{i})\in\mathcal{B}(T)$ and
by $\epsilon_{i,j}$ the element $\Psi_{g}(\epsilon_{i,j})\in\Phi(T)$. We
denote by $\Phi^{+}(T)=\Psi_{g}(\Phi^{+}(T_{m}))$.

Notice that, by eq.(3.0.1), in the character group $X_{*}(T)$ we have

$F(\epsilon_{i})=q\epsilon_{w(i)}.$

Consider now $\alpha\in\mathbb{N}^{I}$, put $m=|\alpha|\coloneqq\sum_{i\in
I}\alpha_{i}$ and consider $\operatorname{GL}_{\alpha}$ as a subgroup of
$\operatorname{GL}_{|\alpha|}$ through the block diagonal embedding. Fix an
$F$-stable maximal torus
$T\subseteq\operatorname{GL}_{\alpha}\subseteq\operatorname{GL}_{m}$. Notice
that $T$ is a maximal torus for $\operatorname{GL}_{\alpha}$ and
$\operatorname{GL}_{m}$. Consider then the basis $\mathcal{B}(T)$ as above.

######  Remark 3.0.4.

For $i\in I$, let
$\pi_{i}:\operatorname{GL}_{\alpha}\to\operatorname{GL}_{\alpha_{i}}$ the
canonical projection. For a maximal torus
$T\subseteq\operatorname{GL}_{\alpha}$, denote by $T_{i}\coloneqq\pi_{i}(T)$.
Notice that

$\displaystyle T\subseteq\prod_{i\in I}T_{i}\subseteq\prod_{i\in
I}\operatorname{GL}_{\alpha_{i}}.$

As $T$ is a maximal torus, we have thus an equality
$T=\displaystyle\prod_{i\in I}T_{i}$. For dimensional reasons, we deduce that,
for each $i\in I$, $T_{i}$ is a maximal torus of
$\operatorname{GL}_{\alpha_{i}}$. From the identity
$T=\displaystyle\prod_{i\in I}T_{i}$, we deduce that there is an isomorphism
$X_{*}(T)=\displaystyle\bigoplus_{i\in I}X_{*}(T_{i})$.

Notice that we can choose $g\in\operatorname{GL}_{\alpha}$ such that
$gT_{m}g^{-1}=T$. We deduce therefore that putting
$\mathcal{B}_{i}(T)=\mathcal{B}(T)\cap X_{*}(T_{i})$, we obtain a partition

$\displaystyle\mathcal{B}(T)=\bigsqcup_{i\in I}\mathcal{B}_{i}(T)$

such that each $\mathcal{B}_{i}(T)$ is a basis of $X_{*}(T_{i})$ and
$\mathcal{B}_{i}(T)$ is $w$-stable for every $i\in I$.

###  3.1 Levi subgroups and parabolic subgroups of general linear groups

Recall that a parabolic subgroup $P\subseteq G$ is a subgroup containing a
Borel subgroup and denote by $U_{P}\subseteq P$ its unipotent radical. A Levi
factor of $P$ is a reductive subgroup $L\subseteq P$ such that $P=LU_{P}$,
where $U_{P}$ is the unipotent radical of $P$. A Levi subgroup of a parabolic
subgroup is called a Levi subgroup of $G$.

Recall that, for any Levi subgroup $L\subseteq G$, there exists a maximal
torus $T\subseteq G$ such that $T\subseteq L$. Moreover, the Levi subgroup $L$
can be described in terms of the root systems $\Phi(T)$ as follows.

Consider the subset $\Phi_{L}(T)\subseteq\Phi(T)$ defined as

$\Phi_{L}(T)\coloneqq\\{\epsilon\in\Phi(T)\ |\
\operatorname{Ker}(\epsilon)\supseteq Z^{\circ}_{L}\\},$

where $Z_{L}^{\circ}$ is the connected componenent containing the identity of
the center $Z_{L}\subseteq L$.

We have the following Lemma, see [50, Lemma 8.4.2].

######  Lemma 3.1.1.

The subset $\Phi_{L}(T)$ is a root subsystem of $\Phi(T)$ and we have :

  1. 1.

$C_{G}(Z^{\circ}_{L})=L$.

  2. 2.

$\displaystyle
Z_{L}^{\circ}=\bigcap_{\epsilon\in\Phi_{L}(T)}(\operatorname{Ker}(\epsilon)^{\circ})$

  3. 3.

$\displaystyle L=T\displaystyle\prod_{\epsilon\in\Phi_{L}(T)}U_{\epsilon}$.

######  Example 3.1.2.

For $G=\operatorname{GL}_{n}$, Levi subgroups and parabolic subgroups can be
explicitly described as follows. For any $n_{0},\dots,n_{s}\in\mathbb{N}$ such
that $n_{0}+\cdots+n_{s}=n$, the subgroup $L_{n_{0},\dots,n_{s}}$ defined as

$L_{n_{0},\dots,n_{s}}=\begin{pmatrix}\operatorname{GL}_{n_{s}}&0&0&0&0&\dots&0\\\
0&\operatorname{GL}_{n_{s-1}}&0&0&0&\dots&0\\\
0&0&\operatorname{GL}_{n_{s-2}}&0&0&\dots&0\\\
\vdots&\vdots&\vdots&\ddots&\dots&\dots&0\\\
0&0&0&0&0&0&\operatorname{GL}_{n_{0}}\end{pmatrix}$

is a Levi subgroup of $G$. We will denote the group $L_{n_{0},\dots,n_{s}}$
simply by

$\operatorname{GL}_{n_{s}}\times\cdots\times\operatorname{GL}_{n_{0}}\subseteq\operatorname{GL}_{n}.$

Notice that

$Z_{L}=\begin{pmatrix}\lambda_{s}I_{n_{s}}&0&0&0&0&\dots&0\\\
0&\lambda_{s-1}I_{n_{s-1}}&0&0&0&\dots&0\\\
0&0&\lambda_{s-2}I_{n_{s-2}}&0&0&\dots&0\\\
\vdots&\vdots&\vdots&\ddots&\dots&\dots&0\\\
0&0&0&0&0&0&\lambda_{0}I_{n_{0}}\end{pmatrix},$

for $\lambda_{0},\dots,\lambda_{s}\in\overline{\mathbb{F}}^{*}_{q}$. Notice in
particular that $Z_{L}$ is connected.

A parabolic subgroup $P$containing
$\operatorname{GL}_{n_{s}}\times\cdots\times\operatorname{GL}_{n_{0}}$ is
given by the upper block triangular matrices

$P=\begin{pmatrix}\operatorname{GL}_{n_{s}}&*&*&*&*&\dots&*\\\
0&\operatorname{GL}_{n_{s-1}}&*&*&*&\dots&*\\\
0&0&\operatorname{GL}_{n_{s-2}}&*&*&\dots&*\\\
\vdots&\vdots&\vdots&\ddots&\dots&\dots&*\\\
0&0&0&0&0&0&\operatorname{GL}_{n_{0}}\end{pmatrix}.$

It is not difficult to verify that, for any Levi subgroup
$L\subseteq\operatorname{GL}_{n}$, there exist $m_{0},\dots,m_{r}$ such that
$n_{0}+\cdots+n_{r}=n$ and $L$ is conjugated to

$\operatorname{GL}_{n_{r}}\times\cdots\times\operatorname{GL}_{n_{0}}.$

Assume now that $L$ is $F$-stable. Similarly to what we said about $F$-stable
tori in Example 3.0.2, we can show that there exist
$d_{0},\dots,d_{r}\in\mathbb{N}$ and $m_{0},\dots,m_{r}$ such that $L$ is
conjugated by an element of $\operatorname{GL}_{n}(\mathbb{F}_{q})$ to the
group

$(\operatorname{GL}_{m_{0}})_{d_{0}}\times\cdots\times(\operatorname{GL}_{m_{r}})_{d_{r}},$

i.e there is an $\mathbb{F}_{q}$-isomorphism

$(L,F)\cong(\operatorname{GL}_{m_{0}})_{d_{0}}\times\cdots\times(\operatorname{GL}_{m_{r}})_{d_{r}}.$

Notice that in this case we have an isomorphism

$(Z_{L},F)\cong(\mathbb{G}_{m})_{d_{0}}\times\cdots\times(\mathbb{G}_{m})_{d_{r}}.$

###  3.2 Admissible subtori of general linear groups

For $\alpha\in\mathbb{N}^{I}$, put $\displaystyle|\alpha|\coloneqq\sum_{i\in
I}\alpha_{i}$ and consider $\operatorname{GL}_{\alpha}$ as a subgroup of
$\operatorname{GL}_{|\alpha|}$via the block diagonal embedding. Recall that
$I$ can be thought of as the set of vertices of a star-shaped quiver. We
introduce here the definition of the admissible subtori of
$\operatorname{GL}_{\alpha}$.

Admissible subtori will play a significant role in this paper. For instance,
they appear in the classification of the irreducible characters of the finite
group $\operatorname{GL}_{\alpha}(\mathbb{F}_{q})$ (see section 5.7). They are
also a key part of the proof of Theorem 4.1.2, which will be the main
technical result needed to study the cohomology of non-generic character
stacks.

######  Definition 3.2.1.

A subtorus $S$ of $\operatorname{GL}_{\alpha}$ is said admissible if there
exists a Levi subgroup $L_{S}\subseteq\operatorname{GL}_{|\alpha|}$ such that
$Z_{L_{S}}=S$.

######  Example 3.2.2.

For any $\alpha\in\mathbb{N}^{I}$, there is an admissible subtorus
$Z_{\alpha}\subseteq\operatorname{GL}_{\alpha}$, given by $Z_{\alpha}\coloneqq
Z_{\operatorname{GL}_{|\alpha|}}\subseteq\operatorname{GL}_{\alpha}$, i.e the
elements of $Z_{\alpha}$ are of the form $(\lambda I_{\alpha_{i}})_{i\in I}$,
for $\lambda\in\overline{\mathbb{F}}_{q}^{*}$.

We have the following Lemma (see [14, Proposition 3.4.6])

######  Lemma 3.2.3.

For an admissible $S$ and a Levi subgroup $L_{S}$ such that $Z_{L_{S}}=S$, we
have $C_{\operatorname{GL}_{|\alpha|}}(S)=L_{S}$. In particular, the group
$L_{S}$ is unique.

######  Remark 3.2.4.

Notice that from Lemma 3.2.3 above, we have that $S$ is $F$-stable if and only
if $L_{S}$ is $F$-stable.

######  Example 3.2.5.

Put $|I|=1$ and let $S\subseteq\operatorname{GL}_{2}$ be the torus

$S=\Biggl{\\{}\begin{pmatrix}\lambda&0\\\ 0&\lambda^{2}\end{pmatrix},\ \
\lambda\in\overline{\mathbb{F}}_{q}^{*}\Biggr{\\}}.$

Notice that $C_{\operatorname{GL}_{2}}(S)=T_{2}$, where
$T_{2}\subseteq\operatorname{GL}_{2}$ is the torus of diagonal matrices.
However, $Z_{T_{2}}=T_{2}\neq S$. We deduce thus that the torus $S$ is not
admissible.

Consider an admissible subtorus $S\subseteq\operatorname{GL}_{\alpha}$ and the
associated Levi subgroup $L_{S}\subseteq\operatorname{GL}_{|\alpha|}$. The
group $C_{\operatorname{GL}_{\alpha}}(S)$ is a Levi subgroup of
$\operatorname{GL}_{\alpha}$ (see [14, Proposition 3.4.7]) which we will
denote by $\widetilde{L_{S}}$.

Notice that $\widetilde{L_{S}}$ is equal to
$L_{S}\cap\operatorname{GL}_{\alpha}$ as
$C_{\operatorname{GL}_{|\alpha|}}(S)\cap\operatorname{GL}_{\alpha}=C_{\operatorname{GL}_{\alpha}}(S)$.
In particular, there exists a maximal torus
$T\subseteq\operatorname{GL}_{\alpha}$ such that $S\subseteq
T\subseteq\widetilde{L_{S}}$.

Conversely, consider an $F$-stable Levi subgroup
$L\subseteq\operatorname{GL}_{|\alpha|}$ such that there exists a maximal
torus $T\subseteq L\cap\operatorname{GL}_{\alpha}$. As $Z_{L}\subseteq T$, the
center $Z_{L}$ is an admissible subtorus of $\operatorname{GL}_{\alpha}$.

######  Example 3.2.6.

Notice that even if two admissible tori $S,S^{\prime}$ are different, we can
have $\widetilde{L_{S}}=\widetilde{L_{S^{\prime}}}$. Consider for example
$S=Z_{\alpha}$ and $S^{\prime}$ defined as

$S^{\prime}=\\{(\lambda_{i}I_{\alpha_{i}})_{i\in I}\ |\ (\lambda_{i})_{i\in
I}\in(\overline{\mathbb{F}}_{q}^{*})^{I}\\}.$

In general, we have $S\neq S^{\prime}$. However, for any
$\alpha\in\mathbb{N}^{I}$, for both tori we have

$\widetilde{L_{S}}=\widetilde{L_{S^{\prime}}}=\operatorname{GL}_{\alpha}.$

Notice, in particular, that it is not true in general that
$Z_{\widetilde{L_{S}}}=S$.

For each $\alpha\in\mathbb{N}^{I}$, denote by $\mathcal{Z}_{\alpha}$ the
subset of $F$-stable admissible subtori of $\operatorname{GL}_{\alpha}$ and
denote by $\mathcal{Z}$ the set defined as

$\mathcal{Z}\coloneqq\bigsqcup_{\alpha\in\mathbb{N}^{I}}\mathcal{Z}_{\alpha}.$

######  Example 3.2.7.

Consider $I=\\{1,2,3,4\\}$ and $\alpha=(2,1,1,1)\in\mathbb{N}^{I}$. Notice
that $|\alpha|=5$. Consider for example the admissible subtori
$S_{1},S_{2},S_{3}\subseteq\operatorname{GL}_{\alpha}$ given by

$S_{1}=\Biggl{\\{}\left(\begin{pmatrix}\lambda&0\\\
0&\mu\end{pmatrix},\lambda,\lambda,\lambda\right)\ |\
\lambda,\mu\in\overline{\mathbb{F}}_{q}^{*}\Biggr{\\}}$
$S_{2}=\Biggl{\\{}\left(\begin{pmatrix}\lambda&0\\\
0&\mu\end{pmatrix},\lambda,\mu,\lambda\right)\ |\
\lambda,\mu\in\overline{\mathbb{F}}_{q}^{*}\Biggr{\\}}$

and

$S_{3}=\Biggl{\\{}\left(\begin{pmatrix}\lambda&0\\\
0&\mu\end{pmatrix},\gamma,\delta,\eta\right)\ |\
\lambda,\mu,\gamma,\delta,\eta\in\overline{\mathbb{F}}_{q}^{*}\Biggr{\\}}.$

In this case, $L_{S_{1}}$ is
$\operatorname{GL}_{5}(\mathbb{F}_{q})$-conjugated to
$\operatorname{GL}_{4}\times\operatorname{GL}_{1}$ and
$\widetilde{L_{S_{1}}}=T_{2}\times\operatorname{GL}_{1}\times\operatorname{GL}_{1}\times\operatorname{GL}_{1}$,
where $T_{2}\subseteq\operatorname{GL}_{2}$ is the torus of diagonal matrices.
Moreover, we have that $L_{S_{2}}$ is
$\operatorname{GL}_{5}(\mathbb{F}_{q})$-conjugated to
$\operatorname{GL}_{2}\times\operatorname{GL}_{3}$ and
$\widetilde{L_{S_{1}}}=\widetilde{L_{S_{2}}}$.

Lastly, notice that $L_{S_{3}}$ is the maximal torus of diagonal matrices
$T_{5}\subseteq\operatorname{GL}_{5}$, and
$\widetilde{L_{S_{3}}}=\widetilde{L_{S_{1}}}$ too.

For a multitype $\omega=(d_{1},\bm{\lambda}_{1})\dots(d_{r},\bm{\lambda}_{r})$
of size $\alpha$, we denote by $S_{\omega}\in\mathcal{Z}_{\alpha}$ the torus
defined as

$(Z_{|\bm{\lambda}_{1}|})_{d_{1}}\times\cdots\times(Z_{|\bm{\lambda}_{r}|})_{d_{r}}\subseteq\operatorname{GL}_{\alpha}$

where
$(Z_{|\bm{\lambda}_{1}|})_{d_{1}}\times\cdots\times(Z_{|\bm{\lambda}_{r}|})_{d_{r}}$
is considered a subtorus of $\operatorname{GL}_{\alpha}$ via the componentwise
block diagonal embedding. Denote by
$\beta_{j}=|\bm{\lambda}_{j}|\in\mathbb{N}^{I}$, for each $j=1,\dots,r$. For
the Levi subgroup $L_{\omega}\subseteq\operatorname{GL}_{|\alpha|}$ defined as

$L_{\omega}=(\operatorname{GL}_{|\beta_{1}|})_{d_{1}}\times\cdots\times(\operatorname{GL}_{|\beta_{r}|})_{d_{r}}$

embedded block diagonally, we have $Z_{L_{\omega}}=S_{\omega}$, i.e
$S_{\omega}$ is admissible.

We will denote by $\widetilde{L_{\omega}}$ the Levi subgroup of
$\operatorname{GL}_{\alpha}$ defined as $\widetilde{L_{\omega}}\coloneqq
L_{\omega}\cap\operatorname{GL}_{\alpha}$. Notice that the groups
$L_{\omega},S_{\omega},\widetilde{L_{\omega}}$ depend only on the
semisimplification $\omega^{ss}$ of $\omega$.

From the description of the Levi subgroups of $\operatorname{GL}_{|\alpha|}$,
we deduce the following Lemma.

######  Lemma 3.2.8.

For any $F$-stable admissible $E\in\mathcal{Z}_{\alpha}$, there exists a
unique semisimple multitype, which we denote by $[E]$, such that $E$ is
$\operatorname{GL}_{\alpha}(\mathbb{F}_{q})$-conjugated to $S_{[E]}.$

######  Example 3.2.9.

For any $\alpha\in\mathbb{N}^{I}$, we have that

$[Z_{\alpha}]=\omega_{\alpha}.$

Let $\sim$ be the equivalence relation on $\mathcal{Z}_{\alpha}$, induced by
the conjugation by $\operatorname{GL}_{\alpha}(\mathbb{F}_{q})$ and let
$\overline{\mathcal{Z}}\coloneqq\mathcal{Z}/\sim$ be the quotient set. The map
$\mathcal{Z}_{\alpha}\to\mathbb{T}_{\alpha}^{ss}$ given by $S\rightarrow[S]$,
induces thus a bijection

$\overline{\mathcal{Z}}\cong\mathbb{T}_{I}^{ss}.$

For an admissible torus $S$ and $(d,\alpha)\in\mathbb{N}\times\mathbb{N}^{I}$,
we denote by $m_{((d,\alpha),S)}$ the value $[S]((d,\alpha))$ i.e the number
of appearances of $(d,\alpha)$ in the writing

$[S]=\psi_{d_{1}}(\omega_{\alpha_{1}})\ast\cdots\ast\psi_{d_{r}}(\omega_{\alpha_{r}}).$

Lastly, we give the following definition of levels for the admissible
subtori.,

######  Definition 3.2.10.

Given $S\in\mathcal{Z}$ and $V\subseteq\mathbb{N}^{I}$, we say that $S$ is of
level $V$ if $[S]$ is of level $V$ (see Definition 2.2.1).

######  Example 3.2.11.

Consider the tori $S_{1},S_{2},S_{3}$ introduced in Example 3.2.7. The torus
$S_{1}$ is the product $Z_{(1,1,1,1)}\times Z_{(1,0,0,0)}$ embedded
componentwise block diagonally into $\operatorname{GL}_{\alpha}$. The
multitype $[S_{1}]$ is therefore the semisimple multitype

$[S_{1}]=(1,(1^{(1,1,1,1)}))(1,(1^{(1,0,0,0)}))=\omega_{(1,1,1,1)}\ast\omega_{(1,0,0,0)}.$

Similarly, we have

$[S_{2}]=\omega_{(1,0,1,0)}\ast\omega_{(0,1,0,1)}$

and

$[S_{3}]=\omega_{(1,0,0,0)}\ast\omega_{(1,0,0,0)}\ast\omega_{(0,1,0,0)}\ast\omega_{(0,0,1,0)}\ast\omega_{(0,0,0,1)}.$

Notice that for $V=\\{(1,1,1,1),(1,0,0,0)\\}$, we have that $S_{1}$ if of
level $V$, while $S_{2},S_{3}$ are not.

###  3.3 Regular elements and Möbius function for admissible tori

In this paragraph we give to $\mathcal{Z}$ the structure of a locally finite
poset, with the ordering induced by inclusion and we introduce the associated
Möbius function

$\mu:\mathcal{Z}\times\mathcal{Z}\to\mathbb{Z}$

and we recall more generally some properties of the Möbius function of a
locally finite poset. The Möbius function $\mu$ is going to be one the main
technical ingredient in the proof of our Theorem 4.1.2.

####  3.3.1 Poset of $F$-stable admissible subtori

For any two elements $S,S^{\prime}$ of $\mathcal{Z}_{\alpha}$, we say that
$S\leq S^{\prime}$ if $S\subseteq S^{\prime}$. Notice that $Z_{\alpha}\leq S$
for any admissible $S\subseteq\operatorname{GL}_{\alpha}$. For any
$S\in\mathcal{Z}$, we denote by $S^{reg}$ the subset of regular elements of
$S$ defined as

$S^{reg}\coloneqq\\{s\in S\ |\ s\notin S^{\prime}\text{ for any
}S^{\prime}\lneq S,\ S^{\prime}\in\mathcal{Z}\\}.$ (3.3.1)

We have the following disjoint union

$\displaystyle S=\bigsqcup_{S^{\prime}\leq S}(S^{\prime})^{reg}$ (3.3.2)

and so, taking $F$-fixed points,

$\displaystyle S^{F}=\bigsqcup_{S^{\prime}\leq S}((S^{\prime})^{reg})^{F}.$
(3.3.3)

In particular, we have an equality $\displaystyle|S^{F}|=\sum_{S^{\prime}\leq
S}|((S^{\prime})^{reg})^{F}|$. Notice that, if
$[S]=\psi_{d_{1}}(\omega_{\beta_{1}})\ast\cdots\ast\psi_{d_{r}}(\omega_{\beta_{r}})$,
we have

$S^{F}=\prod_{j=1}^{r}(\mathbb{G}_{m})_{d_{j}}(\mathbb{F}_{q})=\prod_{j=1}^{r}\mathbb{F}_{q^{d_{j}}}^{*}$

and therefore we have

$|S^{F}|=P_{[S]}(q).$ (3.3.4)

####  3.3.2 Möbius functions of locally finite posets

For a finite poset $(X,\leq)$ denote by

$\mu_{X}:X\times X\to\mathbb{Z}$

its associated Möbius functions. Recall that $\mu_{X}$ is defined by the
following two properties:

  * •

$\mu(x,x)=1$ for each $x\in X$

  * •

For each $x\lneqq x^{\prime}$, we have

$\sum_{x\leq x^{\prime\prime}\lneqq
x^{\prime}}\mu(x,x^{\prime\prime})=-\mu(x,x^{\prime})$ (3.3.5)

The Möbius function has the following property.

######  Proposition 3.3.1.

Given $f_{1},f_{2}:X\to\mathbb{C}$ such that

$\displaystyle f_{1}(x)=\sum_{x^{\prime}\leq x}f_{2}(x^{\prime}),$

we have an equality

$f_{2}(x)=\sum_{x^{\prime}\leq x}f_{1}(x^{\prime})\mu_{X}(x^{\prime},x).$
(3.3.6)

Lastly, we recall the following standard Lemma about Möbius functions.

######  Lemma 3.3.2.

Let $(X,\leq),(Y,\leq)$ be two locally finite posets and equip $X\times Y$
with the ordering defined as $(x,y)\leq(x^{\prime},y^{\prime})$ if and only if
$x\leq x^{\prime}$ and $y\leq y^{\prime}$.

For the locally finite poset $(X\times Y,\leq)$, we have

$\mu_{X\times
Y}((x,y),(x^{\prime},y^{\prime}))=\mu_{X}((x,x^{\prime}))\mu_{Y}((y,y^{\prime})).$
(3.3.7)

###### Proof.

By induction we can assume that

$\mu_{X\times
Y}((x,y),(x^{\prime\prime},y^{\prime\prime}))=\mu_{X}((x,x^{\prime\prime}))\mu_{Y}((y,y^{\prime\prime}))$
(3.3.8)

for all $(x,y)<(x^{\prime\prime},y^{\prime\prime})<(x^{\prime},y^{\prime})$.
From eq.(3.3.5) we have therefore

$\mu_{X\times
Y}((x,y)(x^{\prime},y^{\prime}))=-\sum_{(x,y)\leq(x^{\prime\prime},y^{\prime\prime})<(x^{\prime},y^{\prime})}\mu_{X}(x,x^{\prime\prime})\mu_{Y}(y,y^{\prime\prime})=$
(3.3.9)

$\sum_{x\leq
x^{\prime\prime}<x^{\prime}}\mu_{X}(x,x^{\prime\prime})\sum_{y\leq
y^{\prime\prime}\leq
y^{\prime}}\mu_{Y}(y,y^{\prime\prime})+\mu_{X}(x,x^{\prime})\sum_{y\leq
y^{\prime\prime}<y^{\prime}}\mu_{Y}(y,y^{\prime\prime}).$ (3.3.10)

By eq.(3.3.5), $\displaystyle\sum_{y\leq y^{\prime\prime}\leq
y^{\prime}}\mu_{Y}(y,y^{\prime\prime})$ and $\sum_{y\leq
y^{\prime\prime}<y^{\prime}}\mu_{Y}(y,y^{\prime\prime})=-\mu_{Y}(y,y^{\prime})$
and therefore

$\mu_{X\times
Y}((x,y)(x^{\prime},y^{\prime}))=\mu_{X}(x,x^{\prime})\sum_{y\leq
y^{\prime\prime}<y^{\prime}}\mu_{Y}(y,y^{\prime\prime})=-\mu_{X}(x,x^{\prime})\mu_{Y}(y,y^{\prime}).$
(3.3.11)

∎

For $x\in X$, denote by $[x,\infty]_{X}\subseteq X$ the poset

$[x,\infty]=\\{x^{\prime}\in X\ |\ x^{\prime}\geq x\\}.$

Notice that, for each $x^{\prime}\in[x,\infty]_{X}$, from eq.(3.3.5), we
deduce that we have:

$\mu_{X}(x,x^{\prime})=\mu_{[x,\infty]_{X}}(x,x^{\prime})$ (3.3.12)

####  3.3.3 Möbius function for admissible subtori

Notice that the ordering $\leq$ endows the set $\mathcal{Z}$ with the
structure of a locally finite poset. We denote by

$\mu(-,-):\mathcal{Z}\times\mathcal{Z}\to\mathbb{Z}$

the associated Möbius function.

######  Example 3.3.3.

Let $f_{1},f_{2}:\mathcal{Z}\to\mathbb{C}$ be the functions defined as

$f_{1}(S)=|S^{F}|$

and

$f_{2}(S)=|(S^{reg})^{F}|.$

By eq.(3.3.6) and eq.(3.3.3), we have the following identity:

$|(S^{reg})^{F}|=\sum_{S^{\prime}\leq
S}|(S^{\prime})^{F}|\mu(S^{\prime},S)=\sum_{S^{\prime}\leq
S}P_{[S^{\prime}]}(q)\mu(S^{\prime},S).$ (3.3.13)

###  3.4 Levi subgroups and graphs

In this Paragraph, we will associate a finite graph $\Gamma_{S}$ to each
admissible subtorus $S$. This construnction will be useful to understand the
Mobius function $\mu:\mathcal{Z}\times\mathcal{Z}\to\mathbb{Z}$ and to develop
the combinatorial arguments of section 3.5, both of which will be key parts of
the proof of Theorem 4.1.2 about Log compatible functions.

####  3.4.1 Notations on graphs

Let $\Gamma$ be a finite graph with set of vertices $M$ and $m=|M|$. We say
that $\Gamma$ is of type $K_{m}$ if it is the complete graph associated to
$M$, i.e each pair of distinct vertices is connected exactly by one edge.

We say that $\Gamma$ is admissible if each of its connected components is of
type $K_{d}$ for some $d$.

######  Remark 3.4.1.

Notice that the property of being admissible for a graph $\Gamma$ can be
stated in the following equivalent way.

For any two $m,m^{\prime}\in M$, there is at most one edge of $\Gamma$ joining
$m$ to $m^{\prime}$ and, if $m_{1},m_{2},m_{3}\in M$ are such that there is an
edge of $\Gamma$ between $m_{1}$ and $m_{2}$ and an edge of $\Gamma$ between
$m_{2}$ and $m_{3}$, there is an edge of $\Gamma$ between $m_{1}$ and $m_{3}$.

####  3.4.2 Graph associated to an admissible torus

Let now $\alpha\in\mathbb{N}^{I}$ and $m=|\alpha|$ and fix an $F$-stable
maximal torus
$T\subseteq\operatorname{GL}_{\alpha}\subseteq\operatorname{GL}_{m}$.

Denote simply by $\mathcal{B},\Phi,\Phi^{+}$ the sets
$\mathcal{B}(T),\Phi(T),\Phi^{+}(T)$ and by $\sigma\in S_{m}$ the permutation
such that $F(\epsilon_{i})=q\epsilon_{\sigma(i)}$ for each
$\epsilon_{i}\in\mathcal{B}$.

For any two admissible graphs $\Gamma,\Gamma^{\prime}$ with set of vertices
$\mathcal{B}$ and sets of edges $\Omega_{\Gamma},\Omega_{\Gamma^{\prime}}$
respectively, we say that $\Gamma\leq\Gamma^{\prime}$ if
$\Omega_{\Gamma}\supseteq\Omega_{\Gamma^{\prime}}$.

We denote by $A(\mathcal{B},\sigma)$ be the poset of admissible and
$\sigma$-stable graph with set of vertices $\mathcal{B}$ and by
$\mu_{\mathcal{B},\sigma}(-,-)$ the associated Möbius function. Moreover, we
will denote the complete graph with vertices $\mathcal{B}$ by
$\Gamma_{\alpha}\in A(\mathcal{B},\sigma)$.

######  Remark 3.4.2.

From Remark 3.4.1, we see that the poset $A(\mathcal{B},\sigma)$ is the the
poset of $\sigma$-stable partitions of the set $\mathcal{B}$ with ordering
given by the reversed inclusion, i.e the fixed point set lattice considered in
[25].

In the latter article, the authors computed certain values of the Möbius
function $\mu_{\mathcal{B},\sigma}$ and in particular the values
$\mu_{\mathcal{B},\sigma}(\Gamma_{\alpha},\Gamma^{\prime})$ for each
$\Gamma^{\prime}$. We will review this result in Proposition 3.4.3.

We prefer to introduce this graph theoretic description, as in our opinion
this can ease the notations and give a more direct and graphic understanding
of the results of this Paragraph about the relationship between admissible
graphs and admissible tori.

Fix now an admissible $\sigma$-stable graph $\Gamma$ with set of vertices
$\mathcal{B}$. Notice that, as $\Gamma$ is $\sigma$-stable, $\sigma$ acts by
permutation on the set of connected components of $\Gamma$. Assume that this
action has $r$ orbits of length $d_{1},\dots,d_{r}$ respectively, which we
denote by $O_{1},\dots,O_{r}$.

For each $j=1,\dots,r$, denote by
$\mathcal{B}^{\Gamma}_{j}\subseteq\mathcal{B}$ the set of vertices contained
in the orbit $O_{j}$. Notice that each $\mathcal{B}^{\Gamma}_{j}$ is
$\sigma$-stable and there is an equality

$\displaystyle\mathcal{B}=\bigsqcup_{j=1}^{r}\mathcal{B}^{\Gamma}_{j}.$

For each $j=1,\dots,r$, choose a partition of $\mathcal{B}^{\Gamma}_{j}$ into
$d_{j}$ subsets

$\mathcal{B}^{\Gamma}_{j}=\mathcal{B}^{\Gamma}_{j,1}\bigsqcup\dots\bigsqcup\mathcal{B}^{\Gamma}_{j,d_{j}}$

such that:

  * •

Each $\mathcal{B}^{\Gamma}_{j,h}$ is given by the vertices of a connected
component belonging to the orbit $O_{j}$

  * •

We have $\sigma(\mathcal{B}^{\Gamma}_{j,h})=\mathcal{B}^{\Gamma}_{j,h+1}$ for
each $h=1,\dots,d_{j}$ (here we consider the indices modulo $d_{j}$).

For each $j=1,\dots,r$, let $\beta_{j}\in\mathbb{N}^{I}$ be the element
defined as

$(\beta_{j})_{i}\coloneqq|\mathcal{B}^{\Gamma}_{j,1}\cap\mathcal{B}_{i}|$

for $i\in I$. We denote by $\omega_{\Gamma}\in\mathbb{T}^{ss}_{\alpha}$ the
semisimple multityped defined as

$\omega_{\Gamma}\coloneqq\psi_{d_{1}}(\omega_{\beta_{1}})\ast\cdots\ast\psi_{d_{r}}(\omega_{\beta_{r}}).$

In [25], it is shown the following Proposition.

######  Proposition 3.4.3.

For each $\Gamma\in A(\mathcal{B},\sigma)$, we have

$\mu_{A(\mathcal{B},\sigma)}(Z_{\alpha},\Gamma)=C^{o}_{\omega_{\Gamma}}.$
(3.4.1)

Denote now by $\Gamma_{j,h}$ the restriction of $\Gamma$ to the set
$\mathcal{B}^{\Gamma}_{j,h}$. Notice that $\Gamma_{j,h}$ is the complete graph
with vertices $\mathcal{B}^{\Gamma}_{j,h}$ and so $\Gamma$ is totally
determined by the subsets $\\{\mathcal{B}^{\Gamma}_{j,1}\\}_{j=1,\dots,r}$.
Notice, in addition, that for each $j=1,\dots,r$, we have
$\sigma^{d_{j}}(\mathcal{B}^{\Gamma}_{j,1})=\mathcal{B}^{\Gamma}_{j,1}$.

We have the following Lemma.

######  Lemma 3.4.4.

There is an equivalence of posets

$[\Gamma,\infty]_{A(\mathcal{B},\sigma)}\cong\prod_{j=1}^{r}[\Gamma_{j,1},\infty]_{A(\mathcal{B}^{\Gamma}_{j,1},\sigma^{d_{j}})}$
(3.4.2)

and, for each $\Gamma^{\prime}\geq\Gamma$, denoting by $\Gamma^{\prime}_{j,h}$
the restriction of $\Gamma^{\prime}$ to $\mathcal{B}^{\Gamma}_{j,h}$, we have

$\mu_{\mathcal{B},\sigma}(\Gamma,\Gamma^{\prime})=\prod_{j=1}^{r}\mu_{\mathcal{B}^{\Gamma}_{j,1},\sigma^{d_{j}}}(\Gamma_{j,1},\Gamma^{\prime}_{j,1}).$
(3.4.3)

###### Proof.

Notice indeed that, given admissible graphs $\Gamma^{\prime}_{j,1}$ with
vertices $\mathcal{B}^{\Gamma}_{j,1}$ for each $j=1,\dots,r$, there exist a
unique $\sigma$-stable and admissible graph $\Gamma^{\prime}$ with vertices
$\mathcal{B}$ containing as subgraphs
$\Gamma^{\prime}_{1,1},\dots,\Gamma^{\prime}_{r,1}$ and such that
$\Gamma^{\prime}\geq\Gamma$.

Eq.(3.4.3) is thus a consequence of eq.(3.4.2) and Lemma 3.3.7.

∎

####  3.4.3 Admissible subtori and admissible graphs

Fix now an admissible torus $S\subseteq T$. Denote by $J_{S}\subseteq\Phi$ the
subset

$J_{S}\coloneqq\\{\epsilon\in\Phi\ |\
S\subseteq\operatorname{Ker}(\epsilon)\\}.$

From Lemma 3.1.1 we deduce that we have

$S=\bigcap_{\epsilon\in J_{S}}\operatorname{Ker}(\epsilon)$

and

$L_{S}=T\prod_{\epsilon\in J_{S}}U_{\epsilon}.$

Notice moreover that the subgroup $S$ is $F$-stable if and only if $J_{S}$ is
$\sigma$ stable.

We now associate the following graph $\Gamma_{S}$ to the admissible torus $S$.

  * •

The set of vertices of $\Gamma_{S}$ is $\mathcal{B}$

  * •

$\Gamma_{S}$ has an edge between vertices $\epsilon_{i}$ and $\epsilon_{j}$ if
and only if $\epsilon_{i,j}\in J_{S}\cap\Phi^{+}$.

We denote by $\Omega_{\Gamma_{S}}$ be the set of edges of $\Gamma_{S}$. The
group $S$ is $F$-stable if and only if $\Gamma_{S}$ is $\sigma$-invariant.

######  Example 3.4.5.

Let $I=\\{\cdot\\}$ and $T$ be the torus of diagonal matrices
$T\subseteq\operatorname{GL}_{m}$. In this case, $\sigma$ is trivial.

The graph $\Gamma_{T}$ is thus the graph with no edges and $m$ vertices, while
the graph $\Gamma_{Z_{\operatorname{GL}_{m}}}$ associated to
$Z_{\operatorname{GL}_{m}}$ is the complete graph with $m$ vertices $K_{m}$.

######  Example 3.4.6.

For any $I$ and any $\alpha\in\mathbb{N}^{I}$, notice that
$\Gamma_{Z_{\alpha}}=\Gamma_{\alpha}$.

We can now state the following Lemma, relating admissible graphs and subtori.

######  Lemma 3.4.7.

For any admissible torus $S$, the graph $\Gamma_{S}$ is admissible.
Conversely, for any $\sigma$-stable admissible graph $\Gamma$ with set of
vertices $\mathcal{B}$, there is a unique $F$-stable admissible torus
$S\subseteq T$ such that $\Gamma_{S}=\Gamma$.

Here $\sigma$-stable means that $\Gamma$ has an edge between $\epsilon_{i}$
and $\epsilon_{j}$ if and only if it has an edge between
$\epsilon_{\sigma(i)}$ and $\epsilon_{\sigma(j)}$.

###### Proof.

For an admissible subtorus $S$, notice that if
$\epsilon_{i,j},\epsilon_{j,h}\in J_{S}$ then $\epsilon_{i,h}\in J_{S}$. From
Remark 3.4.1 we deduce that $\Gamma_{S}$ is admissible.

Consider now an admissible $\Gamma$ and the subset

$J_{\Gamma}\coloneqq\\{\epsilon_{j,h}\in\Phi\ |\ \text{there is an edge of
}\Gamma\text{ which has vertices }\epsilon_{j},\epsilon_{h}\\}.$

From [14, Corollary 3.3.4], the subset $J_{\Gamma}$ is a root subsystem and,
from [50, Lemma 8.4.2], we have that the torus

$S_{\Gamma}\coloneqq\bigcap_{\epsilon\in
J_{\Gamma}}\operatorname{Ker}(\epsilon)$ (3.4.4)

is admissible and $F$-stable with

$L_{S_{\Gamma}}=T\prod_{\epsilon\in J_{\Gamma}}U_{\epsilon}.$ (3.4.5)

It is not difficult to check that the graph associated to $S_{\Gamma}$ is
$\Gamma$.

∎

Let $S,S^{\prime}$ be two admissible subtori such that $S\supseteq
T,S^{\prime}\supseteq T$. From Lemma 3.4.7, we deduce the following
Proposition

######  Proposition 3.4.8.

Given $S,S^{\prime}\subseteq T$, we have that $S\leq S^{\prime}$ if and only
if $\Gamma_{S}\leq\Gamma_{S^{\prime}}$.

From eq.(3.3.5), Proposition 3.4.8 and Lemma 3.4.7, we deduce the following
Lemma.

######  Lemma 3.4.9.

For any $S,S^{\prime}\in\mathcal{Z}$ such that $S,S^{\prime}\subseteq T$, we
have an equality

$\mu_{A(\mathcal{B},\sigma)}(\Gamma_{S},\Gamma_{S^{\prime}})=\mu(S,S^{\prime})$
(3.4.6)

Consider now an admissible graph $\Gamma\in A(\mathcal{B},\sigma)$ and the
admissible $F$-stable torus $S_{\Gamma}$ associated to $\Gamma$. We have the
following proposition.

######  Proposition 3.4.10.

With the notations introduced above, the multitype associated to $S_{\Gamma}$
is

$[S_{\Gamma}]=\psi_{d_{1}}(\omega_{\beta_{1}})\ast\cdots\ast\psi_{d_{r}}(\omega_{\beta_{r}}),$

i.e we have

$[S_{\Gamma}]=\omega_{\Gamma}.$

Fix now $\Gamma^{\prime},\Gamma\in A(\mathcal{B},\sigma)$ such that
$\Gamma\leq\Gamma^{\prime}$, and denote by $S=S_{\Gamma}$ and
$S^{\prime}=S_{\Gamma^{\prime}}$. Notice that we have $S\subseteq S^{\prime}$.
The torus $S$ is $\operatorname{GL}_{\alpha}(\mathbb{F}_{q})$ conjugated to
$\displaystyle\prod_{j=1}^{r}(Z_{\beta_{j}})_{d_{j}}$. The admissible graphs
$\Gamma^{\prime}_{j,1}$ correspond to admissible tori
$S^{\prime}_{j}\subseteq\operatorname{GL}_{\beta_{j}}$, for each
$j=1,\dots,r$.

From eq.(3.4.6) and eq.(3.4.3), we deduce the following equality

$\displaystyle\mu(S,S^{\prime})=\prod_{j=1}^{r}\mu(Z_{\beta_{j}},S^{\prime}_{j})=\prod_{j=1}^{r}C^{o}_{[S^{\prime}_{j}]}.$
(3.4.7)

######  Example 3.4.11.

Consider the set $I=\\{1,2,3,4\\}$, the dimension vector $\alpha=(2,1,1,1)$
and the admissible tori $S_{1},S_{2},S_{3}\in\mathcal{Z}_{\alpha}$ of Example
3.2.7. Notice that $S_{1},S_{2},S_{3}$ are all contained in the maximal torus
$T=T_{2}\times\mathbb{G}_{m}\times\mathbb{G}_{m}\times\mathbb{G}_{m}$, where
$T_{2}\subseteq\operatorname{GL}_{2}$ is the maximal torus of diagonal
matrices and more precisely that $T=S_{3}$.

With the notations just introduced, we have
$\mathcal{B}=\\{\epsilon_{1},\epsilon_{2},\epsilon_{3},\epsilon_{4},\epsilon_{5}\\}$,
$\sigma$ is the identity and

$\mathcal{B}_{1}=\\{\epsilon_{1},\epsilon_{2}\\}\ \ \
\mathcal{B}_{2}=\\{\epsilon_{3}\\}$

and

$\mathcal{B}_{3}=\\{\epsilon_{4}\\}\ \ \ \mathcal{B}_{4}=\\{\epsilon_{5}\\}.$

The graph $\Gamma_{S_{1}}$ associated to the torus $S_{1}$ is

${\epsilon_{1}}$${\epsilon_{3}}$${\epsilon_{4}}$${\epsilon_{5}}$${\epsilon_{2}}$

The graph $\Gamma_{S_{2}}$ associated to the torus $S_{2}$ is

${\epsilon_{1}}$${\epsilon_{3}}$${\epsilon_{4}}$${\epsilon_{5}}$${\epsilon_{2}}$

The graph associated to $S_{3}$ is

${\epsilon_{1}}$${\epsilon_{3}}$${\epsilon_{4}}$${\epsilon_{5}}$${\epsilon_{2}}$

Notice that $\Gamma_{S_{1}},\Gamma_{S_{2}}\leq\Gamma_{S_{3}}$ and we have
corresponding inclusions $S_{1},S_{2}\subseteq S_{3}$

######  Example 3.4.12.

Consider $I=\\{1,2\\}$ and the dimension vector
$\alpha=(2,2)\in\mathbb{N}^{I}$, i.e
$\operatorname{GL}_{\alpha}=\operatorname{GL}_{2}\times\operatorname{GL}_{2}$
and let $T\subseteq\operatorname{GL}_{\alpha}$ be the torus

$T=T_{\epsilon}\times
T_{\epsilon}\subseteq\operatorname{GL}_{2}\times\operatorname{GL}_{2},$

where $T_{\epsilon}\subseteq\operatorname{GL}_{2}$ is the torus of Example
2.4.2. In this case, we have
$\mathcal{B}=\\{\epsilon_{1},\epsilon_{2},\epsilon_{3},\epsilon_{4}\\}$ with

$\mathcal{B}_{1}=\\{\epsilon_{1},\epsilon_{2}\\}\ \ \
\mathcal{B}_{2}=\\{\epsilon_{3},\epsilon_{4}\\},$

and $\sigma$ is the permutation $\sigma=(12)(34)\in S_{4}$.

Let $S\subseteq\operatorname{GL}_{\beta}$ be the admissible subtorus

$S=\Biggl{\\{}\left(\begin{pmatrix}\lambda&0\\\
0&\lambda\end{pmatrix},\begin{pmatrix}\mu&0\\\ 0&\mu\end{pmatrix}\right)\ |\
\lambda,\mu\in\overline{\mathbb{F}_{q}}^{*}\Biggr{\\}}.$

Notice that $S\subseteq T$ and the graph $\Gamma_{S}$ is given by

${\epsilon_{1}}$${\epsilon_{3}}$${\epsilon_{2}}$${\epsilon_{4}}$

Notice that $\Gamma_{S}$ has two connected components which are both
stabilized by $\sigma$. With the notations introduced before, we have
therefore two orbits $O_{1},O_{2}$ with
$\mathcal{B}^{\Gamma_{S}}_{1}=\mathcal{B}_{1}$ and
$\mathcal{B}^{\Gamma_{S}}_{2}=\mathcal{B}_{2}$ and $d_{1}=d_{2}=1$. Denote by
$\Gamma_{S,1},\Gamma_{S,2}$ the restriction of $\Gamma$ to
$\mathcal{B}_{1},\mathcal{B}_{2}$ respectively.

Notice moreover that the associated elements $\beta_{1},\beta_{2}$ are given
by

$\beta_{1}=(2,0)\ \ \ \beta_{2}=(0,2)$

and, from Lemma 3.4.10 we find

$[S]=\omega_{\beta_{1}}\ast\omega_{\beta_{2}}.$

Notice indeed that the torus $S$ is $Z_{\beta_{1}}\times Z_{\beta_{2}}$
embedded block diagonally in $\operatorname{GL}_{\alpha}$.

Moreover, from eq.(3.4.7), we deduce that

$\mu(S,T)=\mu_{\mathcal{B}_{1},\sigma}(\Gamma_{S,1},\Gamma_{T_{\epsilon}})\mu_{\mathcal{B}_{2},\sigma}(\Gamma_{S,2},\Gamma_{T_{\epsilon}})=\mu(Z_{2},T_{\epsilon})^{2},$
(3.4.8)

where $Z_{2}=Z_{\operatorname{GL}_{2}}$. Notice that eq.(3.4.8) can be checked
directly from the definition of the Möbius function $\mu$.

Notice indeed that we have

$\\{Z_{\beta_{1}}\times Z_{\beta_{2}},Z_{\beta_{1}}\times
T_{\epsilon},T_{\epsilon}\times
Z_{\beta_{2}}\\}=\\{S^{\prime\prime}\in\mathcal{Z}_{\alpha}\ |\
Z_{\beta_{1}}\times Z_{\beta_{2}}\leq S^{\prime\prime}\lneqq
T_{\epsilon}\times T_{\epsilon}\\}.$

From. eq(3.3.5), we deduce that we have

$\mu(Z_{\beta_{1}}\times Z_{\beta_{2}},Z_{\beta_{1}}\times
T_{\epsilon})=\mu(Z_{\beta_{1}}\times Z_{\beta_{2}},T_{\epsilon}\times
Z_{\beta_{2}})=\mu(Z_{2},T_{\epsilon})=-1$

and thus from eq.(3.3.5) that we have

$\mu(S,T_{\epsilon}\times
T_{\epsilon})=-(-1-1+1)=1=\mu(Z_{2},T_{\epsilon})^{2}.$

Consider now the admissible and $\sigma$-stable graph $\Gamma^{\prime}$ given
by

${\epsilon_{1}}$${\epsilon_{3}}$${\epsilon_{2}}$${\epsilon_{4}}$

and denote by $S^{\prime}=S_{\Gamma^{\prime}}$. The torus $S^{\prime}$ is
given by

$S^{\prime}=\Biggl{\\{}\left(\dfrac{1}{x^{q}-x}\begin{pmatrix}ax^{q}-bx&-a+b\\\
(a-b)xx^{q}&-ax+bx^{q}\end{pmatrix},\dfrac{1}{x^{q}-x}\begin{pmatrix}ax^{q}-bx&-a+b\\\
(a-b)xx^{q}&-ax+bx^{q}\end{pmatrix}\right)\ |\
a,b\in\overline{\mathbb{F}_{q}}^{*}\Biggr{\\}}.$ (3.4.9)

Notice that in this case, the graph $\Gamma^{\prime}$ has $2$ connected
components, which are swapped by $\sigma$. We have therefore a single orbit
$O_{1}$ of length $d_{1}=2$ with

$\mathcal{B}^{\Gamma^{\prime}}_{1,1}=\\{\epsilon_{1},\epsilon_{3}\\}\ \ \
\mathcal{B}^{\Gamma^{\prime}}_{1,2}=\\{\epsilon_{2},\epsilon_{4}\\}.$

Notice that the associated dimension vector $\beta_{1}^{\prime}$ is
$\beta_{1}^{\prime}=(1,1)$. Proposition 3.4.10 states therefore that
$S^{\prime}$ is $\operatorname{GL}_{\alpha}(\mathbb{F}_{q})$-conjugated to the
torus

$(Z_{(1,1)})_{2}\subseteq\operatorname{GL}_{\alpha},$

which is also directly seen by the expression of $S^{\prime}$ in eq.(3.4.9).

Notice that

$\sigma^{2}=\mathrm{Id}:\mathcal{B}^{\Gamma^{\prime}}_{1,1}\to\mathcal{B}^{\Gamma^{\prime}}_{1,1}.$

From eq.(3.4.3), we find therefore

$\mu(S^{\prime},T)=\mu_{\mathcal{B}^{\Gamma^{\prime}}_{1,1},\mathrm{Id}}(\Gamma^{\prime}_{1,1},\Gamma_{T_{2}})=\mu(Z_{(1,1)},Z_{(1,0)}\times
Z_{(0,1)}).$

###  3.5 Inclusion of admissible subtori

Let $\omega_{1},\omega_{2}\in\mathbb{T}^{ss}_{\alpha}$ . Fix a maximal torus
$T$ such that $S_{\omega_{2}}\subseteq T\subseteq\operatorname{GL}_{\alpha}.$
Define the set $P_{\omega_{1},\omega_{2}}$ to be

$P_{\omega_{1},\omega_{2}}\coloneqq\\{S\in\mathcal{Z}_{\alpha}\ \ |\
[S]=\omega_{1}\ ,S\leq S_{\omega_{2}}\\}.$

In this paragraph we give a combinatorial description of
$P_{\omega_{1},\omega_{2}}$ which will be used in the proof of Theorem 4.1.2.

Assume that

$\omega_{1}=\psi_{d_{1}}(\omega_{\beta_{1}})\ast\cdots\ast\psi_{d_{r}}(\omega_{\beta_{r}})$

and

$\omega_{2}=\psi_{d^{\prime}_{1}}(\omega_{\beta^{\prime}_{1}})\ast\cdots\ast\psi_{d_{t}^{\prime}}(\omega_{\beta^{\prime}_{t}})$

respectively.

Up to reordering the factors in the product
$\omega_{1}=\prod_{j=1}^{r}\psi_{d_{j}}(\omega_{\beta_{j}})$, we can assume
that there exists a strictly increasing sequence
$i_{1}<\dots<i_{k}\in\\{1,\dots,r\\}$ such that:

  * •

$(d_{j},\beta_{j})=(d_{1},\beta_{1})$ for $j=1,\dots,i_{1}$

  * •

$(d_{j},\beta_{j})=(d_{i_{p}},\beta_{i_{p}})$ for all $i_{p-1}<j\leq i_{p}$
for $p\in\\{2,\dots,k\\}$

Notice that $i_{1}=m_{((d_{1},\beta_{1}),\omega_{1})}$ and, for each
$h\in\\{2,\dots,k\\}$, we have $\displaystyle
i_{h}-\sum_{p=1}^{h-1}i_{p}=m_{((d_{i_{h}},\beta_{i_{h}}),\omega_{1})}$. Let
$M_{\omega_{1},\omega_{2}}$ be the set of partitions of $\\{1,\dots,t\\}$ into
$r$ non-empty disjoint subsets $X_{1},\dots,X_{r}$ with the following
properties:

  * •

If $h$ belongs to $X_{i}$, then $d_{i}|d^{\prime}_{h}$

  * •

For every $j=1,\dots,r$, it holds $\displaystyle\sum_{h\in
X_{j}}\dfrac{d^{\prime}_{h}}{d_{j}}\beta^{\prime}_{h}=\beta_{j}$.

We will denote the element of $M_{\omega_{1},\omega_{2}}$ associated to the
subsets $X_{1},\dots,X_{r}$ by $(X_{1},\dots,X_{r})$.

Consider now the group $W^{\prime}_{\omega_{1}}$ defined by

$W^{\prime}_{\omega_{1}}\coloneqq
S_{m_{((d_{i_{1}},\beta_{i_{1}}),\omega_{1})}}\times\cdots\times
S_{m_{((d_{i_{k}},\beta_{i_{k}}),\omega_{1})}}.$

The set $M_{\omega_{1},\omega_{2}}$ is endowed with an action of the group
$W^{\prime}_{\omega_{1}}$ defined by the following rule. Consider elements
$\sigma=(\sigma_{1},\dots,\sigma_{k})\in W^{\prime}_{\omega_{1}}$ and
$(X_{1},\dots,X_{r})\in M_{\omega_{1},\omega_{2}}$. We define

$\sigma\cdot(X_{1},\dots,X_{r})\coloneqq(X_{\sigma_{1}(1)},\dots,X_{\sigma_{1}(i_{1})},X_{\sigma_{2}(i_{1}+1)},\dots,X_{\sigma_{k}(r)}).$

Notice that $W_{\omega_{1}}^{\prime}$ acts freely on
$M_{\omega_{1},\omega_{2}}$. We denote by
$\overline{M_{\omega_{1},\omega_{2}}}$ the quotient set
$M_{\omega_{1},\omega_{2}}/W^{\prime}_{\omega_{1}}$. We will now define the
following morphism

$\pi_{\omega_{1},\omega_{2}}:P_{\omega_{1},\omega_{2}}\to\overline{M_{\omega_{1},\omega_{2}}}.$

We denote by $\Gamma^{\prime}$ the graph associated to $S_{\omega_{2}}$ with
respect to the torus $T$. Let
$\\{\mathcal{B}_{j,h}^{\Gamma^{\prime}}\\}_{\begin{subarray}{c}j=1,\dots,t\\\
h=1,\dots,d_{j}^{\prime}\end{subarray}}$ be the partition of the set
$\mathcal{B}$ introduced in Paragraph section 3.4 above for the graph
$\Gamma^{\prime}$.

Consider an $F$-stable admissible torus $S\subseteq S_{\omega_{2}}$ with
$[S]=\omega_{1}$ and the corresponding $\sigma$-stable graph
$\Gamma\leq\Gamma^{\prime}$, i.e
$\Omega_{\Gamma}\supseteq\Omega_{\Gamma^{\prime}}$.

Let $O_{1},\dots,O_{r}$ be the $r$ orbits for the action of $\sigma$ on the
connected components of $\Gamma$ of length $d_{1},\dots,d_{r}$ respectively
and assume to have fixed representatives $\Gamma_{1,1},\dots,\Gamma_{r,1}$ for
each of the orbits.

For each $j=1,\dots,r$, there exists a subset $X_{j}\subseteq\\{1,\dots,t\\}$
and, for each $l\in X_{j}$, a subset
$Z_{l}\subseteq\\{1,\dots,d^{\prime}_{l}\\}$, such that $\Gamma_{j,1}$ is the
complete graph with vertices

$\displaystyle\bigsqcup_{l\in X_{j}}\bigsqcup_{z\in
Z_{l}}\mathcal{B}^{\Gamma^{\prime}}_{l,z}.$

Notice that the subsets $X_{j}$ do not depend on the choice of the
representatives $\Gamma_{1,1},\dots,\Gamma_{r,1}$ and form a partition of the
set $\\{1,\dots,t\\}$. The partition $(X_{1},\dots,X_{r})$ belongs to
$M_{\omega_{1},\omega_{2}}$.

Indeed, since the orbit $O_{j}$ has length $d_{j}$, we must have that

$\sigma^{s}\left(\displaystyle\bigsqcup_{l\in X_{j}}\bigsqcup_{z\in
Z_{l}}\mathcal{B}^{\Gamma^{\prime}}_{l,z}\right)\cap\displaystyle\bigsqcup_{l\in
X_{j}}\bigsqcup_{z\in Z_{l}}\mathcal{B}^{\Gamma^{\prime}}_{l,z}=\varnothing$

for any $0<s\leq d_{j}-1$ and

$\sigma^{d_{j}}\left(\displaystyle\bigsqcup_{l\in X_{j}}\bigsqcup_{z\in
Z_{l}}\mathcal{B}^{\Gamma^{\prime}}_{l,z}\right)=\displaystyle\bigsqcup_{l\in
X_{j}}\bigsqcup_{z\in Z_{l}}\mathcal{B}^{\Gamma^{\prime}}_{l,z}.$

Recall that
$\sigma(\mathcal{B}^{\Gamma^{\prime}}_{l,z})=\mathcal{B}^{\Gamma^{\prime}}_{l,z+1}$,
where the index $z$ of $\mathcal{B}^{\Gamma^{\prime}}_{l,z}$ is considered
modulo $d_{l}^{\prime}$. We deduce therefore that $Z_{l}$ is such that

$(Z_{l}+s)\cap Z_{s}=\varnothing\mod d_{j}$

for each $0<s\leq d_{j}-1$ and

$Z_{l}+d_{j}=Z_{j}\mod d_{j}.$

This implies that $d_{j}|d_{l}^{\prime}$ and that there exists $a_{l}\in
Z_{l}$ such that

$Z_{l}=\left\\{a_{l}+d_{j}k\ |\
k=1,\dots,\dfrac{d_{l}^{\prime}}{d_{j}}\right\\}.$

In particular, it holds that $|Z_{l}|=\dfrac{d_{l}^{\prime}}{d_{j}}$, from
which we deduce that

$\sum_{l\in X_{j}}\dfrac{d_{l}^{\prime}}{d_{j}}\beta^{\prime}_{l}=\beta_{j}.$

We define then

$\pi_{\omega_{1},\omega_{2}}(S)\coloneqq[(X_{1},\dots,X_{r})]$

where $[(X_{1},\dots,X_{r})]$ is the class of the element
$(X_{1},\dots,X_{r})$ in the quotient $\overline{M_{\omega_{1},\omega_{2}}}$.

Notice that the morphism $\pi_{\omega_{1},\omega_{2}}$ is well-defined, i.e
does not depend on the choice of an ordering of the orbits $O_{1},\dots,O_{r}$
and it is surjective as we are taking the class $[(X_{1},\dots,X_{r})]$ in the
quotient $\overline{M_{\omega_{1},\omega_{2}}}$ for the action of
$W_{\omega_{1}}^{\prime}$.

From the description of the subsets $Z_{l}$ given above, we deduce that, for
each $[(X_{1},\dots,X_{r})]$, the fiber
$\pi_{\omega_{1},\omega_{2}}^{-1}([(X_{1},\dots,X_{r})])$ has cardinality

$|\pi^{-1}_{\omega_{1},\omega_{2}}([(X_{1},\dots,X_{r})])|=\prod_{j=1}^{r}d_{j}^{|X_{j}|-1}.$
(3.5.1)

##  4 Log-compatible functions and plethystic identities

In this Section, we recall the Definition of a Log compatible family, first
introduced by Letellier [36] and we prove our main Theorem 4.1.2 about these
families, which will be the key tool of this paper to compute the E-series of
non-generic character stacks.

Consider a family of rational functions indexed by multypes
$\\{F_{\omega}(t)\\}_{\omega\in\mathbb{T}_{I}}\subseteq{\mathbb{Q}}(t)$. For
any $V\subseteq\mathbb{N}^{I}$, we define the rational function
$F_{\alpha,V}(t)\in{\mathbb{Q}}(t)$ in the following way:

$F_{\alpha,V}(t)\coloneqq\sum_{\omega\in\mathbb{T}_{\alpha}}\dfrac{F_{\omega}(t)}{w(\omega)}\left(\sum_{\begin{subarray}{c}S^{\prime}\leq
S_{\omega}\\\ S^{\prime}\text{ of level
}V\end{subarray}}P_{[S^{\prime}]}(t)\mu(S^{\prime},S_{\omega})\right).$
(4.0.1)

For $V=\\{\alpha\\}$ we will use the notation $F_{\alpha,gen}(t)\coloneqq
F_{\alpha,\\{\alpha\\}}(t)$. Notice that

$F_{\alpha,gen}(t)=\sum_{\omega\in\mathbb{T}_{\alpha}}\dfrac{F_{\omega}(t)}{w(\omega)}(t-1)\mu(Z_{\alpha},S_{\omega})==\sum_{\omega\in\mathbb{T}_{\alpha}}\dfrac{F_{\omega}(t)}{w(\omega)}(t-1)C^{o}_{\omega^{ss}}.$

###  4.1 Plethysm and log compatibility: main result

We give the following definition of a Log-compatible family
$\\{F_{\omega}(t)\\}_{\omega\in\mathbb{T}_{I}}$.

######  Definition 4.1.1.

We say that $\\{F_{\omega}(t)\\}_{\omega\in\mathbb{T}_{I}}$ is Log compatible
if for any $\alpha\in\mathbb{N}^{I}$, $\omega\in\mathbb{T}_{\alpha}$ and for
every multitypes $\nu_{1},\dots,\nu_{r}$ and integers $d_{1},\dots,d_{r}$ such
that $\psi_{d_{1}}(\nu_{1})\ast\cdots\ast\psi_{d_{r}}(\nu_{r})=\omega$, we
have

$\prod_{j=1}^{r}F_{\nu_{j}}(t^{d_{j}})=F_{\omega}(t).$

Fix now an element $\alpha\in\mathbb{N}^{I}$ and a subset
$V\subseteq\mathbb{N}^{I}$. We have the following theorem:

######  Theorem 4.1.2.

For a Log compatible family $\\{F_{\omega}(t)\\}_{\omega\in\mathbb{T}_{I}}$,
we have the following equality:

$\operatorname{Coeff}_{\alpha}\left(\operatorname{Exp}\left(\sum_{\beta\in
V}F_{\beta,gen}(t)y^{\beta}\right)\right)=F_{\alpha,V}(t)$ (4.1.1)

###### Proof.

There is a unique morphism $\Theta$ of $\lambda$-rings

$\Theta:\mathcal{K}^{ss}_{I}\to{\mathbb{Q}}(t)$

obtained by extending

$\Theta(\omega_{\alpha})=F_{\alpha,gen}(t).$

By Lemma 2.3.4, Identity (4.1.1) is equivalent to the following Identity

$\sum_{\begin{subarray}{c}\nu\in\mathbb{T}^{ss}_{\alpha}\\\ \text{level
}V\end{subarray}}\dfrac{\Theta(\nu)}{w(\nu)}=F_{\alpha,V}(t)$ (4.1.2)

The RHS of eq.(4.1.2) is given by

$F_{V,\alpha}(t)=\sum_{\omega\in\mathbb{T}_{\alpha}}\dfrac{F_{\omega}(t)}{w(\omega)}\left(\sum_{\begin{subarray}{c}S\subseteq
S_{\omega}\\\ \text{ level
}V\end{subarray}}P_{[S]}(t)\mu(S,S_{\omega})\right).$ (4.1.3)

Consider now $\nu\in\mathbb{T}^{ss}_{I}$ and
$d_{1},\dots,d_{r}\in\mathbb{N}_{>0}$ and
$\beta_{1},\dots,\beta_{r}\in\mathbb{N}^{I}$ such that

$\nu=\psi_{d_{1}}(\omega_{\beta_{1}})\ast\cdots\ast\psi_{d_{r}}(\omega_{\beta_{r}}).$

The value of $\dfrac{\Theta(\nu)}{w(\nu)}$ is given by

$\dfrac{1}{w(\nu)}\prod_{j=1}^{r}\left(\sum_{\omega_{j}\in\mathbb{T}_{\beta_{j}}}\dfrac{F_{\omega_{\beta_{j}}}(t^{d_{j}})}{w(\omega_{j})}(t^{d_{j}}-1)\mu(Z_{\beta_{j}},S_{\omega_{j}})\right)=$
(4.1.4)

$\sum_{\omega\in\mathbb{T}_{\alpha}}\dfrac{F_{\omega}(t)P_{\nu}(t)}{w(\nu)}\left(\sum_{\begin{subarray}{c}\omega_{1}\in\mathbb{T}_{\beta_{1}},\dots,\omega_{r}\in\mathbb{T}_{\beta_{r}}\\\
\psi_{d_{1}}(\omega_{1})\ast\cdots\ast\psi_{d_{r}}(\omega_{r})=\omega\end{subarray}}\dfrac{1}{w(\omega_{1})\cdots
w(\omega_{r})}\prod_{j=1}^{r}\mu(Z_{\beta_{j}},S_{\omega_{j}})\right)$ (4.1.5)

Fix now a multitype $\omega\in\mathbb{T}_{\alpha}$ with
$\omega=(d^{\prime}_{1},\bm{\lambda}_{1})\cdots(d_{t}^{\prime},\bm{\lambda}_{t})$
for multipartitions $\bm{\lambda}_{1},\dots,\bm{\lambda}_{t}$ and its
associated admissible torus $S_{\omega}\subseteq\operatorname{GL}_{\alpha}$.

Denote by $H_{\nu,\omega}$ the set defined by

$H_{\nu,\omega}\coloneqq\\{(\omega_{1},\dots,\omega_{r})\in\mathbb{T}_{\beta_{1}}\times\cdots\times\mathbb{T}_{\beta_{r}}\
|\ \psi_{d_{1}}(\omega_{1})\ast\cdots\ast\psi_{d_{r}}(\omega_{r})=\omega\\}$

and by $\delta_{\nu}:H_{\nu,\omega}\to\mathbb{Z}$ the function defined as

$\delta_{\nu}((\omega_{1},\dots,\omega_{r}))\coloneqq\prod_{j=1}^{r}\mu(Z_{\beta_{j}},S_{\omega_{j}}).$

Let $M_{\nu,\omega^{ss}}$ be the set introduced in eq. 3.4.7. Consider the
following function

$f_{\nu,\omega}:M_{\nu,\omega}\to H_{\nu,\omega}$

defined as:

$f_{\nu,\omega}((X_{1},\dots,X_{r}))=(\omega_{1},\dots,\omega_{r})$

where

$\omega_{i}(d,\bm{\lambda})=\\#\\{h\in X_{i}\text{ such that
}\left(\dfrac{d^{\prime}_{h}}{d_{i}},\bm{\lambda}_{h}\right)=(d,\bm{\lambda})\\}$

for every $(d,\bm{\lambda})\in\mathbb{N}\times\mathcal{P}^{I}$.

The function $f_{\nu,\omega}$ is surjective and for each
$(\omega_{1},\dots,\omega_{r})$, the cardinality of the fiber is given by

$|f^{-1}_{\nu,\omega}(\omega_{1},\dots,\omega_{r})|=\prod_{(d,\bm{\lambda})\in\mathbb{N}\times\mathcal{P}^{I}}\dfrac{\omega(d,\bm{\lambda})!}{\omega_{1}(d,\bm{\lambda})!\cdots\omega_{r}(d,\bm{\lambda})!}$
(4.1.6)

Notice that for any $(\omega_{1},\dots,\omega_{r})\in H_{\nu,\omega}$, we have
the following equality:

$\prod_{(d,\bm{\lambda})\in\mathbb{N}\times\mathcal{P}^{I}}\dfrac{\omega(d,\bm{\lambda})!}{\omega_{1}(d,\bm{\lambda})!\cdots\omega_{r}(d,\bm{\lambda})!}=\dfrac{w(\omega)}{w(\omega_{1})\cdots
w(\omega_{r})}\dfrac{\prod_{j=1}^{r}\prod_{h\in
X_{j}}\left(\frac{d^{\prime}_{h}}{d_{j}}\right)}{\prod_{l=1}^{t}d^{\prime}_{l}}.$
(4.1.7)

As $\displaystyle\sum_{j=1}^{r}|X_{j}|=t$, the right hand side of eq.(4.1.7)
is equal to

$\dfrac{w(\omega)}{w(\omega_{1})\cdots
w(\omega_{r})}\dfrac{1}{\prod_{j=1}^{r}d_{j}^{|X_{j}|}}.$ (4.1.8)

For an element $m=(X_{1},\dots,X_{r})\in M_{\nu,\omega^{ss}}$, denote by
$d_{m}=\prod_{j=1}^{r}d_{j}^{|X_{j}|}$.

We can thus rewrite the RHS of eq.(4.1.5) as:

$\dfrac{F_{\omega}(t)}{w(\nu)}\left(\sum_{(\omega_{1},\dots,\omega_{r})\in
H_{\nu,\omega}}\dfrac{1}{w(\omega_{1})\cdots
w(\omega_{r})}\delta_{\nu}((\omega_{1},\dots,\omega_{r}))\right)=$ (4.1.9)

$\dfrac{F_{\omega}(t)}{w(\nu)w(\omega)}\left(\sum_{m\in
M_{\nu,\omega^{ss}}}\dfrac{w(\omega)}{w(\omega_{1})\cdots
w(\omega_{r})}\dfrac{\delta_{\nu}(f_{\nu,\omega}(m))}{|f_{\nu,\omega}^{-1}(m)|}\right)=\dfrac{F_{\omega}(t)}{w(\nu)w(\omega)}\left(\sum_{m\in
M_{\nu,\omega^{ss}}}\delta_{\nu}(f_{\nu,\omega}(m))d_{m}\right).$ (4.1.10)

The set $H_{\nu,\omega}$ is endowed with the following action of
$W^{\prime}_{\nu}$. We use the notations of section 3.5. An element
$\sigma=(\sigma_{1},\dots,\sigma_{k})\in W^{\prime}_{\nu}$ acts on
$(\omega_{1},\dots,\omega_{r})\in H_{\nu,\omega}$ by

$\sigma\cdot(\omega_{1},\dots,\omega_{r})=(\omega_{\sigma_{1}(1)},\dots,\omega_{\sigma_{1}(i_{1})},\omega_{\sigma_{2}(1)},\dots,\omega_{\sigma_{k}(r)}).$

Notice that the function $\delta_{\nu}$ is $W_{\nu}^{\prime}$ invariant and
$f_{\nu,\omega}$ is $W_{\nu}^{\prime}$ equivariant. The function

$\delta_{\nu}\circ f_{S,\omega}:M_{\nu,\omega^{ss}}\to\mathbb{Z}$

is therefore $W_{\nu}^{\prime}$ invariant and descends to a function
$\overline{M_{\nu,\omega^{ss}}}\to\mathbb{Z}$ which we still denote by
$\delta_{\nu}\circ f_{\nu,\omega}$. Notice moreover that the quantity $d_{m}$
is $W^{\prime}_{\nu}$ invariant too and so $d_{m}$ is well defined for an
element $m\in\overline{M_{\nu,\omega^{ss}}}$ too.

The right hand side of eq.(4.1.9) is therefore equal to

$\dfrac{F_{\omega}(t)P_{\nu}(t)}{w(\omega)}\left(\sum_{m\in\overline{M_{\nu,\omega^{ss}}}}\delta_{\nu}(f_{\nu,\omega}(m))\dfrac{d_{m}|W_{\nu}^{\prime}|}{w(\nu)}\right).$
(4.1.11)

Notice that, for any $m\in\overline{M}_{\nu,\omega^{ss}}$, we have

$\dfrac{d_{m}|W_{\nu}^{\prime}|}{w(\nu)}=\prod_{j=1}^{r}\dfrac{d_{j}^{|X_{j}|}}{d_{j}}=\prod_{j=1}^{r}d_{j}^{|X_{j}|-1}$

By eq.(3.5.1), we can thus rewrite the sum of eq.(4.1.11) as

$\dfrac{F_{\omega}(t)P_{\nu}(t)}{w(\omega)}\left(\sum_{S\in
P_{\nu,\omega}}\delta_{\nu}(f_{\nu,\omega}(\pi_{\nu,\omega}(S)))\right).$
(4.1.12)

From the remarks made in section 3.4 and eq. 3.4.7, we see that
$\delta_{\nu}(f_{\nu,\omega}(\pi_{\nu,\omega}(S))=\mu(S,S_{\omega})$ and so,
from eq.(4.1.9), we deduce that $\dfrac{\Theta(\nu)}{w(\nu)}$ is equal to

$\sum_{\omega\in\mathbb{T}_{\alpha}}\dfrac{F_{\omega}(t)P_{\nu}(t)}{w(\omega)}\sum_{S\in
P_{\nu,\omega^{ss}}}\mu(S,S_{\omega})$ (4.1.13)

Summing over the $\nu\in\mathbb{T}^{ss}_{\alpha}$ of level $V$, we have
therefore:

$\operatorname{Coeff}_{\alpha}\left(\operatorname{Exp}\left(\sum_{\beta\in
V}F_{\beta,gen}(t)y^{\beta}\right)\right)=\sum_{\begin{subarray}{c}\nu\in\mathbb{T}^{ss}_{\alpha}\\\
\text{of level }V\end{subarray}}\dfrac{\Theta(\nu)}{w(\nu)}=$ (4.1.14)

$\sum_{\begin{subarray}{c}\nu\in\mathbb{T}^{ss}_{\alpha}\\\ \text{of level
}V\end{subarray}}\sum_{\omega\in\mathbb{T}_{\alpha}}\dfrac{F_{\omega}(t)P_{\nu}(t)}{w(\omega)}\sum_{S\in
P_{\nu,\omega^{ss}}}\mu(S,S_{\omega})=\sum_{\omega\in\mathbb{T}_{\alpha}}\dfrac{F_{\omega}(t)}{w(\omega)}\left(\sum_{\begin{subarray}{c}S\leq
S_{\omega}\\\ \text{ of level
}V\end{subarray}}P_{[S]}(t)\mu(S,S_{\omega})\right)$ (4.1.15)

The right hand side is equal to $F_{\alpha,V}(t)$ by eq.(4.1.3). ∎

######  Remark 4.1.3.

As mentioned before, the notion of a Log compatible family and the definition
of the polynomials $F_{\alpha,gen}(t)$ had already been introduced in [36,
Paragraph 2.1.2]

Letellier [36, Theorem 2.2] used these objects to show the case in which
$V=\mathbb{N}^{I}$ of Theorem 4.1.2. However, his proof involves symmetric
functions and it has a completely different approach from ours and does not
seem to extend immediately to the case of any $V$.

##  5 Computation of convolution product for products of finite general linear
groups

In this section, we will use Theorem 4.1.2 to compute some invariants related
to the convolution product of class functions of
$\operatorname{GL}_{\alpha}(\mathbb{F}_{q})$.

We will start by reviewing some generalities about convolution product for the
class functions of a finite group and the representation theory of a finite
reductive group $G^{F}$.

We fix a prime $\ell$ such that $(\ell,q)=1$ and an isomorphism
$\overline{{\mathbb{Q}}_{\ell}}\cong\mathbb{C}$. We will identify
$\overline{{\mathbb{Q}}_{\ell}}$-vector spaces with $\mathbb{C}$-vector spaces
through this isomorphism in the following.

###  5.1 Convolution product of class functions of a finite group

Let $H$ be a finite group. We denote by $\mathcal{C}(H)$ the set of complex
class functions, i.e the functions $f:H\to\mathbb{C}$ which are constant on
the conjugacy classes of $H$. We denote by $1\in\mathcal{C}(H)$ the constant
function equal to $1$.

For $f,g\in\mathcal{C}(H)$, we denote by $\langle f,g\rangle$ the quantity

$\langle f,g\rangle=\dfrac{1}{|H|}\sum_{h\in H}f(h)\overline{g(h)}.$

Recall that a basis of $\mathcal{C}(H)$ is given by the irreducible characters
of $H$. We will denote the set of irreducible characters of $H$ by $H^{\vee}$.

The vector space $\mathcal{C}(H)$ can be endowed with the following ring
structure $(\mathcal{C}(H),\ast)$. Given two class functions
$f_{1},f_{2}\in\mathcal{C}(H)$, the convolution $f_{1}\ast f_{2}$ is the class
function defined as

$f_{1}\ast f_{2}(g)=\sum_{h\in H}f_{1}(gh)f_{2}(h^{-1}).$

Denote by $\operatorname{Cl}(H)$ the set of conjugacy classes of $H$. For any
$\mathcal{O}\in\operatorname{Cl}(H)$, we denote by
$1_{\mathcal{O}}\in\mathcal{C}(H)$ the characteristic function of
$\mathcal{O}$. For a central element $\eta\in H$, we denote by $1_{\eta}$ the
characteristic function $1_{\\{\eta\\}}$. Notice that for any class function
$f$, there is an equality

$\langle f\ast 1_{\eta},1_{e}\rangle=\dfrac{f(\eta)}{|H|}.$

Recall now that

$\displaystyle 1_{e}=\sum_{\chi\in H^{\vee}}\dfrac{\chi(1)}{|H|}\chi.$ (5.1.1)

We have therefore

$\dfrac{f(\eta)}{|H|}=\sum_{\chi\in H^{\vee}}\langle f\ast
1_{\eta},\chi\rangle\dfrac{\chi(1)}{|H|}.$ (5.1.2)

For any two class functions $f_{1},f_{2}:H\to\mathbb{C}$ and an irreducible
character $\chi\in H^{\vee}$, we have

$\langle f_{1}\ast f_{2},\chi\rangle=\langle f_{1},\chi\rangle\langle
f_{2},\chi\rangle\dfrac{|H|}{\chi(1)}$ (5.1.3)

(see for example [30, Theorem 2.13]). In particular, from eq.(5.1.2), we
deduce the identity:

$\dfrac{f(\eta)}{|H|}=\sum_{\chi\in H^{\vee}}\langle f,\chi\rangle\langle
1_{\eta},\chi\rangle=\sum_{\chi\in H^{\vee}}\langle
f,\chi\rangle\dfrac{\chi(\eta)}{\chi(1)}\dfrac{\chi(1)}{|H|}.$ (5.1.4)

###  5.2 Deligne-Lusztig induction

In this Section, we review the definition of the Deligne-Lusztig induction for
finite reductive groups. This construnction, which allows to give a geometric
description of the irreducible characters of finite reductive groups, was
first introduced in [13]. We follow mostly the book [14].

Consider an $F$-stable Levi subgroup $L$ of $G$, a parabolic subgroup $P$
having $L$ as Levi factor and denote by $U_{P}$ the unipotent radical of $P$.
Recall that there is an isomorphism

$P/U_{P}\cong L$

and denote by $\pi_{L}:P\to L$ the associated quotient map.

Notice that in general $P$ is not $F$-stable. We can find an $F$-stable
parabolic subgroup $P\supseteq L$ if and only if $\epsilon_{L}=\epsilon_{G}$.

Denote by $\mathcal{L}$ the Lang map $\mathcal{L}:G\to G$ given by
$\mathcal{L}(g)=g^{-1}F(g)$. The variety
$X_{L}\coloneqq\mathcal{L}^{-1}(U_{P})$ has a left $G^{F}$-action and a right
$L^{F}$-action by multiplication on the left/right respectively.

This action induces an action on the compactly supported étale cohomology
groups $H^{i}_{c}(X_{L},\overline{{\mathbb{Q}}}_{\ell})$ and so endows the
virtual vector space

$\displaystyle
H^{*}_{c}(X_{L},\overline{{\mathbb{Q}}}_{\ell})\coloneqq\bigoplus_{i\geq
0}(-1)^{i}H^{i}_{c}(X_{L},\overline{{\mathbb{Q}}}_{\ell})$

with the structure of a virtual $G^{F}$-representation-$L^{F}$. For an
$L^{F}$-representation $M$, we define the Deligne-Lusztig induction
$R^{G}_{L}(M)$ as the virtual $G^{F}$-representation given by

$R^{G}_{L}(M)=H^{*}_{c}(X_{L},\overline{{\mathbb{Q}}}_{\ell})\otimes_{\mathbb{C}[L^{F}]}M.$

We will denote by $R^{G}_{L}$ the induced linear map

$R^{G}_{L}:\mathcal{C}(L^{F})\to\mathcal{C}(G^{F}).$

######  Remark 5.2.1.

In the case that will interest us in this article, it will always be true that
the functor $R^{G}_{L}$ does not depend on the choice of the parabolic
subgroup $P\supseteq L$ (see for example [6]).

Consider now a split Levi subgroup $L$, i.e

$(L,F)\cong\operatorname{GL}_{n_{0}}\times\cdots\times\operatorname{GL}_{n_{s}},$

with $n_{0}+\cdots+n_{s}=n$. In this case, we can take an $F$-stable parabolic
subgroup $P\supseteq L$. The variety $X_{L}$ is a $U_{P}$-principal bundle
over the finite variety $G^{F}/U^{F}$ and
$H_{c}^{*}(X_{L},\overline{{\mathbb{Q}}}_{\ell})$ is therefore concentrated in
degree $2\dim(U_{P})$, see [14] for more details.

We have thus an equality

$R^{G}_{L}(M)=\mathbb{C}[G^{F}/U^{F}]\otimes_{\mathbb{C}[L^{F}]}M$

for every $L^{F}$-representation $M$.

In the split case, we can give the following equivalent description of this
functor. For an $L^{F}$-representation $M$, denote by
$\operatorname{Infl}_{L^{F}}^{P^{F}}(M)$ the natural lift to a
$P^{F}$-representation via the quotient map $\pi_{L}$. In [14, Proposition
5.18 (1)], it is shown the following Lemma:

######  Lemma 5.2.2.

We have an isomorphis of functors:

$R^{G}_{L}\cong\operatorname{Ind}_{P^{F}}^{G^{F}}(\operatorname{Infl}_{L^{F}}^{P^{F}}).$

We end by recalling the following properties of Deligne-Lusztig induction.

######  Lemma 5.2.3.

For a reductive group $G$ and a Levi subgroup $L$, the following holds:

  1. 1.

Given an $F$-stable Levi subgroup $L^{\prime}\supseteq L$, there is an
isomorphism of functors: $R^{G}_{L^{\prime}}(R^{L^{\prime}}_{L})\cong
R^{G}_{L}$.

  2. 2.

Assume there exist reductive groups $G_{1},G_{2}$ and Levi subgroups
$L_{1},L_{2}$ such that $G=G_{1}\times G_{2}$ and $L=L_{1}\times L_{2}$. For
an $L_{1}^{F}$-representation $M_{1}$ and an $L_{2}^{F}$-representation
$M_{2}$, there is a natural isomorphism $R^{G}_{L}(M_{1}\boxtimes
M_{2})=R^{G_{1}}_{L_{1}}(M_{1})\boxtimes R^{G_{2}}_{L_{2}}(M_{2})$.

###### Proof.

The first point is shown in [14, Proposition 9.1.8]. For the second point, we
can choose parabolic subgroups $G_{1}\supseteq P_{1}\supseteq L_{1}$ and
$G_{2}\supseteq P_{2}\supseteq L_{2}$ such that $P_{1},P_{2}$ have as Levi
factor $L_{1},L_{2}$ respectively. Notice thus that $P=P_{1}\times
P_{2}\subseteq G$ is a parabolic subgroup having $L$ as Levi factor and
$U_{P}=U_{P_{1}}\times U_{P_{2}}$. We have therefore

$\mathcal{L}^{-1}(U_{P})=\mathcal{L}^{-1}(U_{P_{1}})\times\mathcal{L}^{-1}(U_{P_{1}})$

and therefore

$H^{*}_{c}(\mathcal{L}^{-1}(U_{P}),\overline{{\mathbb{Q}}_{\ell}})\cong
H^{*}_{c}(\mathcal{L}^{-1}(U_{P_{1}}),\overline{{\mathbb{Q}}_{\ell}})\otimes
H^{*}_{c}(\mathcal{L}^{-1}(U_{P_{2}}),\overline{{\mathbb{Q}}_{\ell}}),$

as $\mathbb{C}[L_{1}^{F}]\otimes\mathbb{C}[L_{2}^{F}]$-modules, from which we
deduce that

$R^{G}_{L}(M_{1}\boxtimes M_{2})=R^{G_{1}}_{L_{1}}(M_{1})\boxtimes
R^{G_{2}}_{L_{2}}(M_{2}).$

∎

######  Remark 5.2.4.

Consider an $F$-stable Levi subgroup $L^{\prime}\supseteq L$ and a linear
character $\theta:(L^{\prime})^{F}\to\mathbb{C}^{*}$. By restriction, we can
consider it as a character $\theta:L^{F}\to\mathbb{C}^{*}$.

For any $f\in\mathcal{C}(L^{F})$, we have an identity
$R^{L^{\prime}}_{L}(\theta f)=\theta R^{L^{\prime}}_{L}(f)$ and therefore, by
Lemma 5.2.3(1), an equality $R^{G}_{L}(\theta f)=R^{G}_{L^{\prime}}(\theta
R^{L^{\prime}}_{L}(f))$.

###  5.3 Unipotent characters

Fix a group $G$ and an $F$-stable maximal torus as above. We follow the
notations of [14, Chapter 11].

Denote by $W$ the Weyl group $W_{G}(T)$ and by $\widetilde{W}$ the semidirect
group $W\rtimes\langle F\rangle$, where $\langle F\rangle$ is the group
generated by the finite order automorphism induced by $F$ on $W$.

Denote by $\mathcal{C}(WF)$ the vector space of function $f:W\to\mathbb{C}$
constant on $F$-conjugacy classes. Equivalenty, a function
$f\in\mathcal{C}(WF)$ can be seen as a function on the coset
$WF\subseteq\widetilde{W}$, invariant under $W$-conjugation.

The vector space $\mathcal{C}(WF)$ is endowed with the Hermitian product
defined as

$\displaystyle\langle f,g\rangle_{WF}=\dfrac{1}{|W|}\sum_{w\in
WF}f(w)\overline{g(w)},$

for $f,g\in\mathcal{C}(WF).$

######  Example 5.3.1.

Let $G=(\operatorname{GL}_{n})_{d}$ and $T$ be the torus of Example 3.0.3. In
this case, the morphism $\psi:\mathcal{C}(S_{n})\to\mathcal{C}(WF)$ defined by
$\psi(f)(\sigma_{1},\dots,\sigma_{d})=f(\sigma_{d}\cdots\sigma_{1})$ is an
isomorphism. It is not difficult to see that the isomorphism $\psi$ is an
isometry too.

In [14, Chapter 11.6], it is shown how to associate to each
$\chi\in(W^{\vee})^{F}$ a function in $\mathcal{C}(WF)$, which we denote by
$\tilde{\chi}.$ This association is isometric with respect to the natural
Hermitian products on both sides. The elements
$\\{\tilde{\chi}\\}_{\chi\in(W^{\vee})^{F}}$ form moreover a basis of
$\mathcal{C}(WF)$.

######  Remark 5.3.2.

Consider the group $G$ and the torus $T$ of Example 5.3.1 above. An
irreducible character $\chi\in W^{\vee}$ is determined by partitions
$\lambda^{1},\dots,\lambda^{d}\in\mathcal{P}_{n}$ such that
$\chi=\chi^{\lambda^{1}}\boxtimes\cdots\boxtimes\chi^{\lambda^{d}}$. The
character $\chi$ is $F$-stable if and only if
$\lambda^{1}=\cdots=\lambda^{d}=\lambda$, so that $(W^{\vee})^{F}$ is in
bijection with $\mathcal{P}_{n}$.

For $\lambda\in\mathcal{P}_{n}$, consider the associated function
$\widetilde{(\chi^{\lambda})^{\boxtimes d}}\in\mathcal{C}(WF)$. It is possible
to check that $\psi^{-1}(\widetilde{(\chi^{\lambda})^{\boxtimes
d}})=\chi^{\lambda}\in\mathcal{C}(S_{n})$.

In [14, Chapter 11.6] for $f\in\mathcal{C}(WF)$, it is defined a class
function $R_{f}:G^{F}\to\mathbb{C}$ as

$R_{f}\coloneqq\dfrac{1}{|W|}\sum_{w\in W}f(w)R^{G}_{T_{w}}(1).$ (5.3.1)

In locus cit, it shown that the map $f\to R_{f}$ induces an isometry
$\mathcal{C}(WF)\to\mathcal{C}(G^{F})$. In particular, the elements
$\\{R_{\tilde{\chi}}\\}_{\chi\in(W^{\vee})^{F}}$ have norm $1$ and are
pairwise orthogonal in $\mathcal{C}(G^{F})$.

Consider now an $F$-stable Levi subgroup $L\supseteq T$ and the corresponding
Weyl group $W_{L}\coloneqq W_{L}(T)$ which is an $F$-stable subgroup of $W$.
Define the induction map
$\operatorname{Ind}_{W_{L}F}^{WF}:\mathcal{C}(W_{L}F)\to\mathcal{C}(WF)$ as

$\displaystyle\operatorname{Ind}_{W_{L}F}^{WF}(f)(w)=\dfrac{1}{|W_{L}|}\sum_{\begin{subarray}{c}h\in
W_{L}\\\ h^{-1}wF(h)\in W_{L}\end{subarray}}f(h^{-1}wF(h)).$

In [14, Proposition 11.6.6] it is shown the following Lemma.

######  Lemma 5.3.3.

For any $f\in\mathcal{C}(W_{L}F)$, we have:

$R^{G}_{L}(R_{f})=R_{\operatorname{Ind}_{W_{L}F}^{WF}(f)}$ (5.3.2)

Let $G=(\operatorname{GL}_{n})_{d}$ and $T$ be the torus considered above. In
[14, Chapter 11.7], it is shown the following Lemma.

######  Lemma 5.3.4.

For each $\chi\in(W^{\vee})^{F}$, the class function $R_{\tilde{\chi}}$ is an
irreducible character of $G^{F}$.

The irreducible characters of this form are called unipotent characters. In
particular, for every $\lambda\in\mathcal{P}_{n}$, there is a corresponding
irreducible character $R_{\widetilde{\chi_{\lambda}^{\boxtimes d}}}$, which we
will denote by $R_{\lambda}$.

Notice that Lemma 5.3.4 is true for any group $G$ of the form
$(\operatorname{GL}_{n_{1}})_{d_{1}}\times\cdots(\operatorname{GL}_{n_{r}})_{d_{r}}$.
In this case, the unipotent characters, by Lemma 5.2.3(2), are in bijection
with the multipartitions
$\bm{\lambda}\in\mathcal{P}_{n_{1}}\times\cdots\times\mathcal{P}_{n_{r}}$ and
we denote by $R_{\bm{\lambda}}$ the associated irreducible unipotent
character.

For a such a group $G$, all $F$-stable Levi subgroups are still of the form
$(\operatorname{GL}_{n^{\prime}_{1}})_{d^{\prime}_{1}}\times\cdots(\operatorname{GL}_{n^{\prime}_{s}})_{d^{\prime}_{s}}$.
By eq.(5.3.2), we deduce the following Proposition.

######  Proposition 5.3.5.

Let
$G=(\operatorname{GL}_{n_{1}})_{d_{1}}\times\cdots(\operatorname{GL}_{n_{r}})_{d_{r}}$
and $L\subseteq G$ an $F$-stable Levi subgroup such that

$(L,F)\cong(\operatorname{GL}_{n^{\prime}_{1}})_{d^{\prime}_{1}}\times\cdots(\operatorname{GL}_{n^{\prime}_{s}})_{d^{\prime}_{s}}.$

For any
$\bm{\mu}\in\mathcal{P}_{n^{\prime}_{1}}\times\cdots\times\mathcal{P}_{n^{\prime}_{s}}$,
the character $R^{G}_{L}(R_{\bm{\mu}})$ belongs to the vector space spanned by
the unipotent characters of $G^{F}$.

Lastly, consider an $F$-stable Levi subgroup $L\subseteq G$, a unipotent
character $R_{f}\in\mathcal{C}(L^{F})$ and a linear character
$\theta:L^{F}\to\mathbb{C}^{*}$. Fix a central element $\gamma\in G^{F}$.
Notice that $\gamma\in L^{F}$ too. The Mackey’s formula [14, Proposition
10.1.2] implies the following Proposition.

######  Proposition 5.3.6.

We have an equality:

$R^{G}_{L}(\theta R_{f})(\gamma)=R^{G}_{L}(\theta R_{f})(e)\theta(\gamma).$
(5.3.3)

###  5.4 Characters of tori and graphs

Consider the group $\operatorname{GL}_{\alpha}$ and fix an $F$-stable maximal
torus $T\subseteq\operatorname{GL}_{\alpha}$. We follow the notations of
section 3.4.

Recall that we have the dual root system $\Phi^{\vee}\subseteq Y_{*}(T)$ which
is endowed with a canonical bijection $\Phi\leftrightarrow\Phi^{\vee}$ and
that, for each $h\neq j\in\\{1,\dots,m\\}$, we denote by
$\epsilon^{\vee}_{h,j}$ the element associated to $\epsilon_{h,j}$ via this
bijection.

Consider now a character $\theta:T^{F}\to\mathbb{C}^{*}$. In this paragraph,
we show how to associate an admissible graph $\Gamma_{\theta}$ with vertices
$\mathcal{B}$ to the character $\theta$.

In [14, Proposition 11.7.1] it is shown that there exists a canonical short
exact sequence

${1}$${Y_{*}(T)}$${Y_{*}(T)}$${T^{F}}$${1.}$

In particular, the character $\theta:T^{F}\to\mathbb{C}^{*}$ induces by
restriction a morphism $\widetilde{\theta}:Y_{*}(T)\to\mathbb{C}^{*}$.

The graph $\Gamma_{\theta}$ is defined as follows. The set of the vertices of
$\Gamma_{\theta}$ is $\mathcal{B}=\\{\epsilon_{1},\dots,\epsilon_{m}\\}$ and,
for each $h>j$, there is an edge between $\epsilon_{h}$ and $\epsilon_{j}$ if
and only if

$\epsilon_{h,j}^{\vee}\in\operatorname{Ker}(\widetilde{\theta}).$

From Remark 3.4.1, we deduce the following Lemma.

######  Lemma 5.4.1.

For any $\theta:T^{F}\to\mathbb{C}^{*}$, the graph $\Gamma_{\theta}$ is
admissible.

In particular, as remarked in Paragraph section 3.4, there exists a unique
admissible subtorus $S_{\theta}\subseteq\operatorname{GL}_{\alpha}$ such that
$\Gamma_{S_{\theta}}=\Gamma_{\theta}$.

We will denote by $L_{\theta}=C_{\operatorname{GL}_{m}}(S_{\theta})$ and by
$\widetilde{L_{\theta}}=L_{\theta}\cap\operatorname{GL}_{\alpha}$. The Levi
subgroup $L_{\theta}$ is the connected centralizer of $\theta$, as introduced
in [13, Definition 5.19].

######  Example 5.4.2.

Consider the subset $I=\\{1,2,3,4\\}$, the dimension vector $\alpha=(2,1,1,1)$
and the torus $T$ of diagonal matrices of Example 3.4.11. Notice that

$T^{F}=\Biggl{\\{}\left(\begin{pmatrix}\lambda&0\\\
0&\mu\end{pmatrix},\gamma,\delta,\eta\right)\ |\
\lambda,\mu,\gamma,\delta,\eta\in\ \mathbb{F}_{q}^{*}\Biggr{\\}}.$

Consider
$(\beta_{1},\beta_{2})\in\operatorname{Hom}(\mathbb{F}_{q},\mathbb{C}^{*})^{2}$
and let

$\theta_{\beta_{1},\beta_{2}}:T^{F}\to\mathbb{C}^{*}$

defined as

$\theta_{\beta_{1},\beta_{2}}(\lambda,\mu,\gamma,\delta,\eta)=\beta_{1}(\lambda\gamma\delta\eta)\beta_{2}(\mu).$

If $\beta_{1}\neq\beta_{2}$, the graph $\Gamma_{\theta_{\beta_{1},\beta_{2}}}$
is

${\epsilon_{1}}$${\epsilon_{3}}$${\epsilon_{4}}$${\epsilon_{5}}$${\epsilon_{2}}$

and the admissible torus $S_{\theta_{\beta_{1},\beta_{2}}}$ is therefore the
torus $S_{1}$ of Example 3.2.7.

If $\beta_{1}=\beta_{2}$, the graph $\Gamma_{\theta_{\beta_{1},\beta_{1}}}$ is

${\epsilon_{1}}$${\epsilon_{3}}$${\epsilon_{4}}$${\epsilon_{5}}$${\epsilon_{2}}$

and the admissible torus $S_{\theta_{\beta_{1},\beta_{1}}}$ is $Z_{\alpha}$.

###  5.5 Characters of tori and Levi subgroups of finite general linear group

####  5.5.1 Reduced characters and connected centralizers

Let now $G=\operatorname{GL}_{n}$ (i.e $|I|=1$) and consider an $F$-stable
Levi subgroup $L\subseteq\operatorname{GL}_{n}$ and a linear character
$\theta:L^{F}\to\mathbb{C}^{*}$. The Levi subgroup $L$ contains an $F$-stable
maximal torus $T$. From $\theta,T$, we determine the connected centralizer
$L_{\theta}$ as above such that $L_{\theta}\supseteq L$. The character
$\theta$ is called reduced if $L_{\theta}=L$.

######  Remark 5.5.1.

While the connected centralizer $L_{\theta}$ does depend on the choice of
torus $T$, from [13, Proposition 5.11(ii), Proposition 5.20] we deduce that
for two $F$-stable maximal tori $T,T^{\prime}\subseteq L$ and corresponding
connected centralizers $L_{\theta},L^{\prime}_{\theta}$, there exists an
element $g\in\operatorname{GL}_{n}(\mathbb{F}_{q})$ such that
$gL_{\theta}g^{-1}=L_{\theta}^{\prime}$.

For any two positive integers $r,d$ such that $r|d$, the norm map
$N_{\mathbb{F}^{*}_{q^{d}}/\mathbb{F}^{*}_{q^{r}}}:\mathbb{F}^{*}_{q^{d}}\to\mathbb{F}_{q^{r}}^{*}$
induces by precomposition an injective homomorphism

$\Gamma_{r,d}\coloneqq\operatorname{Hom}(\mathbb{F}_{q^{r}}^{*},\mathbb{C}^{*})\to\operatorname{Hom}(\mathbb{F}^{*}_{q^{d}},\mathbb{C}^{*}).$

We denote by $\Gamma$ the inductive limit via these maps

$\Gamma\coloneqq\varinjlim\operatorname{Hom}(\mathbb{F}_{q^{d}}^{*},\mathbb{C}^{*}).$

Notice that, for any $n\geq 1$, we can view
$\operatorname{Hom}(\mathbb{F}^{*}_{q^{n}},\mathbb{C}^{*})$ as a subgroup of
$\Gamma$ through the universal maps of the limit. The Frobenius morphism acts
by precomposition on each term
$\operatorname{Hom}(\mathbb{F}_{q^{d}}^{*},\mathbb{C}^{*})$ (i.e
$F(\gamma)=\gamma\circ F$) and so defines a morphism $F:\Gamma\to\Gamma$.

Let now $G=\operatorname{GL}_{n}$ and consider the Levi subgroup

$L=(\operatorname{GL}_{n_{1}})_{d_{1}}\times\cdots\times(\operatorname{GL}_{n_{r}})_{d_{r}}$

with $n_{1},\dots,n_{r},$ $d_{1},\dots,d_{r}$ positive integers such that
$d_{1}n_{1}+\cdots+d_{r}n_{r}=n$ and let $T$ be the maximal torus

$(T_{n_{1}})_{d_{1}}\times\cdots(T_{n_{r}})_{d_{r}}.$

The group $L^{F}$ is isomorphic to
$\operatorname{GL}_{n_{1}}(\mathbb{F}_{q^{d_{1}}})\times\cdots\times\operatorname{GL}_{n_{r}}(\mathbb{F}_{q^{d_{r}}})$.
A character $\theta:L^{F}\to\mathbb{C}^{*}$ is therefore given by an element
$(\theta_{1},\dots,\theta_{r})\in\operatorname{Hom}(\mathbb{F}_{q^{d_{1}}}^{*},\mathbb{C}^{*})\times\cdots\times\operatorname{Hom}(\mathbb{F}_{q^{d_{r}}}^{*},\mathbb{C}^{*})$
such that

$\displaystyle\theta(M_{1},\dots,M_{r})=\prod_{j=1}^{r}\theta_{j}(\det(M_{j}))$

with $M_{j}\in\operatorname{GL}_{n_{j}}(\mathbb{F}_{q^{d_{j}}})$. We have the
following Lemma

######  Lemma 5.5.2.

The character $\theta$ is reduced if and only if the $F$-orbits of
$\theta_{1},\dots,\theta_{r}$ inside $\Gamma$ have length $d_{1},\dots,d_{r}$
respectively and are pairwise disjoint.

###### Proof.

Notice that, for any $h\in\\{1,\dots,n\\}$ there exist unique
$i_{h}\in\\{1,\dots,r\\}$ and $j_{h}\in\\{1,\dots,d_{h}\\}$ such that

$\sum_{s=1}^{i_{h}-1}d_{s}n_{s}<h\leq\sum_{s=1}^{i_{h}}d_{s}n_{s}$

and

$\sum_{s=1}^{i_{h}-1}d_{s}n_{s}+n_{i_{h}}(j_{h}-1)<h\leq\sum_{s=1}^{i_{h}-1}d_{s}n_{s}+n_{i_{h}}j_{h}.$

For $h_{1},h_{2}\in\\{1,\dots,n\\}$, we have that

$\widetilde{\theta}(\epsilon_{h_{1},h_{2}}^{\vee})=1$

if and only if

$\theta_{i_{h_{1}}}^{j_{h_{1}}}=\theta_{i_{h_{2}}}^{j_{h_{2}}}$

as elements of $\Gamma$, from which we deduce the Lemma above.

∎

Consider now two Levi subgroups $L\subseteq\operatorname{GL}_{n}$ and
$L^{\prime}\subseteq\operatorname{GL}_{n^{\prime}}$ and the product Levi
subgroup $M=L\times L^{\prime}\subseteq\operatorname{GL}_{m}$ embedded block
diagonally, where $m=n+n^{\prime}$.

Assume that
$L=(\operatorname{GL}_{n_{1}})_{d_{1}}\times\cdots\times(\operatorname{GL}_{n_{r}})_{d_{r}}$
and
$L^{\prime}=(\operatorname{GL}_{n^{\prime}_{1}})_{d_{1}^{\prime}}\times\cdots\times(\operatorname{GL}_{n^{\prime}_{s}})_{d^{\prime}_{s}}$
and consider two reduced characters $\theta:L^{F}\to\mathbb{C}^{*}$ and
$\theta^{\prime}:(L^{\prime})^{F}\to\mathbb{C}^{*}$ corresponding to
$(\theta_{1},\dots,\theta_{r}),(\theta_{1}^{\prime},\dots,\theta^{\prime}_{s})$
where
$\theta_{i}\in\operatorname{Hom}(\mathbb{F}_{q^{d_{i}}}^{*},\mathbb{C}^{*}),\theta^{\prime}_{j}\in\operatorname{Hom}(\mathbb{F}_{q^{d^{\prime}_{j}}}^{*},\mathbb{C}^{*})$
for $i=1,\dots,r,j=1,\dots,s$.

Consider the character

$\gamma=\theta\times\theta^{\prime}:M^{F}\to\mathbb{C}^{*}.$

Its connected centralizer $M_{\gamma}$ admits the following description. For
$i\in\\{1,\dots,r\\}$, consider the subset $J_{i}\subseteq\\{1,\dots,s\\}$
defined by

$J_{i}\coloneqq\bigl{\\{}j\in\\{1,\dots,s\\}\ |\ d^{\prime}_{j}=d_{i}\text{
and the }F\text{-orbits inside }\Gamma\text{ of
}\theta_{i},\theta^{\prime}_{j}\text{ have nonempty intersection}\bigr{\\}}.$

Notice that either $J_{i}=\varnothing$ or $J_{i}=\\{j_{i}\\}$ for an element
$j_{i}\in\\{1,\dots,s\\}$, as the characters $\theta,\theta^{\prime}$ are both
reduced. Denote by $I^{\prime}\subseteq\\{1,\dots,r\\}$ the subset
$I^{\prime}\coloneqq\\{i\ |\ J_{i}=\varnothing\\}$ and by
$J^{\prime}\subseteq\\{1,\dots,s\\}$ the subset $\displaystyle
J^{\prime}=\\{1,\dots,s\\}-\bigsqcup_{i=1}^{r}J_{i}$.

A similar argument to the one used to prove Lemma 5.5.2 shows that a connected
centralizer $M_{\gamma}$ is $\operatorname{GL}_{n}(\mathbb{F}_{q})$\-
conjugated to the Levi subgroup

$M^{\prime}_{\gamma}=\displaystyle\prod_{i\in
I^{\prime}}(\operatorname{GL}_{n_{i}})_{d_{i}}\prod_{j\in
J^{\prime}}(\operatorname{GL}_{n^{\prime}_{j}})_{d^{\prime}_{j}}\prod_{i\in(I^{\prime})^{c}}(\operatorname{GL}_{n_{i}+n^{\prime}_{j_{i}}})_{d_{i}}.$

Via this conjugation, the character $\gamma$ corresponds to the character
$\gamma:(M^{\prime}_{\gamma})^{F}\to\mathbb{C}^{*}$ associated to
$((\theta_{i})_{i\in I^{\prime}},(\theta^{\prime}_{j})_{j\in
J^{\prime}},(\theta_{i})_{i\in(I^{\prime})^{c}})$.

####  5.5.2 Characters of Levi subgroups

Consider $L,T\subseteq\operatorname{GL}_{m}$ and
$\theta:L^{F}\to\mathbb{C}^{*}$ as before. As remarked at the end of section
5.4, the character $\theta$ can be extended to the connected centralizer
$\theta:L_{\theta}^{F}\to\mathbb{C}^{*}$ and $L_{\theta}\supseteq L$.

Conversely, for each character $\gamma:T^{F}\to\mathbb{C}^{*}$ such that
$L_{\gamma}\supseteq L$, the character $\gamma$ can be first extended to
$\gamma:L_{\gamma}^{F}\to\mathbb{C}^{*}$ and then restricted to obtain a
linear character $\gamma:L^{F}\to\mathbb{C}^{*}$.

We deduce therefore the following Proposition.

######  Proposition 5.5.3.

There is a bijection:

$\operatorname{Hom}(L^{F},\mathbb{C}^{*})\leftrightarrow\\{\gamma\in\operatorname{Hom}(T^{F},\mathbb{C}^{*})\
|\ L_{\gamma}\supseteq L\\}.$ (5.5.1)

Consider now $G=\operatorname{GL}_{\alpha}$, an admissible torus $S$, the
associated Levi subgroups