Alvis–Curtis Duality and a Generalized Mullineux Involution

Alvis–Curtis Duality for Finite General Linear Groups and a Generalized
Mullineux Involution††This paper is a contribution to the Special Issue on the
Representation Theory of the Symmetric Groups and Related Topics. The full
collection is available at https://www.emis.de/journals/SIGMA/symmetric-
groups-2018.html

Olivier DUDAS † and Nicolas JACON ‡

O. Dudas and N. Jacon

† Université Paris Diderot, UFR de Mathématiques, Bâtiment Sophie Germain,
† 5 rue Thomas Mann, 75205 Paris CEDEX 13, France<EMAIL_ADDRESS>
‡ Université de Reims Champagne-Ardenne, UFR Sciences exactes et naturelles,
‡ Laboratoire de Mathématiques EA 4535, Moulin de la Housse BP 1039, 51100
Reims, France<EMAIL_ADDRESS>
Received June 17, 2017, in final form January 22, 2018; Published online
January 30, 2018

We study the effect of Alvis–Curtis duality on the unipotent representations
of $\mathrm{GL}_{n}(q)$ in non-defining characteristic $\ell$. We show that
the permutation induced on the simple modules can be expressed in terms of a
generalization of the Mullineux involution on the set of all partitions, which
involves both $\ell$ and the order of $q$ modulo $\ell$.

Mullineux involution; Alvis–Curtis duality; crystal graph; Harish-Chandra
theory

20C20; 20C30; 05E10

##  1 Introduction

Let $\mathfrak{S}_{n}$ be the symmetric group on $n$ letters. It is well known
that the complex irreducible representations of $\mathfrak{S}_{n}$ are
naturally labelled by the set of partitions of $n$. Tensoring with the sign
representation induces a permutation of the irreducible characters which
corresponds to the conjugation of partitions. An analogous involution can be
considered for representations in positive characteristic $p>0$. In this case,
the irreducible representations are parametrized by $p$-regular partitions and
the permutation induced by tensoring with the sign representation has a more
complicated combinatorial description. The explicit computation of this
involution $\mathsf{M}_{p}$ was first conjectured by Mullineux in [26] and
proved by Ford–Kleshchev in [14]. Their result was later generalized to
representations of Hecke algebras at a root of unity by Brundan [3] with a
view to extending the definition of $\mathsf{M}_{p}$ to the case where $p$ is
any positive integer.

For representations of a finite group of Lie type $G$, the Alvis–Curtis
duality $\mathsf{D}_{G}$ functor (see [10]) provides an involution of the same
nature. For example, the Alvis–Curtis dual of a complex unipotent character of
$\mathrm{GL}_{n}(q)$ parametrized by a partition is, up to a sign, the
character parametrized by the conjugate partition. Unlike the case of
symmetric groups, the duality $\mathsf{D}_{G}$ does not necessarily map
irreducible representations to irreducible representations, but only to
complexes of representations. Nevertheless, Chuang–Rouquier showed in [7] how
to single out a specific composition factor in the cohomology of these
complexes, yielding an involution $\mathsf{d}_{G}$ on the set of irreducible
representations of $G$ in non-defining characteristic $\ell\geq 0$.

The purpose of this paper is to explain how to compute this involution
$\mathsf{d}_{G}$ using the Harish-Chandra theory and the representation theory
of Hecke algebras, see Theorem 2.5. We illustrate our method on the
irreducible unipotent representations of $\mathrm{GL}_{n}(q)$, which are
parametrized by partitions of $n$. This yields an explicit involution
$\mathsf{M}_{e,\ell}$ on the set of partitions of $n$ which depends both on
$\ell$ and on the order $e$ of $q$ modulo $\ell$ (with the convention that
$e=\ell$ if $\ell\,|\,q-1$), see Theorem 3.7.

######  Theorem 1.1.

Assume that $\ell\nmid q$. Let $S(\lambda)$ be the simple unipotent module of
$\mathrm{GL}_{n}(q)$ over $\mathbb{F}_{\ell}$ parametrized by the partition
$\lambda$. Then

$\displaystyle\mathsf{d}_{G}(S(\lambda))=S(\mathsf{M}_{e,\ell}(\lambda)).$

The involution $\mathsf{M}_{e,\ell}$ is a generalization of the original
Mullineux involution since it is defined on the set of all partitions, and
coincides with $\mathsf{M}_{e}$ on the set of $e$-regular partitions. Such a
generalization already appeared in a work of Bezrukavnikov [2] and Losev [23]
on wall-crossing functors for representations of rational Cherednik algebras,
but in the case where $\ell\gg 0$.

Kleshchev showed in [20] that the Mullineux involution $\mathsf{M}_{p}$ can
also be interpreted in the language of crystals for Fock spaces in affine type
$A$, which are certain colored oriented graphs whose vertices are labeled by
partitions. More precisely, the image by $\mathsf{M}_{p}$ of a $p$-regular
partition $\lambda$ is obtained by changing the sign of each arrow in a path
from the empty partition to $\lambda$ in the graph. We propose a definition of
several higher level crystal operators on the Fock space which give a similar
description for our generalized involution $\mathsf{M}_{e,\ell}$, see
Proposition 4.8.

The paper is organized as follows. In Section 2 we introduce the Alvis–Curtis
duality for finite reductive groups and show how to compute it within a given
Harish-Chandra series using a similar duality for the corresponding Hecke
algebra. Section 3 illustrates our method in the case of finite general linear
groups. We give in Section 3.3 the definition of a generalized version of the
Mullineux involution and show in Theorem 3.7 that it is the shadow of the
Alvis–Curtis duality for $\mathrm{GL}_{n}(q)$. The final section is devoted to
an interpretation of our result in the context of the theory of crystal
graphs.

##  2 Alvis–Curtis duality

In this section we investigate the relation between the Alvis–Curtis duality
for a finite reductive group within a Harish-Chandra series and a similar
duality in the Hecke algebra associated to the series.

###  2.1 Notation

Let $\mathbf{G}$ be a connected reductive algebraic group defined over an
algebraic closure of a finite field of characteristic $p$, together with an
endomorphism $F$, a power of which is a Frobenius endomorphism. Given an
$F$-stable closed subgroup $\mathbf{H}$ of $\mathbf{G}$, we will denote by $H$
the finite group of fixed points $\mathbf{H}^{F}$. The group $G$ is a _finite
reductive group_.

We will be interested in the modular representations of $G$ in non-defining
characteristic. We fix a prime number $\ell$ different from $p$ and an
$\ell$-modular system $(K,\mathcal{O},k)$ which is assumed to be large enough
for $G$, so that the algebras $KG$ and $kG$ split. Throughout this section
$\Lambda$ will denote any ring among $K$, $\mathcal{O}$ and $k$.

Given a finite-dimensional $\Lambda$-algebra $A$, we denote by
$A\textsf{-}\mathsf{mod}$ (resp. $\mathsf{mod}\text{-}A$) the category of
finite-dimensional left (resp. right) $A$-modules. The corresponding bounded
derived category will be denoted by $D^{b}(A\textsf{-}\mathsf{mod})$ (resp.
$D^{b}(\mathsf{mod}\text{-}A)$) or simply $D^{b}(A)$ when there is no risk of
confusion. We will identify the Grothendieck group of the abelian category
$A\textsf{-}\mathsf{mod}$ with the Grothendieck group of the triangulated
category $D^{b}(A)$. It will be denoted by $K_{0}(A)$. We will write $[M]$ for
the class of an $A$-module $M$ in $K_{0}(A)$.

###  2.2 Harish-Chandra induction and restriction

Given an $F$-stable parabolic subgroup $\mathbf{P}$ of $\mathbf{G}$ with Levi
decomposition $\mathbf{P}=\mathbf{L}\mathbf{U}$, where $\mathbf{L}$ is
$F$-stable, we denote by

$\textstyle{\Lambda
L\textsf{-}\mathsf{mod}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{R}_{L}^{G}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\Lambda
G\textsf{-}\mathsf{mod}}$$\scriptstyle{{}^{*}\mathsf{R}_{L}^{G}}$

the Harish-Chandra induction and restriction functors. Under the assumption on
$\ell$, they form a biajdoint pair of exact functors. The natural
transformations given by the adjunction (unit and counit) are denoted as
follows

$\displaystyle\operatorname{Id}\xrightarrow{\overline{\eta}^{L\subset
G}}\mathsf{R}_{L}^{G}{}^{*}\mathsf{R}_{L}^{G}\xrightarrow{\underline{\varepsilon}^{L\subset
G}}\operatorname{Id}\qquad\text{and}\qquad\operatorname{Id}\xrightarrow{\underline{\eta}^{L\subset
G}}{}^{*}\mathsf{R}_{L}^{G}\mathsf{R}_{L}^{G}\xrightarrow{\overline{\varepsilon}^{L\subset
G}}\operatorname{Id},$

where $\operatorname{Id}$ denotes here the identity functor. When it is clear
from the context, we will usually drop the superscript $L\subset G$.

Let $\mathbf{Q}$ be another $F$-stable parabolic subgroup with $F$-stable Levi
decomposition $\mathbf{Q}=\mathbf{M}\mathbf{V}$. If
$\mathbf{P}\subset\mathbf{Q}$ and $\mathbf{L}\subset\mathbf{M}$ then the
Harish-Chandra induction and restriction functors satisfy
$\mathsf{R}_{M}^{G}\circ\mathsf{R}_{L}^{M}\simeq\mathsf{R}_{L}^{G}$ and
${}^{*}\mathsf{R}_{L}^{M}\circ{}^{*}\mathsf{R}_{M}^{G}\simeq{}^{*}\mathsf{R}_{L}^{G}$.
Together with the counit $\underline{\varepsilon}^{L\subset
M}\colon\mathsf{R}_{L}^{M}{}^{*}\mathsf{R}_{L}^{M}\longrightarrow 1$, this
gives a natural transformation

$\displaystyle\varphi^{L\subset M\subset G}\colon\
\mathsf{R}_{L}^{G}{}^{*}\mathsf{R}_{L}^{G}\longrightarrow\mathsf{R}_{M}^{G}{}^{*}\mathsf{R}_{M}^{G}.$

If $\mathbf{R}$ is any other $F$-stable parabolic subgroup containing
$\mathbf{Q}$ with an $F$-stable Levi complement $\mathbf{N}$ containing
$\mathbf{M}$ then the natural isomorphism giving the transitivity of Harish-
Chandra induction and restriction can be chosen so that $\varphi^{M\subset
N\subset G}\circ\varphi^{L\subset M\subset G}=\varphi^{L\subset N\subset G}$
(see for example [4, Section 4] or [12, Section III.7.2]).

###  2.3 Alvis–Curtis duality functor

We now fix a Borel subgroup $\mathbf{B}$ of $\mathbf{G}$ containing a maximal
torus $\mathbf{T}$, both of which are assumed to be $F$-stable. Let $\Delta$
be the set of simple roots defined by $\mathbf{B}$. The $F$-stable parabolic
subgroups containing $\mathbf{B}$ are parametrized by $F$-stable subsets of
$\Delta$. They have a unique Levi complement containing $\mathbf{T}$. Such
Levi subgroups and parabolic subgroups are called _standard_. Given $r\geq 0$,
we denote by $\mathcal{L}_{r}$ the (finite) set of $F$-stable standard Levi
subgroups corresponding to a subset $I\subset\Delta$ satisfying
$|I/F|=|\Delta/F|-r$. In particular $\mathcal{L}_{0}=\\{\mathbf{G}\\}$ and
$\mathcal{L}_{|\Delta/F|}=\\{\mathbf{T}\\}$. Following [5, 10, 12] we can form
the complex of exact functors

$\displaystyle
0\longrightarrow\mathsf{R}_{T}^{G}{}^{*}\mathsf{R}_{T}^{G}\longrightarrow\cdots\longrightarrow\bigoplus_{\mathbf{L}\in\mathcal{L}_{2}}\mathsf{R}_{L}^{G}{}^{*}\mathsf{R}_{L}^{G}\longrightarrow\bigoplus_{\mathbf{L}\in\mathcal{L}_{1}}\mathsf{R}_{L}^{G}{}^{*}\mathsf{R}_{L}^{G}\longrightarrow\operatorname{Id}\longrightarrow
0,$

where $\operatorname{Id}$ is in degree $0$. It yields a functor
$\mathsf{D}_{G}$ on the bounded derived category $D^{b}(\Lambda G)$ of
finitely generated $\Lambda G$-modules, called the _Alvis–Curtis duality
functor_. Note that the original definition of the duality by Alvis and Curtis
[1, 8] refers to the linear endomorphism on $K_{0}(KG)$ induced by this
functor. The complex above was introduced by Deligne–Lusztig in [10].

######  Theorem 2.1 (Cabanes–Rickard [5]).

The functor $\mathsf{D}_{G}$ is a self-equivalence of $D^{b}(\Lambda G)$
satisfying

$\displaystyle\mathsf{D}_{G}\circ\mathsf{R}_{L}^{G}\simeq\mathsf{R}_{L}^{G}\circ\mathsf{D}_{L}[r]$
(2.1)

for every $\mathbf{L}\in\mathcal{L}_{r}$.

Note that any quasi-inverse of $\mathsf{D}_{G}$ will also satisfy the relation
(2.1), up to replacing $r$ by $-r$.

Chuang–Rouquier deduced in [7] from (2.1) that the equivalence induced by
$\mathsf{D}_{G}$ is _perverse_ with respect to the cuspidal depth (see below
for the definition). For the reader’s convenience we recall here their
argument.

Assume that $\Lambda$ is a field. The simple $\Lambda G$-modules are
partitioned into Harish-Chandra series, see [19]. Given a simple $\Lambda
G$-module $S$, there exists $\mathbf{L}$ in $\mathcal{L}_{s}$ and a simple
cuspidal $\Lambda L$-module $X$ such that $S$ appears in the head (or
equivalently in the socle) of $\mathsf{R}_{L}^{G}(X)$. The pair
$(\mathbf{L},X)$ is unique up to $G$-conjugation. We say that $S$ lies in the
Harish-Chandra series of the cuspidal pair $(\mathbf{L},X)$ and we call $s$
the _cuspidal depth_ of $S$. In particular, the cuspidal modules are the
modules with cuspidal depth zero.

######  Proposition 2.2 (Chuang–Rouquier [7]).

Assume that $\Lambda$ is a field. Let $S$ be a simple $\Lambda G$-module of
cuspidal depth $s$. Then

  * $(i)$

$H^{i}(\mathsf{D}_{G}(S))=0$ for $i>0$ and $i<-s$.

  * $(ii)$

The composition factors of $H^{i}(\mathsf{D}_{G}(S))$ have depth at most $s$.

  * $(iii)$

Among all the composition factors of $\bigoplus_{i}H^{i}(\mathsf{D}_{G}(S))$,
there is a unique composition factor of depth $s$. It is a submodule of
$H^{-s}(\mathsf{D}_{G}(S))$, and it lies in the same Harish-Chandra series as
$S$.

###### Proof.

We denote by $(L,X)$ a cuspidal pair associated with $S$. Given any other
cuspidal pair $(M,Y)$ with $\mathbf{M}\in\mathcal{L}_{r}$ and $n\in\mathbb{Z}$
we have

$\displaystyle\operatorname{Hom}_{D^{b}(\Lambda
G)}\big{(}\mathsf{R}_{M}^{G}(Y),\mathsf{D}_{G}(S)[n]\big{)}\simeq\operatorname{Hom}_{D^{b}(\Lambda
G)}\big{(}\mathsf{D}_{G}^{-1}(\mathsf{R}_{M}^{G}(Y)),S[n]\big{)}$
$\displaystyle\hphantom{\operatorname{Hom}_{D^{b}(\Lambda
G)}\big{(}\mathsf{R}_{M}^{G}(Y),\mathsf{D}_{G}(S)[n]\big{)}}{}\simeq\operatorname{Hom}_{D^{b}(\Lambda
G)}\big{(}\mathsf{R}_{M}^{G}(\mathsf{D}_{M}^{-1}(Y))[-r],S[n]\big{)}$
$\displaystyle\hphantom{\operatorname{Hom}_{D^{b}(\Lambda
G)}\big{(}\mathsf{R}_{M}^{G}(Y),\mathsf{D}_{G}(S)[n]\big{)}}{}\simeq\operatorname{Hom}_{D^{b}(\Lambda
M)}\big{(}Y,{}^{*}\mathsf{R}_{M}^{G}(S)[r+n]\big{)},$ (2.2)

where in the last equality we used that $\mathsf{D}_{M}^{-1}(Y)\simeq Y$ since
$Y$ is cuspidal. In particular, it is zero when $r<-n$ or when $M$ does not
contain a $G$-conjugate of $L$, so in particular when $r>s$.

Take $n$ to be the smallest integer such that $H^{n}(\mathsf{D}_{G}(S))\neq 0$
and consider a simple $\Lambda G$-module $T$ in the socle of
$H^{n}(\mathsf{D}_{G}(S))$. Let $(M,Y)$ be the cuspidal pair above which $T$
lies. Then the composition $\mathsf{R}_{M}^{G}(Y)\twoheadrightarrow
T\hookrightarrow H^{n}(\mathsf{D}_{G}(S))$ yields a non-zero element in

$\displaystyle\operatorname{Hom}_{D^{b}(\Lambda
G)}\big{(}\mathsf{R}_{M}^{G}(Y),\mathsf{D}_{G}(S)[n]\big{)}.$

From (2.2) we must have $-n\leq r\leq s$ which proves (i). Furthermore, if
$n=-s$ then $r$ and $s$ are equal, and in that case

$\displaystyle\operatorname{Hom}_{D^{b}(\Lambda
G)}\big{(}\mathsf{R}_{M}^{G}(Y),\mathsf{D}_{G}(S)[-s]\big{)}\simeq\operatorname{Hom}_{D^{b}(\Lambda
M)}\big{(}Y,{}^{*}\mathsf{R}_{M}^{G}(S)\big{)}\simeq\operatorname{Hom}_{\Lambda
M}\big{(}Y,{}^{*}\mathsf{R}_{M}^{G}(S)\big{)}.$

By the Mackey formula, ${}^{*}\mathsf{R}_{M}^{G}(S)$ is isomorphic to a direct
sum of $G$-conjugates of $X$. Therefore if $T$ lies in the socle of
$H^{-s}(\mathsf{D}_{G}(S))$ then $Y$ is $G$-conjugate to $X$ which means that
$T$ and $S$ lie in the same Harish-Chandra series.

Now if we replace $Y$ by its projective cover $P_{Y}$ in (2.2) we get

$\displaystyle\operatorname{Hom}_{D^{b}(\Lambda
G)}\big{(}\mathsf{R}_{M}^{G}(P_{Y}),\mathsf{D}_{G}(S)[n]\big{)}\simeq\operatorname{Hom}_{D^{b}(\Lambda
M)}\big{(}P_{Y},D_{M}\big{(}{}^{*}\mathsf{R}_{M}^{G}(S)\big{)}[r+n]\big{)},$

which again is zero unless $r\leq s$ or unless $r=s$ and $(M,Y)$ is conjugate
to $(L,X)$. In that latter case we have

$\displaystyle\operatorname{Hom}_{D^{b}(\Lambda
G)}\big{(}\mathsf{R}_{L}^{G}(P_{X}),\mathsf{D}_{G}(S)[n]\big{)}\simeq\operatorname{Hom}_{D^{b}(\Lambda
L)}\big{(}P_{X},\mathsf{D}_{L}\big{(}{}^{*}\mathsf{R}_{L}^{G}(S)\big{)}[n+s]\big{)}$
$\displaystyle\hphantom{\operatorname{Hom}_{D^{b}(\Lambda
G)}\big{(}\mathsf{R}_{L}^{G}(P_{X}),\mathsf{D}_{G}(S)[n]\big{)}}{}\simeq\operatorname{Hom}_{D^{b}(\Lambda
L)}\big{(}P_{X},{}^{*}\mathsf{R}_{L}^{G}(S)[n+s]\big{)},$

since ${}^{*}\mathsf{R}_{L}^{G}(S)$ is a sum of conjugates of $X$. Therefore
the composition factors of $D_{G}(S)$ lying in the Harish-Chandra series of
$(L,X)$ can only appear in degree $-s$. They appear with multiplicity one
since $\mathsf{D}_{G}$ is a self-equivalence, which proves (iii). ∎

Using the property that $\mathsf{D}_{G}$ is a perverse equivalence (given in
Proposition 2.2) we can define a bijection on the set of simple $\Lambda
G$-modules as follows: given a simple $\Lambda G$-module $S$ with cuspidal
depth $s$ we define $\mathsf{d}_{G}(S)$ to be the unique simple $\Lambda
G$-module with depth $s$ which occurs as a composition factor in the
cohomology of $\mathsf{D}_{G}(S)$. Note that $\mathsf{d}_{G}(S)$ and $S$ lie
in the same Harish-Chandra series.

###  2.4 Compatibility with Hecke algebras

Given a cuspidal pair $(L,X)$ of $G$, we can form the endomorphism algebra

$\displaystyle\mathcal{H}_{G}(L,X)=\operatorname{End}_{G}\big{(}\mathsf{R}_{L}^{G}(X)\big{)}.$

By [17, Theorem 2.4], the isomorphism classes of simple quotients of
$\mathsf{R}_{L}^{G}(X)$ (the Harish-Chandra series of $(L,X)$) are
parametrized by the simple representations of $\mathcal{H}_{G}(L,X)$. The
structure of this algebra was studied for example in [17, Section 3]; it is in
general very close to be a Iwahori–Hecke algebra of a Coxeter group.

Let $\mathbf{M}$ be an $F$-stable standard Levi subgroup of $\mathbf{G}$
containing $\mathbf{L}$. Since $\mathsf{R}_{M}^{G}$ is fully-faithful,
$\mathcal{H}_{M}(L,X)=\operatorname{End}_{G}(\mathsf{R}_{L}^{M}(X))$ embedds
naturally as a subalgebra of $\mathcal{H}_{G}(L,X)$. To this embedding one can
associate the induction and restriction functors

$\textstyle{\mathsf{mod}\textsf{-}\mathcal{H}_{M}(L,X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Ind}_{\mathcal{H}_{M}}^{\mathcal{H}_{G}}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{mod}\textsf{-}\mathcal{H}_{G}(L,X)}$$\scriptstyle{\operatorname{Res}_{\mathcal{H}_{M}}^{\mathcal{H}_{G}}}$

between the categories of right modules. The purpose of this section is to
compare the Alvis–Curtis duality functor $\mathsf{D}_{G}$ for the group with a
similar functor $\mathsf{D}_{\mathcal{H}}$ of the Hecke algebra. From now on
we shall fix the cuspidal pair $(L,X)$, and we will denote simply
$\mathcal{H}_{G}$ and $\mathcal{H}_{M}$ the endomorphism algebras of
$\mathsf{R}_{L}^{G}(X)$ and $\mathsf{R}_{L}^{M}(X)$ respectively.

Let $Y$ be a (non-necessarily cuspidal) $\Lambda M$-module. We consider the
natural transformation $\Theta_{M,Y}$ defined so that the following diagram
commutes:

$\textstyle{\operatorname{Hom}_{M}(Y,{}^{*}\mathsf{R}_{M}^{G}(-))\otimes_{\operatorname{End}_{M}(Y)}\operatorname{Hom}_{M}({}^{*}\mathsf{R}_{M}^{G}\mathsf{R}_{M}^{G}(Y),Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
mult$\scriptstyle{\sim}$$\textstyle{\operatorname{Hom}_{M}({}^{*}\mathsf{R}_{M}^{G}\mathsf{R}_{M}^{G}(Y),{}^{*}\mathsf{R}_{M}^{G}(-))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\operatorname{Hom}_{G}(\mathsf{R}_{M}^{G}(Y),-)\otimes_{\operatorname{End}_{M}(Y)}\operatorname{End}_{G}(\mathsf{R}_{M}^{G}(Y))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\qquad\Theta_{M,Y}}$$\textstyle{\operatorname{Hom}_{G}(\mathsf{R}_{M}^{G}(Y),\mathsf{R}_{M}^{G}{}^{*}\mathsf{R}_{M}^{G}(-)).}$

In other words, $\Theta_{M,Y}$ is defined on the objects by

$\displaystyle\Theta_{M,Y}(f\otimes
h)=\mathsf{R}_{M}^{G}{}^{*}\mathsf{R}_{M}^{G}(f)\circ\mathsf{R}_{M}^{G}(\underline{\eta}_{Y})\circ
h.$

Using this description one can check that $\Theta_{M,Y}$ is compatible with
the right action of $\operatorname{End}_{G}(\mathsf{R}_{M}^{G}(Y))$. Therefore
it is a well-defined natural transformation between functors from $\Lambda
G\textsf{-}\mathsf{mod}$ to
$\mathsf{mod}\textsf{-}\operatorname{End}_{G}(\mathsf{R}_{M}^{G}(Y))$.

Let $\mathbf{N}$ be another standard $F$-stable Levi subgroup of $\mathbf{G}$
with $\mathbf{M}\subset\mathbf{N}$. Recall from Section 2.2 the natural
transformation $\varphi^{M\subset N\subset
G}\colon\mathsf{R}_{M}^{G}{}^{*}\mathsf{R}_{M}^{G}\longrightarrow\mathsf{R}_{N}^{G}{}^{*}\mathsf{R}_{N}^{G}$
which was needed for the construction of $\mathsf{D}_{G}$. By composition it
induces a natural transformation

$\displaystyle\operatorname{Hom}_{G}\big{(}\mathsf{R}_{M}^{G}(Y),\mathsf{R}_{M}^{G}{}^{*}\mathsf{R}_{M}^{G}(-)\big{)}\longrightarrow\operatorname{Hom}_{G}\big{(}\mathsf{R}_{M}^{G}(Y),\mathsf{R}_{N}^{G}{}^{*}\mathsf{R}_{N}^{G}(-)\big{)}.$

######  Proposition 2.3.

The following diagram is commutative

$\textstyle{\operatorname{Hom}_{G}\big{(}\mathsf{R}_{M}^{G}(Y),-\big{)}\otimes_{\operatorname{End}_{M}(Y)}\operatorname{End}_{G}\big{(}\mathsf{R}_{M}^{G}(Y)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\qquad\quad\Theta_{M,Y}}$$\textstyle{\operatorname{Hom}_{G}\big{(}\mathsf{R}_{M}^{G}(Y),\mathsf{R}_{M}^{G}{}^{*}\mathsf{R}_{M}^{G}(-)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{G}\big{(}\mathsf{R}_{M}^{G}(Y),-\big{)}\otimes_{\operatorname{End}_{N}\big{(}\mathsf{R}_{M}^{N}(Y)\big{)}}\operatorname{End}_{G}\big{(}\mathsf{R}_{M}^{G}(Y)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\operatorname{Hom}_{G}\big{(}\mathsf{R}_{M}^{G}(Y),\mathsf{R}_{N}^{G}{}^{*}\mathsf{R}_{N}^{G}(-)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\operatorname{Hom}_{G}\big{(}\mathsf{R}_{N}^{G}(\mathsf{R}_{M}^{N}(Y)),-\big{)}\otimes_{\operatorname{End}_{N}\big{(}\mathsf{R}_{M}^{N}(Y)\big{)}}\operatorname{End}_{G}\big{(}\mathsf{R}_{N}^{G}\big{(}\mathsf{R}_{M}^{N}(Y)\big{)}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\qquad\quad\qquad\Theta_{N,\mathsf{R}_{M}^{N}(Y)}}$$\textstyle{\operatorname{Hom}_{G}\big{(}\mathsf{R}_{N}^{G}\big{(}\mathsf{R}_{M}^{N}(Y)\big{)},\mathsf{R}_{N}^{G}{}^{*}\mathsf{R}_{N}^{G}(-)\big{)}.}$

###### Proof.

Since $\mathsf{R}_{M}^{N}$ is faithful one can see $\operatorname{End}_{M}(Y)$
as a subalgebra of $\operatorname{End}_{N}(\mathsf{R}_{M}^{N}(Y))$. Then the
first vertical map on the left-hand side of the diagram is the canonical
projection. We will write
$t\colon\mathsf{R}_{M}^{G}\mathop{\longrightarrow}\limits^{\sim}\mathsf{R}_{N}^{G}\mathsf{R}_{M}^{N}$
(resp.
$t^{*}\colon{}^{*}\mathsf{R}_{M}^{G}\mathop{\longrightarrow}\limits^{\sim}{}^{*}\mathsf{R}_{M}^{N}{}^{*}\mathsf{R}_{N}^{G}$)
for the isomorphism of functors coming from the transitivity of Harish-Chandra
induction (resp. restriction).

Let $Z$ be a $\Lambda G$-module,
$h\in\operatorname{End}(\mathsf{R}_{M}^{G}(Y))$ and
$f\in\operatorname{Hom}_{G}(\mathsf{R}_{M}^{G}(Y),Z)$. The commutativity of
the diagram is equivalent to the relation

$\displaystyle\mathsf{R}_{N}^{G}{}^{*}\mathsf{R}_{N}^{G}\big{(}f\circ
t_{Y}^{-1}\big{)}\circ\mathsf{R}_{N}^{G}\big{(}\big{(}\underline{\eta}^{N\subset
G}\big{)}_{\mathsf{R}_{M}^{N}(Y)}\big{)}\circ\big{(}t_{Y}\circ h\circ
t_{Y}^{-1}\big{)}$ $\displaystyle\qquad{}=\big{(}\varphi^{M\subset N\subset
G}\big{)}_{Z}\circ\mathsf{R}_{M}^{G}{}^{*}\mathsf{R}_{M}^{G}(f)\circ\mathsf{R}_{M}^{G}\big{(}\big{(}\underline{\eta}^{M\subset
G}\big{)}_{Y}\big{)}\circ h\circ t_{Y}^{-1}.$

Since $\big{(}\varphi^{M\subset N\subset
G}\big{)}_{Z}\circ\mathsf{R}_{M}^{G}{}^{*}\mathsf{R}_{M}^{G}(f)=\mathsf{R}_{N}^{G}{}^{*}\mathsf{R}_{N}^{G}(f)\circ\big{(}\varphi^{M\subset
N\subset G}\big{)}_{\mathsf{R}_{M}^{G}(Y)}$ it is enough to show that

$\displaystyle\big{(}\varphi^{M\subset N\subset
G}\big{)}_{\mathsf{R}_{M}^{G}(Y)}\circ\mathsf{R}_{M}^{G}\big{(}\big{(}\underline{\eta}^{M\subset
G}\big{)}_{Y}\big{)}=\mathsf{R}_{N}^{G}{}^{*}\mathsf{R}_{N}^{G}\big{(}t_{Y}^{-1}\big{)}\circ\mathsf{R}_{N}^{G}\big{(}\big{(}\underline{\eta}^{N\subset
G}\big{)}_{\mathsf{R}_{M}^{N}(Y)}\big{)}\circ t_{Y}.$ (2.3)

This comes from the following commutative diagram

$\textstyle{\mathsf{R}_{M}^{G}{}^{*}\mathsf{R}_{M}^{G}\mathsf{R}_{M}^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t}$$\scriptstyle{\varphi^{M\subset
N\subset
G}}$$\textstyle{\mathsf{R}_{N}^{G}\mathsf{R}_{M}^{N}{}^{*}\mathsf{R}_{M}^{G}\mathsf{R}_{M}^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{*}}$$\textstyle{\mathsf{R}_{N}^{G}\mathsf{R}_{M}^{N}{}^{*}\mathsf{R}_{M}^{N}{}^{*}\mathsf{R}_{N}^{G}\mathsf{R}_{M}^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
14.22636pt\underline{\varepsilon}^{M\subset
N}}$$\scriptstyle{t}$$\textstyle{\framebox{$\mathsf{R}_{N}^{G}{}^{*}\mathsf{R}_{N}^{G}\mathsf{R}_{M}^{G}$}}$$\textstyle{\framebox{$\mathsf{R}_{M}^{G}$}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{\eta}^{M\subset
G}}$$\scriptstyle{t}$$\textstyle{\mathsf{R}_{N}^{G}\mathsf{R}_{M}^{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{\eta}^{M\subset
G}}$$\scriptstyle{\underline{\eta}^{M\subset
N}}$$\textstyle{\mathsf{R}_{N}^{G}\mathsf{R}_{M}^{N}{}^{*}\mathsf{R}_{M}^{N}{}^{*}\mathsf{R}_{N}^{G}\mathsf{R}_{N}^{G}\mathsf{R}_{M}^{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
22.76219pt\underline{\varepsilon}^{M\subset
N}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{R}_{N}^{G}{}^{*}\mathsf{R}_{N}^{G}\mathsf{R}_{N}^{G}\mathsf{R}_{M}^{N}}$$\scriptstyle{t^{-1}}$$\textstyle{\mathsf{R}_{N}^{G}\mathsf{R}_{M}^{N}{}^{*}\mathsf{R}_{M}^{N}\mathsf{R}_{M}^{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{\varepsilon}^{M\subset
N}}$$\scriptstyle{\underline{\eta}^{N\subset
G}}$$\textstyle{\mathsf{R}_{N}^{G}\mathsf{R}_{M}^{N}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{\eta}^{N\subset
G}}$

where for simplicity we did not write the identity natural transformations.
The only non-trivial commutative subdiagram is the central one, which comes
from the relation

$\displaystyle\big{(}{}^{*}\mathsf{R}_{M}^{N}\cdot\underline{\eta}^{N\subset
G}\cdot\mathsf{R}_{M}^{N}\big{)}\circ\underline{\eta}^{M\subset N}=(t^{*}\cdot
t)\circ\underline{\eta}^{N\subset G},$

which we assume to hold by our choice of $t$ and $t^{*}$. For more on the
compatibility of the unit, counit, $t$ and $t^{*}$ see [12, Part 3]. Now the
relation (2.3) comes from the equality between two natural transformations
between the functors $\mathsf{R}_{M}^{G}$ and
$\mathsf{R}_{N}^{G}{}^{*}\mathsf{R}_{N}^{G}\mathsf{R}_{M}^{G}$, given by the
top and the bottom arrows respectively. ∎

In the particular case where $Y=\mathsf{R}_{L}^{G}(X)$ with $(L,X)$ being a
cuspidal pair, the natural transformation $\Theta_{M,\mathsf{R}_{L}^{M}(X)}$
becomes

$\displaystyle\operatorname{Ind}_{\mathcal{H}_{M}}^{\mathcal{H}_{G}}\operatorname{Res}_{\mathcal{H}_{M}}^{\mathcal{H}_{G}}\big{(}\operatorname{Hom}_{G}(\mathsf{R}_{L}^{G}(X),-)\big{)}\xrightarrow{\Theta_{M,\mathsf{R}_{L}^{M}(X)}}\operatorname{Hom}_{G}(\mathsf{R}_{L}^{G}(X),\mathsf{R}_{M}^{G}{}^{*}\mathsf{R}_{M}^{G}(-)).$

It can be seen as a way to intertwine the endofunctor
$\operatorname{Ind}_{\mathcal{H}_{M}}^{\mathcal{H}_{G}}\operatorname{Res}_{\mathcal{H}_{M}}^{\mathcal{H}_{G}}$
of $\mathsf{mod}\textsf{-}\mathcal{H}_{G}$ and the endofunctor
$\mathsf{R}_{M}^{G}{}^{*}\mathsf{R}_{M}^{G}$ of $\Lambda
G\textsf{-}\mathsf{mod}$ via the functor
$\operatorname{Hom}_{G}(\mathsf{R}_{L}^{G}(X),-)$, as shown in the following
diagram

---
$\textstyle{kG\textsf{-}\mathsf{mod}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Hom}_{G}(\mathsf{R}_{L}^{G}(X),-)}$$\scriptstyle{\mathsf{R}_{M}^{G}{}^{*}\mathsf{R}_{M}^{G}}$$\textstyle{\mathsf{mod}\textsf{-}\mathcal{H}_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Ind}_{\mathcal{H}_{M}}^{\mathcal{H}_{G}}\operatorname{Res}_{\mathcal{H}_{M}}^{\mathcal{H}_{G}}}$

$\Longleftarrow$

$\textstyle{kG\textsf{-}\mathsf{mod}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Hom}_{G}(\mathsf{R}_{L}^{G}(X),-)}$$\textstyle{\mathsf{mod}\textsf{-}\mathcal{H}_{G}.}$

In this diagram the central double arrow represents the natural transformation
$\Theta_{M,\mathsf{R}_{L}^{M}(X)}$. We give a condition for this
transformation to be an isomorphism. Note that similar results were obtained
by Dipper–Du in [11, Theorem 1.3.2] in the case of general linear groups and
by Seeber [27] with coinduction instead of induction.

######  Proposition 2.4.

Assume that $\Lambda$ is a field. Let $(L,X)$ be a cuspidal pair of $G$, and
$\mathbf{M}$ be a standard Levi of $\mathbf{G}$ containing $\mathbf{L}$.
Assume that any cuspidal pair of $M$ which is $G$-conjugate to $(L,X)$ is
actually $M$-conjugate to $(L,X)$. Then $\Theta_{M,\mathsf{R}_{L}^{M}(X)}$ is
an isomorphism.

###### Proof.

By definition of $\Theta_{M,\mathsf{R}_{L}^{M}(X)}$ it is enough to show that
the multiplication map

$\textstyle{\operatorname{Hom}_{M}\big{(}\mathsf{R}_{L}^{M}(X),{}^{*}\mathsf{R}_{M}^{G}(-)\big{)}\otimes_{\operatorname{End}_{M}\big{(}\mathsf{R}_{L}^{M}(X)\big{)}}\operatorname{Hom}_{M}\big{(}{}^{*}\mathsf{R}_{M}^{G}\mathsf{R}_{L}^{G}(X),\mathsf{R}_{L}^{M}(X)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{M}\big{(}{}^{*}\mathsf{R}_{M}^{G}\mathsf{R}_{L}^{G}(X),{}^{*}\mathsf{R}_{M}^{G}(-)\big{)}}$

is an isomorphism of $k$-vector spaces. Now by the Mackey formula [4,
Proposition 1.5] we have

$\displaystyle{}^{*}\mathsf{R}_{M}^{G}\mathsf{R}_{L}^{G}(X)\simeq\bigoplus_{\begin{subarray}{c}x\in
Q\backslash G/P\\\
{}^{x}\mathbf{L}\subset\mathbf{M}\end{subarray}}\mathsf{R}_{{}^{x}L}^{M}(X^{x}).$

Under the assumption on $(L,X)$, any cuspidal pair $({}^{x}L,X^{x})$ with
${}^{x}L\subset M$ is conjugate to $(L,X)$ under $M$. In particular
$\mathsf{R}_{{}^{x}L}^{M}(X^{x})\simeq\mathsf{R}_{L}^{M}(X)$ and we deduce
that each of the composition maps

$\textstyle{\operatorname{Hom}_{M}\big{(}\mathsf{R}_{L}^{M}(X),{}^{*}\mathsf{R}_{M}^{G}(-)\big{)}\otimes_{\operatorname{End}_{M}\big{(}\mathsf{R}_{L}^{M}(X)\big{)}}\operatorname{Hom}_{M}\big{(}\mathsf{R}_{{}^{x}L}^{M}(X^{x}),\mathsf{R}_{L}^{M}(X)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{M}\big{(}\mathsf{R}_{{}^{x}L}^{M}(X^{x}),{}^{*}\mathsf{R}_{M}^{G}(-)\big{)}}$

is an isomorphism. ∎

As in the case of the finite group $G$ we can form a complex of functors
coming from induction and restriction in $\mathcal{H}_{G}$. Given $r\geq 0$,
we denote by $\mathcal{L}_{r}(\mathbf{L})$ the subset of $\mathcal{L}_{r}$ of
standard Levi subgroups containing $\mathbf{L}$. The complex of functors

$\displaystyle
0\longrightarrow\operatorname{Ind}_{\mathcal{H}_{L}}^{\mathcal{H}_{G}}\operatorname{Res}_{\mathcal{H}_{L}}^{\mathcal{H}_{G}}\longrightarrow\cdots\longrightarrow\bigoplus_{\mathbf{M}\in\mathcal{L}_{1}(\mathbf{L})}\operatorname{Ind}_{\mathcal{H}_{M}}^{\mathcal{H}_{G}}\operatorname{Res}_{\mathcal{H}_{M}}^{\mathcal{H}_{G}}\longrightarrow\operatorname{Id}\longrightarrow
0,$

where $\operatorname{Id}$ is in degree $0$, induces a triangulated functor
$\mathsf{D}_{\mathcal{H}_{G}}$ in
$D^{b}(\mathsf{mod}\textsf{-}\mathcal{H}_{G})$ whenever each term of the
complex is exact. For that property to hold we need to assume that
$\mathcal{H}_{G}$ is flat over each subalgebra of the form $\mathcal{H}_{M}$.
Combining Propositions 2.3 and 2.4 we get

######  Theorem 2.5.

Assume that $\Lambda$ is a field. Let $\mathbf{L}$ be an $F$-stable standard
Levi subgroup of $\mathbf{G}$ and $X$ be a cuspidal $\Lambda L$-module. Let
$\mathcal{H}_{G}=\operatorname{End}_{G}(\mathsf{R}_{L}^{G}(X))$. Assume that
for every $F$-stable standard Levi subgroup $\mathbf{M}$ of $\mathbf{G}$
containing $\mathbf{L}$ we have:

  * $(i)$

$\mathcal{H}_{G}$ is flat over
$\mathcal{H}_{M}=\operatorname{End}_{M}\big{(}\mathsf{R}_{L}^{M}(X)\big{)}$.

  * $(ii)$

Every cuspidal pair of $M$ which is $G$-conjugate to $(L,X)$ is actually
$M$-conjugate to $(L,X)$.

Then there is a natural isomorphism of endofunctors of
$D^{b}(\mathsf{mod}\textsf{-}\mathcal{H}_{G})$

$\displaystyle\mathsf{D}_{\mathcal{H}_{G}}\big{(}\mathrm{RHom}_{G}\big{(}\mathsf{R}_{L}^{G}(X),-\big{)}\big{)}\overset{\sim}{\to}\mathrm{RHom}_{G}\big{(}\mathsf{R}_{L}^{G}(X),\mathsf{D}_{G}(-)\big{)}.$

##  3 Unipotent representations of $\boldsymbol{\mathrm{GL}_{n}(q)}$

In this section we show how to use Theorem 2.5 to compute
$d_{\mathrm{GL}_{n}(q)}(S)$ for every unipotent simple
$k\mathrm{GL}_{n}(q)$-module $S$. This will involve an involution on the set
of partitions of $n$ generalizing the Mullineux involution [26]. So here
$\mathbf{G}=\mathrm{GL}_{n}(\overline{\mathbb{F}}_{p})$ will be the general
linear group over an algebraic closure of $\mathbb{F}_{p}$, and
$F\colon\mathbf{G}\longrightarrow\mathbf{G}$ the standard Frobenius
endomorphism, raising the entries of a matrix to the $q$-th power.

###  3.1 Partitions

A _partition_ $\lambda$ of $n\in\mathbb{N}$ is a non-increasing sequence
$(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{r})$ of positive integers
which add up to $n$. By convention, $\varnothing$ is the unique partition of
$0$ and is called the empty partition. The set of partitions of $n$ will be
denoted by $\mathcal{P}(n)$, and the set of all partitions by
$\mathcal{P}:=\sqcup_{n\in\mathbb{N}}\mathcal{P}(n)$. We shall also use the
notation $\Lambda=(1^{r_{1}},2^{r2},\ldots,n^{r_{n}})$ where $r_{i}$ denotes
the multiplicity of $i$ in the sequence $\lambda$.

Given $\lambda$ and $\mu$ two partitions of $n_{1}\in\mathbb{N}$ and
$n_{2}\in\mathbb{N}$ respectively, we denote by $\lambda\sqcup\mu$ the
partition of $n_{1}+n_{2}$ obtained by concatenation of the two partitions and
by reordering the parts to obtain a partition. If $\lambda$ is a partition of
$n$ and $k\in\mathbb{N}$, we denote by $\lambda^{k}$ the partition
$\underbrace{\lambda\sqcup\lambda\sqcup\cdots\sqcup\lambda}_{k\text{ times}}$.

Let $d\in\mathbb{N}_{>1}$. A partition $\lambda$ is called $d$-regular if no
part in $\lambda$ is repeated $d$ or more times. Each partition $\lambda$ can
be decomposed uniquely as $\lambda=\mu\sqcup\nu^{d}$ where $\mu$ is
$d$-regular. Then $\lambda$ is $d$-regular if and only if $\nu$ is empty. The
set of $d$-regular partitions of $n$ is denoted by
$\operatorname{Reg}_{d}(n)$. This set has a remarkable involution
$\mathsf{M}_{d}$ called the _Mullineux involution_ which will be defined in
the next section (see Section 4.5 for its interpretation in terms of
crystals).

More generally we shall decompose partitions with respect to two integers.
Recall that $\ell$ is a prime number. If
$\lambda=(1^{r_{1}},2^{r2},\ldots,n^{r_{n}})$ is a partition of $n$, we can
decompose the integers $r_{i}$ as

$\displaystyle r_{i}=r_{i,-1}+dr_{i,0}+d\ell
r_{i,1}+d\ell^{2}r_{i,2}+\cdots+d\ell^{n}r_{i,n}$

with $0\leq r_{i,-1}<d$ and $0\leq r_{i,j}<\ell$ for all $j\geq 0$. If we
define the partition
$\lambda_{(j)}=(1^{r_{1,j}},2^{r_{2,j}},\ldots,n^{r_{n,j}})$, then

$\displaystyle\lambda=\lambda_{(-1)}\sqcup(\lambda_{(0)})^{d}\sqcup(\lambda_{(1)})^{d\ell}\sqcup\cdots\sqcup(\lambda_{(n)})^{d\ell^{n}},$

where $\lambda_{(-1)}$ is $d$-regular and $\lambda_{(j)}$ is $\ell$-regular
for all $j\geq 0$. This decomposition is called the _$d$ -$\ell$-adic
decomposition of $\lambda$_ [11].

###  3.2 Hecke algebras of type $\boldsymbol{A}$ and the Mullineux involution

Let $q\in k^{\times}$ and $m\geq 1$. We denote by
$\mathcal{H}_{q}(\mathfrak{S}_{m})$ the Iwahori–Hecke algebra of the symmetric
group $\mathfrak{S}_{m}$ over $k$, with parameter $q$. It has a $k$-basis
$\\{T_{w}\\}_{w\in\mathfrak{S}_{m}}$ satisfying the following relations, for
$w\in\mathfrak{S}_{m}$ and $s=(i,i+1)$ a simple reflection:

$\displaystyle T_{w}T_{s}=\begin{cases}T_{ws}&\text{if $\ell(ws)>\ell(w)$
(i.e., if $w(i)<w(i+1)$)},\\\ qT_{ws}+(q-1)T_{w}&\text{otherwise}.\end{cases}$

In particular the basis elements corresponding to the simple reflections
generate $\mathcal{H}_{q}(\mathfrak{S}_{m})$ as an algebra, and they satisfy
the relation $(T_{s}-q)(T_{s}+1)=0$.

Let us consider the integer

$\displaystyle e:=\min\big{\\{}i\geq
0\,|\,1+q+q^{2}+\cdots+q^{i-1}=0\big{\\}}\in\mathbb{N}_{>1}.$

It is equal to the order of $q$ in $k^{\times}$ when $q\neq 1$, and to
$\ell=\mathrm{char}\,k$ when $q=1$. Then the set of simple
$\mathcal{H}_{q}(\mathfrak{S}_{m})$-modules is parametrized by the set of
$e$-regular partitions of $m$. Given an $e$-regular partition $\lambda$, we
will denote by $D(\lambda)$ the corresponding simple module. Note that when
$q=1$, the Hecke algebra $\mathcal{H}_{q}(\mathfrak{S}_{m})$ coincides with
the group algebra of $\mathfrak{S}_{m}$ over $k$, whose irreducible
representations are parametrized by $\ell$-regular partitions of $m$.

The map $\alpha\colon T_{w}\longmapsto(-q)^{\ell(w)}(T_{w^{-1}})^{-1}$ is an
algebra automorphism of $\mathcal{H}_{q}(\mathfrak{S}_{m})$ of order $2$. This
follows from the fact that $-qT_{s}^{-1}$ satisfies the same quadratic
equation as $T_{s}$. The automorphism $\alpha$ induces a permutation
$\alpha^{*}$ on the set of simple $\mathcal{H}_{q}(\mathfrak{S}_{m})$-modules,
and therefore on the set of $e$-regular partitions. In other words, the exists
an involution $\mathsf{M}_{e}$ on $\operatorname{Reg}_{e}(n)$, called the
_Mullineux involution_ , such that for any $e$-regular partition $\lambda$

$\displaystyle\alpha^{*}(D(\lambda))\simeq D(\mathsf{M}_{e}(\lambda)).$

In the case where $q=1$, the involution $\alpha$ is just the multiplication by
the sign representation $\varepsilon$ in the group algebra of
$\mathfrak{S}_{m}$ and $D(\mathsf{M}_{\ell}(\lambda))\simeq
D(\lambda)\otimes\varepsilon$.

The involution $\mathsf{M}_{\ell}$ was introduced by Mullineux, who suggested
in [26] a conjectural explicit combinatorial algorithm to compute it. This
conjecture was subsequently proved by Ford–Kleshchev in [14]. An
interpretation in terms of crystals was later given by Kleshchev [20] (for the
case $q=1$) and by Brundan [3] for the general case. We will review their
result in Section 4.5.

######  Remark 3.1.

When $e>m$, every partition of $m$ is $e$-regular. The Hecke algebra is
actually semi-simple in that case and the Mullineux involution
$\mathsf{M}_{e}$ corresponds to conjugating partitions.

Recall that $\mathfrak{S}_{m}$ has a structure of a Coxeter group, where the
simple reflections are given by the transpositions $(i,i+1)$. Given a
parabolic subgroup $\mathfrak{S}$ of $\mathfrak{S}_{m}$, one can consider the
subalgebra of $\mathcal{H}_{q}(\mathfrak{S}_{m})$ generated by
$\\{T_{w}\\}_{w\in\mathfrak{S}}$. It corresponds to the Hecke algebra
$\mathcal{H}_{q}(\mathfrak{S})$ of $\mathfrak{S}$ with parameter $q$, and
$\mathcal{H}_{q}(\mathfrak{S}_{m})$ is flat as a module over that subalgebra.
It is even free, with basis given by the elements $T_{w}$ where $w$ runs over
a set of representatives of $\mathfrak{S}\backslash\mathfrak{S}_{m}$ with
minimal length. Therefore, following Section 2.4 (see also [22]) we can use
the induction and restriction functors to define a duality functor
$\mathsf{D}_{\mathcal{H}}$ on the bounded derived category of finitely
generated $\mathcal{H}_{q}(\mathfrak{S}_{m})$-modules. For Hecke algebras this
duality functor is actually a shifted Morita equivalence.

######  Theorem 3.2 (Linckelmann–Schroll [22]).

Given a finitely generated right $\mathcal{H}_{q}(\mathfrak{S}_{m})$-module
$X$ we have

$\displaystyle\mathsf{D}_{\mathcal{H}}(X)\simeq\alpha^{*}(X)[-m+1]$

in $D^{b}(\mathsf{mod}\textsf{-}\mathcal{H}_{q}(\mathfrak{S}_{m}))$.

In particular when $X=D(\lambda)$ is simple we get
$\mathsf{D}_{\mathcal{H}}(D(\lambda))\simeq D(\mathsf{M}_{e}(\lambda))[-m+1]$.

###  3.3 Harish-Chandra series of $\boldsymbol{\mathrm{GL}_{n}(q)}$ and a
generalized Mullineux involution

From now on $\mathbf{G}=\mathrm{GL}_{n}(\overline{\mathbb{F}}_{p})$ is the
general linear group over an algebraic closure of $\mathbb{F}_{p}$, and
$F\colon\mathbf{G}\longrightarrow\mathbf{G}$ the standard Frobenius
endomorphism, raising the entries of a matrix to the $q$-th power. The
corresponding finite reductive group is $G=\mathrm{GL}_{n}(q)$.

Recall that the unipotent simple $kG$-modules are parametrized by partitions
of $n$. We will denote by $S(\lambda)$ the simple $kG$-module corresponding to
the partition $\lambda$. We review now the results in [11, 16] on the
partition of the unipotent representations into Harish-Chandra series (see
also [4, Section 19]). This classification depends on both $\ell$, and the
integer $e>1$ defined as the order of $q$ modulo $\ell$ (with the convention
that $e=\ell$ if $q\equiv 1$ modulo $\ell$).

By [11, Corollary 4.3.13] $S(\lambda)$ is cuspidal if and only if $\lambda=1$
or $\lambda=1^{e\ell^{i}}$ for some $i\geq 0$. In particular, given $n\geq 1$
there is at most one cuspidal unipotent simple $k\mathrm{GL}_{n}(q)$-module,
and there is exactly one if and only if $n=1$ or $n=e\ell^{i}$ for some $i\geq
0$. By considering products of such representations one can construct any
cuspidal pair of $G$. To this end, we introduce the set

$\displaystyle\mathcal{N}(n)=\big{\\{}{\bf
m}=(m_{-1},m_{0},\ldots,m_{n})\,|\,n=m_{-1}+em_{0}+e\ell
m_{1}+\cdots+e\ell^{n}m_{n}\big{\\}}.$

Given ${\bf m}\in\mathcal{N}(n)$ we define the standard Levi subgroup

$\displaystyle
L_{\mathbf{m}}=\mathrm{GL}_{1}(q)^{m_{-1}}\times\mathrm{GL}_{e}(q)^{m_{0}}\times\cdots\times\mathrm{GL}_{e\ell^{n}}(q)^{m_{n}}.$

It has a unique cuspidal unipotent simple module $X_{\mathbf{m}}$, and all the
cuspidal pairs $(L_{\mathbf{m}},X_{\mathbf{m}})$ of $G$ are obtained this way
for various ${\mathbf{m}}\in\mathcal{N}(n)$.

The simple $kG$-modules lying in the corresponding Harish-Chandra series are
parametrized by the irreducible representations of the endomorphism algebra
$\mathcal{H}_{\mathbf{m}}=\operatorname{End}_{G}\big{(}\mathsf{R}_{L_{\mathbf{m}}}^{G}(X_{\mathbf{m}})\big{)}$.
By [11, Section 3.4] (see also [4, Lemma 19.24]), there is a natural
isomorphism of algebras

$\displaystyle\mathcal{H}_{\mathbf{m}}\simeq\mathcal{H}_{q}(\mathfrak{S}_{m_{-1}})\otimes
k\mathfrak{S}_{m_{0}}\otimes\cdots\otimes k\mathfrak{S}_{m_{n}},$ (3.1)

where $\mathcal{H}_{q}(\mathfrak{S}_{m_{-1}})$ is the Iwahori–Hecke algebra of
$\mathfrak{S}_{m_{-1}}$ introduced in the previous section. Therefore the
simple modules of $\mathcal{H}_{\mathbf{m}}$ are parametrized by tuples of
partitions

$\displaystyle\boldsymbol{\lambda}=(\lambda_{(-1)},\lambda_{(0)},\ldots,\lambda_{(n)}),$

where $\lambda_{(-1)}$ is an $e$-regular partition of $m_{-1}$ and each
$\lambda_{(i)}$ for $i\geq 0$ is an $\ell$-regular partition of $m_{i}$.

By the tensor product theorem of Dipper–Du [11, Corollary 4.3.11] (see also
[4, Theorem 19.20]), the simple module $S(\lambda)$ of $G$ attached to this
multipartition $\boldsymbol{\lambda}$ is given by

$\displaystyle\lambda=\lambda_{(-1)}\sqcup(\lambda_{(0)})^{e}\sqcup(\lambda_{(1)})^{e\ell}\sqcup\cdots\sqcup(\lambda_{(n)})^{e\ell^{n}}.$
(3.2)

In other words,

$\displaystyle\operatorname{Hom}_{kG}\big{(}R_{L_{\bf m}}^{G}(X_{\bf
m}),S(\lambda)\big{)}\simeq D(\boldsymbol{\lambda})\qquad\text{in}\
\mathsf{mod}\text{-}\mathcal{H}_{\bf m}.$ (3.3)

Conversely, any partition $\lambda$ of $n$ can be uniquely decomposed as (3.2)
using the $e$-$\ell$-adic decomposition defined in Section 3.1. The tuple
$\mathbf{m}=(|\lambda_{(-1)}|,|\lambda_{(0)}|,\ldots,|\lambda_{(n)}|)$ will be
denoted by $\mathsf{hc}(\lambda)$, thus defining a map
$\mathsf{hc}\colon\mathcal{P}(n)\longrightarrow\mathcal{N}(n)$. With this
notation, the simple $kG$-modules lying in the Harish-Chandra of $(L_{\bf
m},X_{\bf m})$ are parametrized by $\mathsf{hc}^{-1}(\mathbf{m})$.

Motivated by the isomorphism (3.1) and the tensor product theorem of Dipper–Du
we define a version of the Mullineux involution as follows.

######  Definition 3.3.

Let $\lambda$ be a partition of $n$, written as
$\lambda=\lambda_{(-1)}\sqcup(\lambda_{(0)})^{e}\sqcup\cdots\sqcup(\lambda_{(n)})^{e\ell^{n}}$
where $\lambda_{(-1)}$ is $e$-regular and each $\lambda_{(i)}$ for $i\geq 0$
is $\ell$-regular. The _generalized Mullineux involution_ on $\lambda$ is
defined by

$\displaystyle\mathsf{M}_{e,\ell}(\lambda)=\mathsf{M}_{e}(\lambda_{(-1)})\sqcup(\mathsf{M}_{\ell}(\lambda_{(0)}))^{e}\sqcup\cdots\sqcup(\mathsf{M}_{\ell}(\lambda_{(n)}))^{e\ell^{n}}.$

######  Example 3.4.

It is interesting to note the following particular cases.

  1. (a)

If $\lambda$ is an $e$-regular partition then
$\mathsf{M}_{e,\ell}(\lambda)=\mathsf{M}_{e}(\lambda)$ is the ordinary
Mullineux involution.

  2. (b)

If $e\ell>n$ then $\lambda_{(i)}=\varnothing$ for $i>0$. In addition, $\ell$
is bigger than the size of $\lambda_{(0)}$ and therefore
$\mathsf{M}_{\ell}(\lambda_{(0)})=\lambda_{(0)}^{t}$ is just the conjugate of
$\lambda_{(0)}$. Consequently we get

$\displaystyle\mathsf{M}_{e,\ell}(\lambda)=\mathsf{M}_{e}(\lambda_{(-1)})\sqcup({\lambda_{(0)}}^{t})^{e}.$

Quite remarkably, this involution already appears in the work of Bezrukavnikov
[2] and Losev [23, Corollary 5.7] on the wall-crossing bijections for
representations of rational Cherednik algebras.

Let $\operatorname{Irr}_{k}\big{(}G|(L_{\mathbf{m}},X_{\mathbf{m}})\big{)}$ be
the set of isomorphism classes of simple $kG$-modules lying in the Harish-
Chandra series associated with $(L_{\mathbf{m}},X_{\mathbf{m}})$. Note that
$M_{e,\ell}$ preserves $\mathsf{hc}^{-1}(\mathbf{m})$, therefore $S(\lambda)$
and $S(\mathsf{M}_{e,\ell}(\lambda))$ lie in the same Harish-Chandra series.
By construction, the involution $\mathsf{M}_{e,\ell}$ is the unique operation
which makes the following diagram commutative:

$\textstyle{\mathsf{hc}^{-1}(\mathbf{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{M}_{e,\ell}}$$\textstyle{\operatorname{Irr}_{k}\big{(}G|(L_{\mathbf{m}},X_{\mathbf{m}})\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Hom}_{G}\big{(}\mathsf{R}_{L_{\mathbf{m}}}^{G}(X_{\mathbf{m}}),-\big{)}}$$\textstyle{\operatorname{Irr}\mathcal{H}_{\mathbf{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{*}}$$\textstyle{\mathsf{hc}^{-1}(\mathbf{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Irr}_{k}\big{(}G|(L_{\mathbf{m}},X_{\mathbf{m}})\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Hom}_{G}\big{(}\mathsf{R}_{L_{\mathbf{m}}}^{G}(X_{\mathbf{m}}),-\big{)}}$$\textstyle{\operatorname{Irr}\mathcal{H}_{\mathbf{m}}.}$

###  3.4 Computation of $\mathsf{d}_{G}(S)$

Fix ${\bf m}=(m_{-1},m_{0},\ldots,m_{n})\in\mathcal{N}(n)$ and let
$(L_{\mathbf{m}},X_{\mathbf{m}})$ be the corresponding cuspidal pair, as
defined in Section 3.3. One cannot apply Theorem 2.5 directly to $G$ and
$L_{\mathbf{m}}$ since the assumption (ii) might not be satisfied for every
intermediate Levi subgroup between $L_{\mathbf{m}}$ and $G$. For example, if
${\bf m}=(e,1)$ then the Levi subgroups
$(\mathrm{GL}_{1}(q))^{e}\times\mathrm{GL}_{e}(q)$ and
$\mathrm{GL}_{e}(q)\times(\mathrm{GL}_{1}(q))^{e}$ are conjugate under
$\mathrm{GL}_{2e}(q)$ but not under
$\mathrm{GL}_{e}(q)\times\mathrm{GL}_{e}(q)$. To solve this problem we
consider, instead of $\mathbf{G}$, the standard Levi subgroup

$\displaystyle\mathbf{G}_{\mathbf{m}}=\mathrm{GL}_{m_{-1}}\big{(}\overline{\mathbb{F}}_{p}\big{)}\times\mathrm{GL}_{em_{0}}\big{(}\overline{\mathbb{F}}_{p}\big{)}\times\mathrm{GL}_{e\ell
m_{1}}\big{(}\overline{\mathbb{F}}_{p}\big{)}\times\cdots\times\mathrm{GL}_{e\ell^{n}m_{n}}\big{(}\overline{\mathbb{F}}_{p}\big{)}.$

Then $\mathbf{L}_{\mathbf{m}}$ is the only standard Levi subgroup of
$\mathbf{G}_{m}$ which is conjugate to $\mathbf{L}_{\mathbf{m}}$. In
particular, assumption (ii) of Theorem 2.5 is satisfied.

######  Lemma 3.5.

The functor $\mathsf{R}_{G_{\mathbf{m}}}^{G}$ induces an isomorphism of
algebras

$\displaystyle\operatorname{End}_{G_{\mathbf{m}}}\big{(}\mathsf{R}_{L_{\mathbf{m}}}^{G_{\mathbf{m}}}(X_{\mathbf{m}})\big{)}\overset{\sim}{\to}\operatorname{End}_{G}\big{(}\mathsf{R}_{L_{\mathbf{m}}}^{G}(X_{\mathbf{m}})\big{)}=\mathcal{H}_{\mathbf{m}}.$

###### Proof.

Since $\mathsf{R}_{G_{\mathbf{m}}}^{G}$ is fully-faithful, the natural map
$\operatorname{End}_{G_{\mathbf{m}}}(\mathsf{R}_{L_{\mathbf{m}}}^{G_{\mathbf{m}}}(X_{\mathbf{m}}))\longrightarrow\operatorname{End}_{G}(\mathsf{R}_{L_{\mathbf{m}}}^{G}(X_{\mathbf{m}}))$
is an embedding of algebras. To conclude it is enough to compute the
dimensions using the Mackey formula. The equality comes from the fact that any
element $g\in G$ which normalizes $L_{\mathbf{m}}$ and $X_{\mathbf{m}}$ is in
fact in $G_{\mathbf{m}}$. ∎

In particular, the simple $kG_{\mathbf{m}}$-modules lying in the Harish-
Chandra series of $(L_{\mathbf{m}},X_{\mathbf{m}})$ are also parametrized by
multipartitions
$\boldsymbol{\lambda}=(\lambda_{(-1)},\lambda_{(0)},\ldots,\lambda_{(n)})$ of
$\mathbf{m}$, where $\lambda_{(-1)}$ is $e$-regular, and each $\lambda_{(i)}$
for $i\geq 0$ is $\ell$-regular. We will write
$S_{\mathbf{m}}(\boldsymbol{\lambda})$ for the simple module corresponding to
$\boldsymbol{\lambda}$, which by definition satisfies

$\displaystyle\operatorname{Hom}_{kG_{\bf m}}(R_{L_{\bf m}}^{G_{\bf m}}(X_{\bf
m}),S_{\bf m}(\boldsymbol{\lambda}))\simeq
D(\boldsymbol{\lambda})\qquad\text{in}\ \mathsf{mod}\text{-}\mathcal{H}_{\bf
m}$ (3.4)

(compare with (3.3)). Then it follows from the tensor product theorem [11,
Corolary 4.3.11] that

$\displaystyle S_{\bf m}(\boldsymbol{\lambda})=S(\lambda_{(-1)})\boxtimes
S\big{(}(\lambda_{(0)})^{e}\big{)}\boxtimes
S\big{(}(\lambda_{(1)})^{e\ell}\big{)}\boxtimes\cdots$

and

$\displaystyle R_{G_{\bf m}}^{G}(S_{\bf m}(\boldsymbol{\lambda}))\simeq
S(\lambda),$ (3.5)

where as in (3.2) we set
$\lambda=\lambda_{(-1)}\sqcup(\lambda_{(0)})^{e}\sqcup(\lambda_{(1)})^{e\ell}\sqcup\cdots\sqcup(\lambda_{(n)})^{e\ell^{n}}$.

The construction of the isomorphism (3.1) given for example in [4, Section 19]
is compatible with induction and restriction. The map
$\mathbf{M}\longmapsto\operatorname{End}_{M}(\mathsf{R}_{L_{\mathbf{m}}}^{M}(X_{\mathbf{m}}))$
gives a one-to-one correspondence between the standard Levi subgroups of
$\mathbf{G}_{m}$ containing $\mathbf{L}$ and the parabolic subalgebras of
$\mathcal{H}_{\mathbf{m}}$. Since $q\neq 0$, $\mathcal{H}_{\mathbf{m}}$ is
flat over each of these subalgebras and Theorem 2.5 can be applied to get

$\displaystyle\mathsf{D}_{\mathcal{H}_{\mathbf{m}}}\big{(}\mathrm{RHom}_{G_{\mathbf{m}}}\big{(}\mathsf{R}_{L_{\mathbf{m}}}^{G_{\mathbf{m}}}(X_{\mathbf{m}}),-\big{)}\big{)}\,\mathop{\longrightarrow}\limits^{\sim}\,\mathrm{RHom}_{G_{\mathbf{m}}}\big{(}\mathsf{R}_{L_{\mathbf{m}}}^{G_{\mathbf{m}}}(X_{\mathbf{m}}),\mathsf{D}_{G_{\mathbf{m}}}(-)\big{)}.$
(3.6)

Recall from Section 2.3 that given a simple $kG$-module $S$, there is unique
composition factor in the cohomology of $\mathsf{D}_{G}(S)$ which lies in the
same Harish-Chandra series as $S$. We denote this composition factor by
$\mathsf{d}_{G}(S)$. Combining (3.6) and Theorem 3.2 we can determine
$\mathsf{d}_{G_{\mathbf{m}}}$ explicitly on the unipotent representations.

######  Proposition 3.6.

Let
$\boldsymbol{\lambda}=(\lambda_{(-1)},\lambda_{(0)},\lambda_{(1)},\ldots,\lambda_{(n)})$
be a multipartition where $\lambda_{(-1)}$ is an $e$-regular partition of
$m_{-1}$ and each $\lambda_{(i)}$ for $i\geq 0$ is an $\ell$-regular partition
of $m_{i}$. Then

$\displaystyle d_{G_{\mathbf{m}}}(S_{\mathbf{m}}(\boldsymbol{\lambda}))\simeq
S_{\mathbf{m}}\big{(}\mathsf{M}_{e}(\lambda_{(-1)}),\mathsf{M}_{\ell}(\lambda_{(0)}),\ldots,\mathsf{M}_{\ell}(\lambda_{(n)})\big{)}.$

###### Proof.

For simplicity we will write $S$ for $S_{\mathbf{m}}(\boldsymbol{\lambda})$
throughout this proof. Let $r$ be the cuspidal depth of the
$kG_{\mathbf{m}}$-module $S$. By Proposition 2.2 and the definition of
$\mathsf{d}_{G_{\mathbf{m}}}(S)$ the natural map

$\displaystyle\operatorname{Hom}_{G_{\mathbf{m}}}\big{(}\mathsf{R}_{L_{\mathbf{m}}}^{G_{\mathbf{m}}}(X_{\mathbf{m}}),\mathsf{d}_{G_{\mathbf{m}}}(S)\big{)}\longrightarrow\operatorname{Hom}_{G_{\mathbf{m}}}\big{(}\mathsf{R}_{L_{\mathbf{m}}}^{G_{\mathbf{m}}}(X_{\mathbf{m}}),H^{-r}(\mathsf{D}_{G_{\mathbf{m}}}(S))\big{)}$

is an isomorphism of right $\mathcal{H}_{\mathbf{m}}$-modules. Now let us
consider the distinguished triangle

$\displaystyle
H^{-r}(\mathrm{D}_{G_{\mathbf{m}}}(S))\longrightarrow\mathsf{D}_{G_{\mathbf{m}}}(S)[-r]\longrightarrow\tau_{>-r}(\mathsf{D}_{G_{\mathbf{m}}}(S))[-r]\rightsquigarrow$

in $D^{b}(kG_{\mathbf{m}})$. We apply the functor
$\operatorname{Hom}_{D^{b}(kG_{\mathbf{m}})}\big{(}\mathsf{R}_{L_{\mathbf{m}}}^{G_{\mathbf{m}}}(X_{\mathbf{m}}),-\big{)}$,
which, by the properties of $\mathsf{D}_{G_{\mathbf{m}}}(S)$ listed in
Proposition 2.2, gives an isomorphism

$\displaystyle\operatorname{Hom}_{G_{\mathbf{m}}}\big{(}\mathsf{R}_{L_{\mathbf{m}}}^{G_{\mathbf{m}}}(X_{\mathbf{m}}),H^{-r}(\mathrm{D}_{G_{\mathbf{m}}}(S))\big{)}\,\mathop{\longrightarrow}\limits^{\sim}\,\operatorname{Hom}_{D^{b}(kG_{\mathbf{m}})}\big{(}\mathsf{R}_{L_{\mathbf{m}}}^{G_{\mathbf{m}}}(X_{\mathbf{m}}),\mathsf{D}_{G_{\mathbf{m}}}(S)[-r]\big{)}.$

Combining this with Theorem 2.5 gives

$\displaystyle\operatorname{Hom}_{G_{\mathbf{m}}}\big{(}\mathsf{R}_{L_{\mathbf{m}}}^{G_{\mathbf{m}}}(X_{\mathbf{m}}),H^{-r}(\mathrm{D}_{G_{\mathbf{m}}}(S))\big{)}\simeq
H^{-r}\big{(}\mathrm{RHom}_{G_{\mathbf{m}}}\big{(}\mathsf{R}_{L_{\mathbf{m}}}^{G_{\mathbf{m}}}(X_{\mathbf{m}}),\mathsf{D}_{G_{\mathbf{m}}}(S)\big{)}\big{)}$
$\displaystyle\hphantom{\operatorname{Hom}_{G_{\mathbf{m}}}\big{(}\mathsf{R}_{L_{\mathbf{m}}}^{G_{\mathbf{m}}}(X_{\mathbf{m}}),H^{-r}(\mathrm{D}_{G_{\mathbf{m}}}(S))\big{)}}{}\simeq
H^{-r}\big{(}\mathsf{D}_{\mathcal{H}}\big{(}\mathrm{RHom}_{G_{\mathbf{m}}}(\mathsf{R}_{L_{\mathbf{m}}}^{G_{\mathbf{m}}}(X_{\mathbf{m}}),S)\big{)}\big{)}$

in $\mathsf{mod}\textsf{-}\mathcal{H}_{\mathbf{m}}$. Now, by Theorem 3.2 the
duality functor $\mathsf{D}_{\mathcal{H}}$ is induced by a shifted Morita
equivalence, obtained by twisting by the algebra automorphism $\alpha$ defined
in Section 3.2. Note that the corresponding shift equals the rank of the
Coxeter group associated with the Hecke algebra $\mathcal{H}_{\mathbf{m}}$,
which also equals the cuspidal depth $r$ of $S$. In particular, we have

$\displaystyle
H^{-r}\big{(}\mathsf{D}_{\mathcal{H}}\big{(}\mathrm{RHom}_{G_{\mathbf{m}}}(\mathsf{R}_{L_{\mathbf{m}}}^{G_{\mathbf{m}}}(X_{\mathbf{m}}),S)\big{)}\big{)}\simeq
H^{-r}\big{(}\alpha^{*}\mathrm{RHom}_{G_{\mathbf{m}}}\big{(}\mathsf{R}_{L_{\mathbf{m}}}^{G_{\mathbf{m}}}(X_{\mathbf{m}}),S\big{)}[r]\big{)}$
$\displaystyle\hphantom{H^{-r}\big{(}\mathsf{D}_{\mathcal{H}}\big{(}\mathrm{RHom}_{G_{\mathbf{m}}}(\mathsf{R}_{L_{\mathbf{m}}}^{G_{\mathbf{m}}}(X_{\mathbf{m}}),S)\big{)}\big{)}}{}\simeq\alpha^{*}H^{0}\big{(}\mathrm{RHom}_{G_{\mathbf{m}}}\big{(}\mathsf{R}_{L_{\mathbf{m}}}^{G_{\mathbf{m}}}(X_{\mathbf{m}}),S\big{)}\big{)}$
$\displaystyle\hphantom{H^{-r}\big{(}\mathsf{D}_{\mathcal{H}}\big{(}\mathrm{RHom}_{G_{\mathbf{m}}}(\mathsf{R}_{L_{\mathbf{m}}}^{G_{\mathbf{m}}}(X_{\mathbf{m}}),S)\big{)}\big{)}}{}\simeq\alpha^{*}\big{(}\operatorname{Hom}_{D^{b}(kG_{\mathbf{m}})}\big{(}\mathsf{R}_{L_{\mathbf{m}}}^{G_{\mathbf{m}}}(X_{\mathbf{m}}),S\big{)}\big{)}\simeq\alpha^{*}D(\boldsymbol{\lambda}).$

Note that the last isomorphism uses (3.4) and the fact that the natural
functor $kG_{\bf m}\rightarrow D^{b}(kG_{\bf m})$ is fully-faithful. Finally,
by definition of the Mullineux involution, the module
$\alpha^{*}D(\boldsymbol{\lambda})$ is the simple
$\mathcal{H}_{\mathbf{m}}$-module labelled by the multipartition
$(\mathsf{M}_{e}(\lambda_{(-1)}),\mathsf{M}_{\ell}(\lambda_{(0)}),\ldots,\mathsf{M}_{\ell}(\lambda_{(n)}))$.
∎

We can finally prove the expected relation between the Alvis–Curtis duality
and our generalization of the Mullineux involution.

######  Theorem 3.7.

Let $e=\min\\{i\geq 0\,|\,1+q+q^{2}+\cdots+q^{i-1}\equiv 0\,{\rm
mod}\,\ell\\}$ and $\lambda$ be a partition of $n$. Then

$\displaystyle\mathsf{d}_{\mathrm{GL}_{n}(q)}(S(\lambda))\simeq
S(\mathsf{M}_{e,\ell}(\lambda)),$

where $\mathsf{M}_{e,\ell}$ is the generalized Mullineux involution defined in
Definition 3.3.

###### Proof.

Let $\boldsymbol{\lambda}$ be the multipartition associated to $\lambda$ as in
(3.2). Given a bounded complex $C$ of representations (of $kG$ or
$kG_{\mathbf{m}}$), recall that $[C]$ denotes its class in the corresponding
Grothendieck group ($K_{0}(kG)$ or $K_{0}(kG_{\mathbf{m}})$, see Section 2.1).
By (2.1) we have

$\displaystyle\big{[}\mathsf{D}_{G}\big{(}\mathsf{R}_{G_{\mathbf{m}}}^{G}(S_{\mathbf{m}}(\boldsymbol{\lambda}))\big{)}\big{]}=\pm\big{[}\mathsf{R}_{G_{\mathbf{m}}}^{G}(\mathsf{D}_{G_{\mathbf{m}}}(S_{\mathbf{m}}(\boldsymbol{\lambda})))\big{]}\qquad\text{in}\
K_{0}(kG).$ (3.7)

Let $\mathcal{C}$ (resp. $\mathcal{C}_{\mathbf{m}}$) be the sublattice of
$K_{0}(kG)$ (resp. $K_{0}(kG_{\mathbf{m}})$) spanned by the classes of simple
modules with cuspidal depth strictly less than $S(\lambda)$ (resp.
$S_{\mathbf{m}}(\boldsymbol{\lambda})$). It follows from Proposition 2.2(ii)
that these lattices are stable under Alvis–Curtis duality. In addition, the
Harish-Chandra induction functor satisfies
$\mathsf{R}_{G_{\mathbf{m}}}^{G}(\mathcal{C}_{\mathbf{m}})\subset\mathcal{C}$.

By Proposition 2.2, we have
$[\mathsf{D}_{G_{\mathbf{m}}}(S_{\mathbf{m}}(\boldsymbol{\lambda}))]\in\pm[\mathsf{d}_{G_{\mathbf{m}}}(S_{\mathbf{m}}(\boldsymbol{\lambda}))]+\mathcal{C}_{\mathbf{m}}$.
We deduce from (3.5) and Proposition 3.6 that

$\displaystyle\big{[}\mathsf{R}_{G_{\mathbf{m}}}^{G}(\mathsf{D}_{G}(S_{\mathbf{m}}(\boldsymbol{\lambda})))\big{]}\in\pm[S(\mathsf{M}_{e,\ell}(\lambda))]+\mathcal{C}.$

On the other hand
$[\mathsf{D}_{G}(S(\lambda))]\in\pm[\mathsf{d}_{G}(S(\lambda))]+\mathcal{C}$,
so that again by (3.5) we have

$\displaystyle\big{[}\mathsf{D}_{G}\big{(}\mathsf{R}_{G_{\mathbf{m}}}^{G}(S_{\mathbf{m}}(\boldsymbol{\lambda}))\big{)}\big{]}\in\pm[\mathsf{d}_{G}(S(\lambda))]+\mathcal{C}$

and we conclude that
$[\mathsf{d}_{G}(S(\lambda))]=[S(\mathsf{M}_{e,\ell}(\lambda))]$ using (3.7) .
∎

######  Remark 3.8.

The simple unipotent $k\mathrm{GL}_{n}(q)$-module associated with the trivial
partition $\lambda{=}(n)$ is the trivial module $k$. In that case the complex
$\mathsf{D}_{G}(k)$ is quasi-isomophic to a module shifted in degree $-n+1$,
by the Solomon–Tits theorem [9, Theorem 66.33]. This module is a
characteristic $\ell$ version of the Steinberg representation. By Theorem 3.7,
its socle is isomorphic to $S(\mathsf{M}_{e}(n))$, which is consistent with
[15].

##  4 Interpretation in terms of crystals

The aim of this section is to give an alternative description of the map
$\mathsf{M}_{e,\ell}$ using the crystal graph theory in the same spirit as for
the original Mullineux involution [20] (see Proposition 4.8).

###  4.1 More on partitions

We fix an integer $d>1$. Given a partition
$\lambda=(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{r}>0)$, its _Young
diagram_ $[\lambda]$ is the set

$\displaystyle[\lambda]=\big{\\{}(a,b)\,|\,1\leq a\leq r,\,1\leq
b\leq\lambda_{a}\big{\\}}\subset\mathbb{N}\times\mathbb{N}.$

The elements of this set are called the nodes of $\lambda$. The $d$-residue of
a node $\gamma\in[\lambda]$ is by definition
$\operatorname{res}_{d}(\gamma)=b-a+d\mathbb{Z}$. For
$j\in\mathbb{Z}/d\mathbb{Z}$, we say that $\gamma$ is a $j$-node if
$\operatorname{res}_{d}(\gamma)=j$. In addition, $\gamma$ is called a
removable $j$-node for $\lambda$ if the set $[\lambda]\setminus\\{\gamma\\}$
is the Young diagram of some partition $\mu$. In this case, we also say that
$\gamma$ is an addable $j$-node for $\mu$. We write
$\mu\overset{j}{\rightarrow}\lambda$ if $[\mu]\subset[\lambda]$ and
$[\lambda]\setminus[\mu]=\\{\gamma\\}$ for a $j$-node $\gamma$.

Let $\gamma=(a,b)$ and $\gamma^{\prime}=(a^{\prime},b^{\prime})$ be two
addable or removable $j$-nodes of the same partition $\lambda$. Then we write
$\gamma>\gamma^{\prime}$ if $a<a^{\prime}$. Let $w_{j}(\lambda)$ be the word
obtained by reading all the addable and removable $j$-nodes in increasing
order and by encoding each addable $j$-node with the letter $A$ and each
removable $j$-node with the letter $R$. Then deleting as many subwords $RA$ in
this word as possible, we obtain a sequence $A\cdots AR\cdots R$. The node
corresponding to the rightmost $A$ (if it exists) is called the good addable
$j$-node and the node corresponding to the leftmost $R$ (if it exists) is
called the good removable $j$-node.

###  4.2 Fock space and Kashiwara operators

Let $\mathcal{F}:=\mathbb{C}\mathcal{P}$ be the $\mathbb{C}$-vector space with
basis given by the set $\mathcal{P}$ of all partitions. There is an action of
the quantum group $\mathcal{U}(\widehat{\mathfrak{sl}}_{d})$ on $\mathcal{F}$
[25] which makes $\mathcal{F}$ into an integrable module of level $1$. The
Kashiwara operators $\widetilde{E}_{i,d}$ and $\widetilde{F}_{i,d}$ are then
defined as follows:

$\displaystyle\widetilde{F}_{i,d}\cdot\lambda=\begin{cases}\mu&\textrm{if
}\lambda\overset{i}{\rightarrow}\mu\text{ and }[\mu]\setminus[\lambda]\text{
is a good addable $i$-node of $\lambda$},\\\ 0&\text{if $\lambda$ has no good
addable $i$-node},\end{cases}$
$\displaystyle\widetilde{E}_{i,d}\cdot\mu=\begin{cases}\lambda&\textrm{if
}\lambda\overset{i}{\rightarrow}\mu\text{ and }[\mu]\setminus[\lambda]\text{
is a good removable $i$-node of $\mu$},\\\ 0&\textrm{if $\mu$ has no good
removable $i$-node}.\end{cases}$

Using these operators one can construct the
$\widehat{\mathfrak{sl}}_{d}$-crystal graph of $\mathcal{F}$, which is the
graph with

  * •

vertices: all the partitions $\lambda$ of $n\in\mathbb{N}$,

  * •

arrows: there is an arrow from $\lambda$ to $\mu$ colored by
$i\in\mathbb{Z}/d\mathbb{Z}$ if and only if
$\widetilde{F}_{i,d}\cdot\lambda=\mu$, or equivalently if and only if
$\lambda=\widetilde{E}_{i,d}\cdot\mu$.

Note that the definition makes sense for $d=\infty$. The corresponding
${\mathfrak{sl}}_{\infty}$-crystal graph coincides with the Young graph, also
known as the branching graph of the complex irreducible representations of
symmetric groups.

The following result can be found for example in [21, Section 2.2].

######  Proposition 4.1.

A partition $\lambda$ is a $d$-regular partition of $n$ if and only if there
exists $(i_{1},\ldots,i_{n})$ $\in(\mathbb{Z}/d\mathbb{Z})^{n}$ such that

$\displaystyle\widetilde{F}_{i_{1},d}\cdots\widetilde{F}_{i_{n},d}\cdot\varnothing=\lambda.$

In other words, the connected component the
$\widehat{\mathfrak{sl}}_{d}$-crystal graph containing the empty partition is
the full subgraph of the $\widehat{\mathfrak{sl}}_{d}$-crystal whose vertices
are labelled by $d$-regular partitions.

The arrows in this component give the branching rule for induction and
restriction in the Hecke algebra of symmetric groups at a primitive $d$-th
root of unity (see [3] for more details). The partitions $\lambda$ for which
we have $\widetilde{E}_{i,d}\cdot\lambda=0$ for all
$i\in\mathbb{Z}/d\mathbb{Z}$ are the _highest weight vertices_. One can
observe that they correspond to partitions of the form $\lambda=\mu^{d}$ for
some partition $\mu$.

There is also a representation theoretic interpretation of the other
components, using the representation theory of the finite general linear
group. Let $e$ be the order of $q$ modulo $\ell$, which we assume to be
different from $1$. Following [18], one can define a _weak Harish-Chandra
theory_ for unipotent representations of $\mathrm{GL}_{n}(q)$ for various $n$.
Recall from Section 3.3 that these unipotent representations are parametrized
by partitions. Consequently, the complexified Grothendieck group of the
category of unipotent representations is naturally isomophic to $\mathcal{F}$.
Under this identification, it follows from [6] that the action of
$\mathcal{U}(\widehat{\mathfrak{sl}}_{e})$ comes from a truncated version of
Harish-Chandra induction and restriction. As in [13], we deduce that:

  * •

A simple unipotent module $S(\lambda)$ is weakly cuspidal if and only if
$\lambda$ labels a highest weight vertex in the
$\widehat{\mathfrak{sl}}_{e}$-crystal graph of $\mathcal{F}$.

  * •

Two simple modules $S(\lambda)$ and $S(\mu)$ of $k\mathrm{GL}_{n}(q)$ lie in
the same weak Harish-Chandra series if and only if there exist a highest
weight vertex $\nu$, $k\in\mathbb{N}$,
$(i_{1},\ldots,i_{k})\in(\mathbb{Z}/e\mathbb{Z})^{k}$ and
$(j_{1},\ldots,j_{k})\in(\mathbb{Z}/e\mathbb{Z})^{k}$ such that

$\displaystyle\widetilde{F}_{i_{1},e}\ldots\widetilde{F}_{i_{k},e}\cdot\nu=\lambda,\qquad\text{and}\qquad\widetilde{F}_{j_{1},e}\ldots\widetilde{F}_{j_{k},e}\cdot\nu=\mu.$

This means that $\lambda$ and $\mu$ are in the same connected component of the
associated $\widehat{\mathfrak{sl}}_{e}$-crystal graph.

###  4.3 Crystals: the case $\boldsymbol{\ell=\infty}$

Let $e\in\mathbb{Z}_{>1}$. Recall from Section 3.1 that any partition
$\lambda$ of $n$ can be decomposed in a unique way as

$\displaystyle\lambda=\lambda_{(-1)}\sqcup(\lambda_{(0)})^{e},$

where $\lambda_{(-1)}$ is $e$-regular.

We claim that the entire $\widehat{\mathfrak{sl}}_{e}$-crystal graph structure
on the Fock space may be recovered from the subgraph with vertices labelled by
$e$-regular partitions. Indeed, it follows from the definition of the
Kashiwara operators that for any partition $\lambda$

$\displaystyle\widetilde{F}_{i,e}\lambda=\mu\iff\widetilde{F}_{i,e}\lambda_{(-1)}=\mu_{(-1)}\qquad\text{and}\qquad\lambda_{(0)}=\mu_{(0)}.$
(4.1)

One can now define crystal operators $\widetilde{F}_{i,\infty,0}$ and
$\widetilde{E}_{i,\infty,0}$ for all $i\in\mathbb{Z}$ by

$\displaystyle\widetilde{F}_{i,\infty,0}\lambda:=\mu\iff\widetilde{F}_{i,\infty}\lambda_{(0)}=\mu_{(0)}\qquad\text{and}\qquad\lambda_{(-1)}=\mu_{(-1)},$
$\displaystyle\widetilde{E}_{i,\infty,0}\lambda:=\mu\iff\widetilde{E}_{i,\infty}\lambda_{(0)}=\mu_{(0)}\qquad\text{and}\qquad\lambda_{(-1)}=\mu_{(-1)}.$

This endows $\mathcal{P}$ with an ${\mathfrak{sl}}_{\infty}$-crystal
structure, which by (4.1) commutes with the
$\widehat{\mathfrak{sl}}_{e}$-crystal structure. Note that the only highest
weight with respect to these two structures is the empty partition. In fact,
these constructions already appear in the work of Losev [23] (where the
${\mathfrak{sl}}_{\infty}$-crystal is called the _Heisenberg crystal_ , see
also [24, Proposition 4.6]).

###  4.4 Crystals: the case $\boldsymbol{\ell\in\mathbb{N}}$

More generally, recall from Section 3.3 that any partition $\lambda$ of $n$
can be decomposed in a unique way as

$\displaystyle\lambda=\lambda_{(-1)}\sqcup(\lambda_{(0)})^{e}\sqcup(\lambda_{(1)})^{e\ell}\sqcup\cdots\sqcup(\lambda_{(n)})^{e\ell^{n}},$
(4.2)

where $\lambda_{(-1)}$ is $e$-regular and each $\lambda_{(i)}$ for $i>-1$ is
$\ell$-regular. For example, with $\mu=(2^{2}.1^{7})$, $e=2$ and $\ell=3$ we
obtain $\mu=(1)\sqcup(2)^{2}\sqcup(1)^{2\times 3}$.

As in the previous section, the $\widehat{\mathfrak{sl}}_{e}$-crystal
operators act on the $e$-regular part of partitions, which makes the
definition of other operators possible. Let us fix $j\in\mathbb{N}$. We set
$\widetilde{F}_{i,\ell,j}\lambda=\mu$ if and only if
$\widetilde{F}_{i,\ell}\lambda_{(j)}=\mu_{(j)}$ and $\lambda_{(l)}=\mu_{(l)}$
for all $l\neq j$. Similarly, $\widetilde{E}_{i,\ell,j}\lambda=\mu$ if and
only if $\widetilde{E}_{i,\ell}\lambda_{(j)}=\mu_{(j)}$ and
$\lambda_{(l)}=\mu_{(l)}$ for all $l\neq j$. In other words,
$\widetilde{F}_{i,\ell,j}$ and $\widetilde{E}_{i,\ell,j}$ are defined as the
usual $\widehat{\mathfrak{sl}}_{\ell}$-crystal operators acting on the
component $\lambda_{(j)}$ in the decomposition (4.2), or equivalently on the
(non-necessarily $\ell$-regular) partition given by
$\lambda_{(j)}\sqcup\lambda_{(j+1)}^{\ell}\sqcup\cdots$.

As a consequence we obtain an $\widehat{\mathfrak{sl}}_{e}$-crystal structure
of level $1$ together with many $\widehat{\mathfrak{sl}}_{\ell}$-crystal
structures (each of them indexed by an integer $j\in\mathbb{N}$, and of level
$e\ell^{j}$) on the set of partitions. The following proposition is clear
using the decomposition of a partition.

######  Proposition 4.2.

The above $\widehat{\mathfrak{sl}}_{e}$-crystal structure and the various
$\widehat{\mathfrak{sl}}_{\ell}$-crystal structures on $\mathcal{P}$ mutually
commute.

We can thus define a graph containing all the information on the
$\widehat{\mathfrak{sl}}_{e}$-crystal structure and
$\widehat{\mathfrak{sl}}_{\ell}$-crystal structures. There is then an obvious
notion of highest weight.

######  Lemma 4.3.

The empty partition is the unique highest weight vertex with respect to the
above crystal structure.

In other words, the corresponding crystal graph is connected.

###### Proof.

Assume that $\lambda$ is a non-empty partition. Then there exists
$r\in\mathbb{Z}_{\geq-1}$ such that $\lambda_{(r)}\neq\varnothing$. Now

  * •

If $r=-1$ then $\lambda_{-1}$ is $e$-regular and we have
$\widetilde{E}_{i,e}\cdot\lambda\neq\varnothing$ for some $i\in\mathbb{Z}$.

  * •

If $r\neq-1$ then $\lambda_{r}$ is $\ell$-regular and we have
$\widetilde{E}_{i,\ell,r}\cdot\lambda\neq\varnothing$ for some
$i\in\mathbb{Z}$.

Thus $\lambda$ is not a highest weight vertex, and $\varnothing$ is the only
highest weight vertex. ∎

######  Example 4.4.

Let $e=2$ and consider the partition $\lambda=(2^{2}.1^{7})$.

  1. (a)

Assume that $\ell=\infty$. Then we have $\lambda_{(-1)}=1$ and
$\lambda_{(0)}=2.1^{3}$. We have

$\displaystyle\widetilde{F}_{2,\infty,0}\cdot\lambda:=(1)\sqcup\big{(}3.1^{3}\big{)}^{2}=3^{2}.1^{7},$
$\displaystyle\widetilde{F}_{0,\infty,0}\cdot\lambda:=(1)\sqcup\big{(}2^{2}.1^{2}\big{)}^{2}=2^{4}.1^{5},$
$\displaystyle\widetilde{F}_{-4,\infty,0}\cdot\lambda:=(1)\sqcup\big{(}2.1^{4}\big{)}^{2}=2^{2}.1^{9}.$

All the others Kashiwara operators $\widetilde{F}_{j,\infty,0}$ act by $0$ on
$\lambda$.

  2. (b)

Assume that $\ell=3$. Then we have $\lambda_{(-1)}=1$, $\lambda_{(0)}=2$ and
$\lambda_{(1)}=1$.

$\displaystyle\widetilde{F}_{2,3,0}\cdot\lambda:=(1)\sqcup(3)^{2}\sqcup
1^{6}=3^{2}.1^{7}$

and the action of the other operators $\widetilde{F}_{j,0,3}$ are $0$. We also
have

$\displaystyle\widetilde{F}_{1,3,1}\cdot\lambda:=(1)\sqcup(2)^{2}\sqcup
2^{6}=2^{8}.1,$
$\displaystyle\widetilde{F}_{2,3,1}\cdot\lambda:=(1)\sqcup(2)^{2}\sqcup(1.1)^{6}=2^{2}.1^{13}.$

One can also consider

$\displaystyle\widetilde{F}_{1,3,2}\cdot\lambda=(1)\sqcup(3)^{2}\sqcup
1^{6}\sqcup 1^{18}=3^{2}.1^{25}$

or, more generally for $k\geq 2$:

$\displaystyle\widetilde{F}_{1,3,k}\cdot\lambda=(1)\sqcup(3)^{2}\sqcup
1^{6}\sqcup 1^{3^{k}\times 2}=3^{2}.1^{2\times 3^{k}+6}.$

######  Remark 4.5.

Assume $\lambda$ is a partition of $n$ such that $e\ell>n$. Then it follows
from the construction that $\lambda_{(k)}=\varnothing$ for all $k>0$. Note
also that $\lambda_{(0)}$ is a partition of rank strictly less than $\ell$,
and therefore it is $\ell$-regular. In that case the action of
$\widetilde{E}_{i,\ell,0}$ and $\widetilde{E}_{i,\infty,0}$ coincide. In
particular, the $\widehat{\mathfrak{sl}}_{\ell}$ and
${\mathfrak{sl}}_{\infty}$-crystal structures coincide for the partitions of
rank less than $n$.

###  4.5 Crystals and the Mullineux involution

Recall that any $d$-regular partition $\lambda$ belongs to the connected
component of the empty partition in the $\widehat{\mathfrak{sl}_{d}}$-crystal
graph. The image of $\lambda$ by the Mullineux involution $\mathsf{M}_{d}$ is
also in that component and it can be computed using the following theorem.

######  Theorem 4.6 (Ford–Kleshchev [14]).

Let $\lambda\in\operatorname{Reg}_{d}(n)$ and let
$(i_{1},\ldots,i_{n})\in(\mathbb{Z}/d\mathbb{Z})^{n}$ such that

$\displaystyle\widetilde{F}_{i_{1},d}\cdots\widetilde{F}_{i_{n},d}\cdot\varnothing=\lambda$

$($see Proposition 4.1). Then there exists $\mu\in\operatorname{Reg}_{d}(n)$
such that

$\displaystyle\widetilde{F}_{-i_{1},d}\cdots\widetilde{F}_{-i_{n},d}\cdot\varnothing=\mu$

and $\mu=\mathsf{M}_{d}(\lambda)$.

######  Remark 4.7.

When $d>n$ every addable node is good, and Theorem 4.6 implies that
$M_{d}(\lambda){=}\lambda^{t}$, the conjugate partition (see also Remark 3.1).

From the definition of $\mathsf{M}_{e,\ell}$ (see Definition 3.3) and the
construction of the various crystal operators, Theorem 4.6 generalizes to the
following situation.

######  Proposition 4.8.

Let $\lambda$ be a partition which we write

$\displaystyle\widetilde{F}_{i_{1},p_{1},k_{1}}\cdots\widetilde{F}_{i_{m},p_{m},k_{m}}\cdot\varnothing=\lambda$

with for all $j=1,\ldots,m$,

  * •

$k_{j}\in\mathbb{Z}_{\geq-1}$,

  * •

$p_{j}=e$ if $k_{j}=-1$ and $p_{j}=\ell$ otherwise,

  * •

$i_{j}\in\mathbb{Z}/p_{j}\mathbb{Z}$.

Then

$\displaystyle\widetilde{F}_{-i_{1},p_{1},k_{1}}\cdots\widetilde{F}_{-i_{m},p_{m},k_{m}}\cdot\varnothing=\mathsf{M}_{e,\ell}(\lambda).$

###### Proof.

This is clear as the usual Mullineux involution on the set of $e$-regular
partitions is given in Theorem 4.6. ∎

######  Remark 4.9.

When $\ell=\infty$, the result remains valid and $\mathsf{M}_{e,\infty}$
coincides with the operation described by Bezrukavnikov in [2] and Losev in
[23] (see Example 3.4(b)).

### Acknowledgements

The authors gratefully acknowledge financial support by the ANR grant GeRepMod
ANR-16-CE40-0010-01. We thank Gunter Malle, Emily Norton and the referees for
their many valuable comments on a preliminary version of the manuscript.

## References

  * [1] Alvis D., The duality operation in the character ring of a finite Chevalley group,
hrefhttps://doi.org/10.1090/S0273-0979-1979-14690-1textitBull. Amer. Math.
Soc. (N.S.) 1 (1979), 907–911.

  * [2] Bezrukavnikov R., Hilbert schemes and stable pairs, unpublished notes.
  * [3] Brundan J., Modular branching rules and the Mullineux map for Hecke algebras of type $A$, Proc. London Math. Soc. 77 (1998), 551–581.
  * [4] Cabanes M., Enguehard M., Representation theory of finite reductive groups, New Mathematical Monographs, Vol. 1, Cambridge University Press, Cambridge, 2004.
  * [5] Cabanes M., Rickard J., Alvis–Curtis duality as an equivalence of derived categories, in Modular Representation Theory of Finite Groups (Charlottesville, VA, 1998), de Gruyter, Berlin, 2001, 157–174.
  * [6] Chuang J., Rouquier R., Derived equivalences for symmetric groups and $\mathfrak{sl}_{2}$-categorification, Ann. of Math. 167 (2008), 245–298, math.RT/0407205.
  * [7] Chuang J., Rouquier R., Perverse equivalences, in preparation.
  * [8] Curtis C.W., Truncation and duality in the character ring of a finite group of Lie type, J. Algebra 62 (1980), 320–332.
  * [9] Curtis C.W., Reiner I., Methods of representation theory, Vol. II, With applications to finite groups and orders, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987.
  * [10] Deligne P., Lusztig G., Duality for representations of a reductive group over a finite field, J. Algebra 74 (1982), 284–291.
  * [11] Dipper R., Du J., Harish-Chandra vertices and Steinberg’s tensor product theorems for finite general linear groups, Proc. London Math. Soc. 75 (1997), 559–599.
  * [12] Dreyfus-Schmidt L., Équivalences perverses splendides, conditions de stabilité et catégorification du complexe de Coxeter, Ph.D. Thesis, Paris Diderot – Paris 7, 2014.
  * [13] Dudas O., Varagnolo M., Vasserot E., Categorical actions on unipotent representations I. Finite unitary groups, arXiv:1509.03269.
  * [14] Ford B., Kleshchev A.S., A proof of the Mullineux conjecture, Math. Z. 226 (1997), 267–308.
  * [15] Geck M., On the modular composition factors of the Steinberg representation, J. Algebra 475 (2017), 370–391.
  * [16] Geck M., Hiss G., Malle G., Cuspidal unipotent Brauer characters, J. Algebra 168 (1994), 182–220.
  * [17] Geck M., Hiss G., Malle G., Towards a classification of the irreducible representations in non-describing characteristic of a finite group of Lie type, Math. Z. 221 (1996), 353–386.
  * [18] Gerber T., Hiss G., Jacon N., Harish-Chandra series in finite unitary groups and crystal graphs, Int. Math. Res. Not. 2015 (2015), 12206–12250, arXiv:1408.1210.
  * [19] Hiss G., Harish-Chandra series of Brauer characters in a finite group with a split $BN$-pair, J. London Math. Soc. 48 (1993), 219–228.
  * [20] Kleshchev A.S., Branching rules for modular representations of symmetric groups. III. Some corollaries and a problem of Mullineux, J. London Math. Soc. 54 (1996), 25–38.
  * [21] Lascoux A., Leclerc B., Thibon J.-Y., Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), 205–263.
  * [22] Linckelmann M., Schroll S., A two-sided $q$-analogue of the Coxeter complex, J. Algebra 289 (2005), 128–134.
  * [23] Losev I., Supports of simple modules in cyclotomic Cherednik categories $\mathcal{O}$, arXiv:1509.00526.
  * [24] Losev I., Rational Cherednik algebras and categorification, in Categorification and Higher Representation Theory, Contemp. Math., Vol. 683, Amer. Math. Soc., Providence, RI, 2017, 1–40, arXiv:1509.08550.
  * [25] Misra K., Miwa T., Crystal base for the basic representation of $U_{q}(\widehat{\mathfrak{sl}}(n))$, Comm. Math. Phys. 134 (1990), 79–88.
  * [26] Mullineux G., Bijections of $p$-regular partitions and $p$-modular irreducibles of the symmetric groups, J. London Math. Soc. 20 (1979), 60–66.
  * [27] Seeber J., On Harish-Chandra induction and the Hom-functor, MSc Thesis, RWTH Aachen University, 2016.