Invariants of maximal tori and unipotent constituents of some quasi-projective characters for finite classical groups

TL;DR

This paper analyzes the decomposition of certain characters of finite classical groups, focusing on orbit characters, their multiplicities on maximal tori, and the unipotent constituents of their products with the Steinberg character.

Contribution

It provides explicit computations of multiplicities and unipotent constituents for orbit and Brauer characters associated with specific weights in finite classical groups.

Findings

Computed multiplicities of trivial characters in orbit characters for all maximal tori.

Determined unipotent constituents of characters formed by orbit characters times the Steinberg character.

Analyzed the structure of characters arising from strongly q-restricted weights.

Abstract

We study the decomposition of certain reducible characters of classical groups as the sum of irreducible ones. Let ${\mathbf G}$ be an algebraic group of classical type with defining characteristic $p>0$, $\mu$ a dominant weight and $W$ the Weyl group of ${\mathbf G}$. Let $G=G(q)$ be a finite classical group, where $q$ is a $p$-power. For a weight $\mu$ of ${\mathbf G}$ the sum $s_\mu$ of distinct weights $w(\mu)$ with $w\in W$ viewed as a function on the semisimple elements of $G$ is known to be a generalized Brauer character of $G$ called an orbit character of $G$. We compute, for certain orbit characters and every maximal torus $T$ of $G$, the multiplicity of the trivial character $1_T$ of $T$ in $s_\mu$. The main case is where $\mu=(q-1)\omega$ and $\omega$ is a fundamental weight of ${\mathbf G}$. Let $St$ denote the Steinberg character of $G$. Then we determine the unipotent characters occurring as constituents of $s_\mu\cdot St$ defined to be 0 at the $p$-singular elements of $G$. Let $\beta_\mu$ denote the Brauer character of a representation of $SL_{n}(q)$ arising from an irreducible representation of ${\mathbf G}$ with highest weight $\mu$. Then we determine the unipotent constituents of the characters $\beta_\mu\cdot St$ for $\mu=(q-1)\omega$, and also for some other $\mu$ (called strongly $q$-restricted). In addition, for strongly restricted weights $\mu$, we compute the \mult of $1_T$ in the restriction $\beta_\mu|_T$ for every maximal torus $T$ of $G$.

Equations

$$(\varphi {|}_T,1_T)=\sum \frac{|W(T)|}{|W_L(T^{\prime })|},$$

$$u(\varphi \cdot St)=\sum _{\mathbf{T}}\frac{(\varphi {|}_T,1_T)}{|W(T)|}{\epsilon }_{\mathbf{G}}{\epsilon }_{\mathbf{T}}{R}_{\mathbf{T},1},$$

$$(St,u(\varphi \cdot St))=(St,\varphi \cdot St)=\sum _{\mathbf{T}}\frac{(\varphi {|}_T,1_T)}{|W(T)|},$$

$$St=\sum _{\mathbf{T}}\frac{1}{|W(T)|}{\epsilon }_{\mathbf{G}}{\epsilon }_{\mathbf{T}}{R}_{\mathbf{T},1}.$$

$$u(\varphi \cdot St)=\sum _{\mathbf{T}}\frac{(\varphi {|}_T,1_T)}{|W(T)|}{\epsilon }_{\mathbf{G}}{\epsilon }_{\mathbf{T}}{R}_{\mathbf{T},1}=\sum _{{\mathbf{T}}^{\prime }}\frac{1}{|W(T^{\prime })|}{\epsilon }_{\mathbf{G}}{\epsilon }_{{\mathbf{T}}^{\prime }}{R}_{{\mathbf{T}}^{\prime },1},$$

$$\sum _{{\mathbf{T}}^{\prime }}\frac{1}{|W(T^{\prime })|}{\epsilon }_{\mathbf{G}}{\epsilon }_{{\mathbf{T}}^{\prime }}{R}_{{\mathbf{T}}^{\prime },1}=(\sum _{{\mathbf{T}}^{\prime }}\frac{1}{|W(T^{\prime })|}{\epsilon }_{\mathbf{L}}{\epsilon }_{{\mathbf{T}}^{\prime }}{R}_{{\mathbf{T}}^{\prime },1}^L{)}^{\#G}.$$

$$r(q-1)(q^{l_1}+\cdots +q^{l_k})=r(q-1)(q^{l_1}+\cdots +q^{l_c}+q^{n-(k-c)}+\cdots +q^{n-1})=$$

$$=r(q-1)(q^{l_1}+\cdots +q^{l_c})+r(q^n-1)-r(q^{n-(k-c)}-1).$$

$$(S{t}^{\prime },\varphi \cdot S{t}^{\prime })=\sum _{{\mathbf{T}}^{\prime }}\frac{(\varphi {|}_{T^{\prime }},1_{T^{\prime }})}{|W(T^{\prime })|}=\sum _{{\mathbf{T}}^{\prime }}\frac{q-2+(\chi {|}_T,1_T)}{|W(T^{\prime })|}=\sum _{\mathbf{T}}\frac{(\chi {|}_T,1_T)}{|W(T)|}+(q-2)\sum _{{\mathbf{T}}^{\prime }}\frac{1}{|W(T^{\prime })|}.$$

Full text

Invariants of maximal tori and unipotent constituents of some quasi-projective characters for finite classical groups

A.E. Zalesski

Department of Physics, Mathematics and Informatics, National Academy of Sciences of Belarus, 66 Nezavisimosti prospekt, Minsk, Belarus

alexandre.zalesski@gmail.com

Dedicated to Efim Zelmanov on occasion of his 60th birthday

Abstract We study the decomposition of certain reducible characters of classical groups as the sum of irreducible ones. Let ${\mathbf{G}}$ be an algebraic group of classical type with defining characteristic $p>0$, $\mu$ a dominant weight and $W$ the Weyl group of ${\mathbf{G}}$. Let $G=G(q)$ be a finite classical group, where $q$ is a $p$-power. For a weight $\mu$ of ${\mathbf{G}}$ the sum $s_{\mu}$ of distinct weights $w(\mu)$ with $w\in W$ viewed as a function on the semisimple elements of $G$ is known to be a generalized Brauer character of $G$ called an orbit character of $G$. We compute, for certain orbit characters and every maximal torus $T$ of $G$, the multiplicity of the trivial character $1_{T}$ of $T$ in $s_{\mu}$. The main case is where $\mu=(q-1)\omega$ and $\omega$ is a fundamental weight of ${\mathbf{G}}$. Let $St$ denote the Steinberg character of $G$. Then we determine the unipotent characters occurring as constituents of $s_{\mu}\cdot St$ defined to be 0 at the $p$-singular elements of $G$. Let $\beta_{\mu}$ denote the Brauer character of a representation of $SL_{n}(q)$ arising from an irreducible representation of ${\mathbf{G}}$ with highest weight $\mu$. Then we determine the unipotent constituents of the characters $\beta_{\mu}\cdot St$ for $\mu=(q-1)\omega$, and also for some other $\mu$ (called strongly $q$-restricted). In addition, for strongly restricted weights $\mu$, we compute the multiplicity of $1_{T}$ in the restriction $\beta_{\mu}|_{T}$ for every maximal torus $T$ of $G$.

*Key words: Finite classical groups, Representation theory *

1. Introduction

The groups $G$ under consideration in this papers are $GL_{n}(q)$, $SL_{n+1}(q)$, $Sp_{2n}(q)$, $SO_{2n+1}(q)$, $q$ odd, $SO^{\pm}_{2n}(q)$, $q$ odd, $Spin^{\pm}_{2n}(q)$, $q$ even. Let $\overline{F}_{q}$ be the algebraic closure of finite field $F_{q}$ of $q$ elements. Let ${\mathbf{G}}$ be the respective algebraic group over $\overline{F}_{q}$, and $W$ the Weyl group of ${\mathbf{G}}$. For the notion of a maximal torus in $G$ see [10, 5]. The maximal tori of $G$, up to $G$-conjugation, are in bijection with the conjugacy classes of $W$ unless $G=SO^{-}_{2n}(q)$, $q$ odd, and $Spin^{-}_{2n}(q)$, $q$ even [5, 3.3.3]. So we denote by $T_{w}$ any maximal torus of $G$ from the class corresponding to $w\in W$.

Let ${\mathbf{T}}$ be the group of diagonal matrices in $GL_{n}(\overline{F}_{q})$. Let $\varepsilon_{i}$ be the mapping sending every diagonal matrix $\mathop{\rm diag}\nolimits(x_{1},\ldots,x_{n})$ to the $i$-th entry $x_{i}$ ($1\leq i\leq n$). There is a natural embedding $GL_{n}(\overline{F}_{q})\rightarrow{\mathbf{G}}$ which identifies ${\mathbf{T}}$ with a maximal torus of ${\mathbf{G}}.$ So $\varepsilon_{1},\ldots,\varepsilon_{n}$ can be viewed as weights of ${\mathbf{G}}$, as well as $\sum a_{i}\varepsilon_{i}$ for $a_{i}\in\mathbb{Z}$. Set $\omega_{i}=\varepsilon_{1}+\cdots+\varepsilon_{i}$ ($1\leq i\leq n$). Then $\omega_{i}$ is a fundamental weight of ${\mathbf{G}}$, unless $i=n$ for ${\mathbf{G}}$ of type $B_{n}$ and $i=n-1,n$ for type $D_{n}$. As $W$ acts on the weights of ${\mathbf{G}}$, we may set $W_{i}=\{w\in W:w(\omega_{i})=\omega_{i}\}$. It is well known that $W_{i}$ is the Weyl group of a certain Levi subgroup ${\mathbf{L}}_{i}$ of ${\mathbf{G}}$. For finite groups $A\subset B$ denote by $1_{A}$ the trivial character of $A$ and by $1_{A}^{B}$ the induced character.

Theorem 1.1**.**

Let ${\mathbf{G}},G,W$ be as above, $G\neq SO^{-}_{2n}(q),Spin^{-}_{2n}(q)$, and for $w\in W$ let $T_{w}$ be a respective maximal torus in $G$. Let $\mu=(q-1)\omega_{i}$, where $i\in\{1,\ldots,n\}$. Then the number of distinct weights $g(\mu)$ $(g\in W)$ such that $g(\mu)(T_{w})=1$ is equal to $1_{W_{i}}^{W}(w)$. This also equals the number of distinct weights $g(\omega_{i})$ $(g\in W)$ such that $g(\omega_{i})(T_{w})\subseteq F_{q}$.

Unipotent characters are introduced by Deligne and Lusztig [9]. For any character $\sigma$ of $G$ we denote by $u(\sigma)$ the “unipotent part” of $\sigma$, which is the sum of all unipotent irreducible constituents of $\sigma$ regarding multiplicities. For the notions of Harish-Chandra induction and the Steinberg character see [10, Ch. 4,9]. If $\tau$ is a character of a Levi subgroup $L$ of $G$ then $\tau^{\#G}$ denotes the Harish-Chandra induced character.

Theorem 1.2**.**

Let ${\mathbf{G}},G,W,\omega_{i},W_{i}$ be as above, $\mu=(q-1)\omega_{i}$ $(1\leq i\leq n)$ and let $s_{\mu}$ be the orbit character of $G$ coresponding to $\mu$. If $G=SO^{-}_{2n}(q)$ or $Spin^{-}_{2n}(q)$, assume $j<n$. Then $u(s_{\mu}\cdot St)=St_{L_{i}}^{\#G}$, where $L_{i}$ is a Levi subgroup of $G$ with Weyl group $W_{i}$ and $St_{i}$ the Steinberg character of $L_{i}$.

For special linear groups we have more precise results. For a dominant weight $\nu$ of ${\mathbf{G}}=SL_{n+1}(\overline{F}_{q})$ one can write $\nu=a_{1}\lambda_{1}+\cdots+a_{n}\lambda_{n}$, where $\lambda_{1},\ldots,\lambda_{n}$ are fundamental weights and $a_{1},\ldots,a_{n}$ are non-negative integers. Then $\nu$ is called $q$-restricted if $0\leq a_{1},\ldots,a_{n}<q$. Let $\nu=\nu_{0}+\nu_{1}p+\cdots+\nu_{m-1}p^{m-1}$ be the “Steinberg expansion” of $\nu$, where $\nu_{0},\ldots,\nu_{m-1}$ are $p$-restricted weights. We say that $\nu$ is strongly $p$-restricted if $a_{1}+\cdots+a_{n}<p$ and strongly $q$-restricted if each weight $\nu_{0},\ldots,\nu_{m-1}$ is strongly $p$-restricted.

Theorem 1.3**.**

Let $\rho_{\nu}$ be an irreducible representation of ${\mathbf{G}}=SL_{n+1}(\overline{F}_{q})$ with highest weight $\nu$ and $d_{0}$ the multiplicity of weight [math] of $\rho_{\nu}$. Let $G=SL_{n+1}(q)$, $q=p^{m}$, and let $\beta_{\nu}$ be the Brauer character of $\rho_{\nu}|_{G}$. Let $T=T_{w}$ be a maximal torus of $G$. Suppose that $\nu$ is strongly $q$-restricted.

$(1)$* Suppose that $\nu\neq(q-1)\lambda_{i}$ for every $i\in\{1,\ldots,n\}$. Then $(\beta_{\nu}|_{T},1_{T})=d_{0}$ and $u(\beta_{\nu}\cdot St)=d_{0}\cdot St$.*

$(2)$* Suppose that $\nu=(q-1)\lambda_{i}$ for some $i\in\{1,\ldots,n\}$, and let $L_{i}$ be a Levi subgroup of $G$ whose Weyl group is $W_{i}$. Then $(\beta_{\nu}|_{T},1_{T})=d_{0}+1_{W_{i}}^{W}(w)$ and $u(\beta_{\nu}\cdot St)=d_{0}\cdot St+St_{L_{i}}^{\#G}$, where $d_{0}\leq 1$.*

As $\beta_{\nu}\cdot St$ is well known to be the Brauer character of a projective $\overline{F}_{q}G$-module, the results of this paper may be useful for study of the decomposition matrix of the above groups for natural characteristic $p$. In some cases the characters $s_{\mu}\cdot St$ are characters of projective modules however the question for which $\mu$ this happens in general remains open, see [13, Ch. 9,10]. For the cases discussed above $s_{\mu}\cdot St$ is expected to be a proper character vanishing at all elements of order multiple to $p$. Such characters are called quasi-projective in [20] and $p$-vanishing in [15].

2. Deligne-Lusztig characters and $L$-controlled functions

Notation Throughout this section $G$ is a finite reductive group in defining characteristic $p$, that is, a group of shape $G={\mathbf{G}}^{F}$, where ${\mathbf{G}}$ is a connected reductive algebraic group, $F$ a Frobenius morphism of ${\mathbf{G}}$ and ${\mathbf{G}}^{F}$ stands for the set of elements fixed by $F$. The action of $F$ on ${\mathbf{G}}$ induces an action on every $F$-stable maximal torus ${\mathbf{T}}$, and hence on $N_{{\mathbf{G}}}({\mathbf{T}})$. A subgroup $T$ of $G$ is called a maximal torus if $T={\mathbf{T}}^{F}$ for an $F$-stable maximal torus ${\mathbf{T}}$ of ${\mathbf{G}}$. (In particular, a choice of a maximal torus $T$ means that one has also chosen ${\mathbf{T}}$. The same convention is used for the term ‘a parabolic subgroup of $G$’.) For an $F$-stable maximal torus ${\mathbf{T}}$ we set $W(T)=(N_{{\mathbf{G}}}({\mathbf{T}})/{\mathbf{T}})^{F}=N_{{\mathbf{G}}}({\mathbf{T}})^{F}/T$ [10, 3.13].

Let $W_{\mathbf{G}}=N_{{\mathbf{G}}}({\mathbf{T}})/{\mathbf{T}}$ be the Weyl group of ${\mathbf{G}}$ (we often drop the subscript and write $W$ for $W_{\mathbf{G}}$ ). As the group $W$ is finite, $F$ induces an automorphism of $W$. The set $\{x\in W:x^{-1}wF(x)\}$ is called the $F$-conjugacy class of $w\in W$, and the set $FC_{W}(w)=\{x\in W:x^{-1}wF(w)=w\}$ is called the $F$-centralizer of $w$. (This is meaningful for any finite group $K$ with an automorphism $F$.) The $G$-conjugacy classes of $F$-stable maximal tori in ${\mathbf{G}}$ are in bijection with the $F$-conjugacy classes in $W$, and we denote by ${\mathbf{T}}_{w}$ a representative of the class corresponding to $w\in W$. We set $T_{w}={\mathbf{T}}^{F}_{w}$. Then $W(T_{w})\cong FC_{W}(w)$ [5, 3.3.6]. The torus ${\mathbf{T}}_{1}$ (that is, for $w=1$) is called the reference torus. The group $G$ is called non-twisted if $F$ acts on $W$ trivially; in this case $W(T_{w})\cong C_{W}(w)$. To every $F$-stable maximal torus ${\mathbf{T}}$ the Deligne-Lusztig theory corresponds a generalized character $R_{{\mathbf{T}},1}$. If $F$-stable maximal tori ${\mathbf{T}}$ and ${\mathbf{T}}^{\prime}$ are $G$-conjugate then $R_{{\mathbf{T}},1}=R_{{\mathbf{T}}^{\prime},1}$. An irreducible character $\chi$ of $G$ is called unipotent if $(\chi,R_{{\mathbf{T}},1})\neq 0$ for an $F$-stable maximal torus ${\mathbf{T}}$.

To every reductive group ${\mathbf{G}}$ with a Frobenius morphism $F$ one corresponds the number $\varepsilon_{\mathbf{G}}=(-1)^{r}$, where $r$ is the $F_{q}$-rank of ${\mathbf{G}}$ (see [10, pp.64,66]). This is meaningful for an $F$-stable maximal torus ${\mathbf{T}}$ of ${\mathbf{G}}$. Recall that $\varepsilon_{{\mathbf{T}}}=\varepsilon_{{\mathbf{T}}^{\prime}}$ if ${\mathbf{T}}^{\prime}$ is $G$-conjugate to ${\mathbf{T}}$.

The notions of parabolic and Levi subgroups in ${\mathbf{G}}$ and $G$ are standard [10]. (So a Levi subgroup of $G$ means a Levi subgroups of a parabolic subgroup of $G.$) The orbit characters $s_{\nu}$ are defined in the introduction, see Humphreys [13, §9.6] for more details.

Let $\phi$ be a class function on $G$. Then $\phi|_{T}$ means the restriction of $\phi$ to $T$, and, if $\theta$ is a character of $T$ then $(\phi|_{T},\theta)$ is the inner product of these functions on $T$.

2.1. $L$-controlled functions

Definition 2.1**.**

Let $G$ be a finite reductive group and $L$ a Levi subgroup of $G.$ A function $\phi$ on G is called L-controlled if for every F-stable maximal torus ${\mathbf{T}}$ of ${\mathbf{G}}$ we have

[TABLE]

where the sum is over representatives of the $L$-conjugacy classes of maximal tori $T^{\prime}$ of $L$ that are $G$-conjugate to $T$. (The right hand side is defined to be zero if $L$ contains no torus $G$-conjugate to $T$.)

Clearly, the values of $\phi$ at the non-semisimple elements are irrelevant for $\phi$ being $L$-controlled. Note that an $L$-controlled function is non-zero.

Lemma 2.2**.**

Let $\phi$ be a generalized character of $G$ and $St$ the Steinberg character. Then

[TABLE]

and

[TABLE]

where in both the sums ${\mathbf{T}}$ ranges over representatives of all $G$-conjugacy classes of F-stable maximal tori of ${\mathbf{G}}$, and $T={\mathbf{T}}^{F}$. In addition, $u(\phi\cdot St)=0$ if and only if $(\phi|_{T},1_{T})=0$ for every maximal torus of $G$.

Proof. Formula (3) is equivalent to formula (2) in [12, p.1911] where one takes $s=1$. For the additional claim see [12, Lemma 2.6].

To prove (3), let ${\mathbf{T}},{\mathbf{T}}^{\prime}$ be $F$-stable maximal tori of ${\mathbf{G}}$, and $T={\mathbf{T}}^{F},T^{\prime}={\mathbf{T}}^{F}$. The Deligne-Lusztig characters $R_{T,1},R_{T^{\prime},1}$ are orthogonal if tori ${\mathbf{T}},{\mathbf{T}}^{\prime}$ are not $G$-conjugate, and $(R_{T,1},R_{T,1})=|W(T)|$ [10, 11.16]. By [5, 7.6.6], we have

[TABLE]

Therefore, $St$ is a unipotent character, and hence $(St,u(\phi\cdot St))=(St,\phi\cdot St)$. So (4) and (2) yield (3).

Theorem 2.3**.**

Let $\phi$ be an L-controlled function on G. Then $u(\phi\cdot St)=St_{L}^{\#G}$.

Proof. By Definition 2.1, $(\phi|_{T},1_{T})=0$ if $T$ is not $G$-conjugate to a torus in $L$, otherwise we have $\frac{(\phi|_{T},1_{T})}{|W(T)|}=\sum_{T^{\prime}}\frac{1}{|W_{L}(T^{\prime})|}$, where the sum is over the $L$-conjugacy classes of $F$-stable maximal tori ${\mathbf{T}}^{\prime}\subset{\mathbf{L}}$ that are $G$-conjugate to ${\mathbf{T}}$. Therefore, (2) implies

[TABLE]

where the right hand side sum is over representatives of the $L$-conjugacy classes of $F$-stable maximal tori ${\mathbf{T}}^{\prime}\subset{\mathbf{L}}$.

Note that $\varepsilon_{{\mathbf{G}}}\varepsilon_{{\mathbf{T}}}=\varepsilon_{{\mathbf{L}}}\varepsilon_{{\mathbf{T}}^{\prime}}$ and $R_{{\mathbf{T}}^{\prime},1}=(R^{L}_{{\mathbf{T}}^{\prime},1})^{\#G}$, see [5, 7.4.4] (here $R^{L}_{{\mathbf{T}}^{\prime},1}$ is the Deligne-Lusztig character of $L$). So

[TABLE]

By formula (4), applied to $L$, $\sum_{{\mathbf{T}}^{\prime}}\frac{1}{|W(T^{\prime})|}\varepsilon_{{\mathbf{L}}}\varepsilon_{{\mathbf{T}}^{\prime}}R^{L}_{{\mathbf{T}}^{\prime},1}=St_{L}$. So the result follows.

Lemma 2.4**.**

Let $A\subset B$ be finite groups, $F$ an automorphism of B such that $F(A)=A$ and $a\in A$. Denote by $f_{A}^{B}(a)$ the number of distinct cosets $gA$ such that $aF(g)\in gA$.

$(1)$* Let $D_{a}$ be the F-conjugacy class of a in B, and $a_{1}=a,a_{2},\ldots,a_{k}\in A$ be representatives of the F-conjugacy classes of elements in $D_{a}\cap A$. Then $f_{A}^{B}(a)=\sum_{i=1}^{k}\frac{|FC_{B}(a)|}{|FC_{A}(a_{i})|}$.*

$(2)$* Let $\tilde{B}=B\cdot\langle F\rangle$ be the semidirect product and $\tilde{A}=A\cdot\langle F\rangle$. Let $\tilde{a}=a\cdot F\in\tilde{A}$. Then $f_{A}^{B}(a)=1_{\tilde{A}}^{\tilde{B}}(\tilde{a})$.*

Proof. (1) Recall that $FC_{B}(a)$ denotes the $F$-centralizer of $a$ in $B$ and similarly $FC_{A}(a_{i})$. Set $X=\{x\in B:x^{-1}aF(x)\in A\}$. Then $|X|=|A|\cdot f_{A}^{B}(a)$. Furthermore, $X$ contains $FC_{B}(a)$, and hence $X$ is the union of cosets $xFC_{B}(a)$. For $c\in A\cap D_{a}$ let $x_{c}\in X$ be such that $x_{c}^{-1}aF(x_{c})=c$. Then all cosets $FC_{B}(a)x_{c}$ are distinct, so $X=\cup_{c\in(A\cap D_{a})}FC_{B}(a)x_{c}$. So $|X|=|FC_{B}(a)|\cdot|A\cap D_{a}|=|FC_{B}(a)|\cdot\sum_{i=1}^{k}\frac{|A|}{|FC_{A}(a_{i})|}$, and the result follows.

(2) Let $BF=FB$ denote the coset of $B$ in $\tilde{B}$ containing $F$. Then $AF\subseteq BF$. Let $b,b^{\prime}\in B$. Then $g^{-1}bF(g)=b^{\prime}$ if and only if $g^{-1}(bF)g=b^{\prime}F$ (as $F(g)$ is $FgF^{-1}$ when the automorphism $F$ is realized as conjugation by $F\in\tilde{B}$. Therefore, $f_{A}^{B}(a)$ equals the number of distinct cosets $gA$ $(g\in B)$ such that $g^{-1}(aF)g\in AF.$

Recall that $1_{\tilde{A}}^{\tilde{B}}(\tilde{a})$ is the number of distinct cosets $h\tilde{A}$ in $\tilde{B}$ such that $\tilde{a}\cdot h\tilde{A}=h\tilde{A}$. As $\tilde{B}=B\tilde{A}$, coset representatives can be chosen in $B$, so this number equals the number of distinct cosets $b\tilde{A}$ with $b\in B$ such that $\tilde{a}\cdot b\tilde{A}=b\tilde{A}$, equivalently, $b^{-1}(aF)b\in\tilde{A}$. As every coset $BF^{i}$ is $\tilde{B}$-invariant and $aF\in BF$, we have $b^{-1}(aF)b\in(BF\cap\tilde{A})=AF$. In addition, the cosets $b\tilde{A},b^{\prime}\tilde{A}$ are distinct if and only if so are the cosets $bA,b^{\prime}A$ in $B$. So $1_{\tilde{A}}^{\tilde{B}}(\tilde{a})$ equals the number of distinct cosets $bA$ in $B$ such that $b^{-1}(aF)b\in AF$.

Let ${\mathbf{G}}$ be a connected reductive algebraic group with Frobenius morphism $F$, ${\mathbf{T}}_{1}$ a reference $F$-stable maximal torus of ${\mathbf{G}}$ and $W_{\mathbf{G}}=N_{{\mathbf{G}}}({\mathbf{T}}_{1})/{\mathbf{T}}_{1}$ the Weyl group of ${\mathbf{G}}$. Denote by $F_{1}$ the automorphism of $W_{\mathbf{G}}$ obtained from the action of $F$ on $N_{{\mathbf{G}}}({\mathbf{T}}_{1})$. Set $\tilde{W}=W_{\mathbf{G}}\cdot\langle F_{1}\rangle$ and $\tilde{W}_{{\mathbf{L}}}=W_{{\mathbf{L}}}\cdot\langle F_{1}\rangle$ for an $F$-stable Levi subgroup of ${\mathbf{G}}$ containing ${\mathbf{T}}_{1}$. In this notation we have

Lemma 2.5**.**

For $w\in W_{\mathbf{G}}$ let ${\mathbf{T}}_{w}\subset{\mathbf{G}}$ be an F-stable maximal torus corresponding to $w$. Then the right hand side of formula $(\ref{co5})$ in Definition 2.1 coincides with $f_{W_{{\mathbf{L}}}}^{W}(w)=1_{\tilde{W}_{{\mathbf{L}}}}^{\tilde{W}}(wF_{1})$. In particular, if $G$ is non-twisted then this coincides with $1_{W_{{\mathbf{L}}}}^{W}(w)$.

Proof. Let $T_{w}={\mathbf{T}}^{F}$ and $T^{\prime}={\mathbf{T}}_{w^{\prime}}$ for $w^{\prime}\in W_{\mathbf{L}}$. Then $|W(T)|/|W_{L}(T^{\prime})|=|FC_{W}(w)|/|FC_{W_{\mathbf{L}}}(w)|$ or 0 if ${\mathbf{T}}$ is not $G$-conjugate to an $F$-stable maximal torus in ${\mathbf{L}}.$ By Lemma 2.4(1) with $A=W_{\mathbf{L}}$, $B=W_{\mathbf{G}}$, $a=w$, the right hand side of (1) in Definition 2.1 equals $f_{W_{\mathbf{L}}}^{W}(w)$, and this is equal to $1_{\tilde{W}_{{\mathbf{L}}}}^{\tilde{W}}(wF_{1})$ by Lemma 2.4(2). For non-twisted groups $F_{1}$ is the trivial automorphism of $W$, whence the result.

3. Maximal tori in classical groups

Let ${\mathbf{G}}$ be a reductive algebraic group, $F$ a Frobenius morphism of ${\mathbf{G}}$ and $G={\mathbf{G}}^{F}$. Let ${\mathbf{T}}_{1}$ be a maximal torus of ${\mathbf{G}}$, $W=N_{\mathbf{G}}({\mathbf{T}}_{1})/{\mathbf{T}}_{1}$ the Weyl group of ${\mathbf{G}}$ and $w\in W$. Set $D_{w}:={\mathbf{T}}_{1}^{w^{-1}\circ F}=\{t\in{\mathbf{T}}_{1}:w^{-1}F(t)=t\}=\{t\in{\mathbf{T}}_{1}:F(t)=w(t)\}$. It is known that $D_{w}$ is conjugate to $T_{w}$ in ${\mathbf{G}}$, see [5, the proof of Proposititon 3.3.6].

For our purpose we need to describe $D_{w}$ explicitly, and we do this in a way which facilitates further computations. So we choose representatives $w$ of the $F$-conjugacy classes of $W$ so that the action of $w$ on ${\mathbf{T}}_{1}$ and also $D_{w}$ to be well described. The choice of $w$ is called canonical, and the group $D_{w}$ is called the canonical $w$-torus.

This is done in terms of the action of $w$ on ${\rm Hom}\,({\mathbf{T}}_{1},\overline{F}^{*}_{q})$, the group of algebraic group homomorphisms from ${\mathbf{T}}_{1}$ into $\overline{F}^{*}_{q}$, the multiplicative group of $\overline{F}_{q}$. The elements of ${\rm Hom}\,({\mathbf{T}}_{1},\overline{F}^{*}_{q})$ are called the weights of ${\mathbf{T}}_{1}$ (and also of ${\mathbf{G}}$). If $\dim{\mathbf{T}}=n$ then ${\rm Hom}\,({\mathbf{T}}_{1},\overline{F}^{*}_{q})\cong\mathop{\mathbb{Z}}\nolimits^{n}$ ($\mathop{\mathbb{Z}}\nolimits$ is the ring of integers and $\mathop{\mathbb{Z}}\nolimits^{n}$ is a free $\mathop{\mathbb{Z}}\nolimits$-module of rank $n$).

We first illustrate our strategy on the example of ${\mathbf{G}}=GL_{n}(\overline{F}_{q})$. In this case $W\cong S_{n}$, where we specify $S_{n}$ to act on the set $\{1,\ldots,n\}$. The conjugacy classes of $W$ are parameterized by partitions $\pi=[n_{1},\ldots,n_{k}]$ of $n$. We fix a canonical representative of the conjugacy class corresponding to $\pi$ assuming that $w(1)=n_{1}$, $w(n_{1}+\cdots+n_{j}+1)=n_{1}+\cdots+n_{j+1}$ for $j=2,\ldots,k-1$ and $w(i)=i-1$ for all other numbers $i$ with $1\leq i\leq n$.

The reference torus ${\mathbf{T}}_{1}$ of ${\mathbf{G}}$ can be chosen to be the group of diagonal matrices. So every $t\in{\mathbf{T}}_{1}$ can be written as $t=\mathop{\rm diag}\nolimits(x_{1},\ldots,x_{n})$, $x_{1},\ldots,x_{n}\in\overline{F}^{\times}_{q}$, or simply $(x_{1},\ldots,x_{n})$. The Weyl group $W$ acts on ${\mathbf{T}}_{1}$ by permuting $(x_{1},\ldots,x_{n})$ in a similar way, that is, $w(x_{1},\ldots,x_{n})=(x_{2},\ldots,x_{n_{1}-1},x_{1},x_{n_{1}+2},\ldots,x_{n_{1}+n_{2}-1},x_{n_{1}+1},\ldots)$ for $w\in W$.

The mappings $\varepsilon_{i}:{\mathbf{T}}_{1}\rightarrow\overline{F}_{q}$ given by $\varepsilon_{i}(t)=x_{i}$ ($i=1,\ldots,n$) are sometimes called the Bourbaki weights. Note that $\{\varepsilon_{1},\ldots,\varepsilon_{n}\}$ is a basis of $\mathop{\mathbb{Z}}\nolimits^{n}$, and writing $t=(x_{1},\ldots,x_{n})\in{\mathbf{T}}_{1}$ is equivalent to saying that $\varepsilon_{i}(t)=x_{i}$. The other weights of ${\mathbf{T}}_{1}$ are integral linear combinations of $\varepsilon_{1},\ldots,\varepsilon_{n}$. If $\lambda=\sum z_{i}\varepsilon_{i}$ ($z_{1},\ldots,z_{n}\in\mathop{\mathbb{Z}}\nolimits$) then $\lambda({\mathbf{T}}_{1})=x_{1}^{z_{1}}\cdots x_{n}^{z_{n}}$. The action of the Weyl group $W$ on the weights of ${\mathbf{T}}_{1}$ is defined by the formula $w(\lambda)(t)=\lambda(w(t))$ [5, p.18]. As $\varepsilon_{j}(t)=x_{j}$, we have $w(\varepsilon_{j})(t)=\varepsilon_{j}(w(t))=x_{i+1}$ unless $i=1$ or $n_{1}+\cdots+n_{m}+1$ for some $m\in\{2,\ldots,k\}$. Whence $w(\varepsilon_{j})=\varepsilon_{w^{-1}(j)}$. In particular, $w(\varepsilon_{i})=\varepsilon_{i+1}$ if $i\neq n_{1}+\cdots+n_{l}$ for some $l\in\{1,\ldots,k\}$. So the action of $W$ on $\varepsilon_{1},\ldots,\varepsilon_{n}$ is dual to that on $\{1,\ldots,n\}$. Observe that the $W$-orbit of $\varepsilon_{1}+\cdots+\varepsilon_{j}$ $(1\leq j\leq n)$ consists of weights $\varepsilon_{m_{1}}+\cdots+\varepsilon_{m_{j}}$, where $1\leq m_{1}<\cdots<m_{j}\leq n$.

3.1. Canonical representatives of the conjugacy classes of $W$

In general, let ${\mathbf{G}}\in\{SL_{n+1}(\overline{F}_{q})$, $Sp_{2n}(\overline{F}_{q}),SO_{2n+1}(\overline{F}_{q})$, $q$ odd, $SO_{2n}(\overline{F}_{q})$, $q$ odd, $Spin_{2n}(\overline{F}_{q})$, $q$ even$\}$. (If $q$ is even then $SO_{2n}(\overline{F}_{q})$ is not a connected algebraic group. So we replace this group by $Spin_{2n}(\overline{F}_{q})$.) Furthermore, there is an injective algebraic group homomorphism $GL_{n}(\overline{F}_{q})\rightarrow{\mathbf{G}}$, which identifies ${\mathbf{T}}_{1}$ with a maximal torus in ${\mathbf{G}}$. This defines the weights $\varepsilon_{1},\ldots,\varepsilon_{n}$ for the above groups ${\mathbf{G}}$.

Suppose first that ${\mathbf{G}}=SO_{2n+1}(\overline{F}_{q})$ with $q$ odd, or $Sp_{2n}(\overline{F}_{q})$. The Weyl group $W$ of ${\mathbf{G}}$ in both the cases is isomorphic to the semidirect product of $S_{n}$ with abelian normal subgroup of order $2^{n}$ acting on ${\mathbf{T}}_{1}$ by sending $(x_{1},\ldots,x_{n})$ to $(x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1})$. Therefore, $W$ acts on $\mathop{\mathbb{Z}}\nolimits^{n}\cong{\rm Hom}\,({\mathbf{T}}_{1},\overline{F}^{\times}_{q})$ by sending every $\varepsilon_{i}$ to $\varepsilon_{j}$ or $-\varepsilon_{j}$ for some $j$.

It is well known [6] that the conjugacy classes of $W$ are parameterized by double partitions $\pi=[n_{1},\ldots,n_{k},n_{k+1},\ldots,n_{k+l}]$, where $n_{1}\geq n_{2}\geq\cdots\geq n_{k}$, $n_{k+1}\geq n_{k+2}\geq\cdots\geq n_{k+l}$ and $n_{1}+\cdots+n_{k+l}=n$. To avoid confusion we shall write $\pi=[n_{1},\ldots,n_{k},n^{*}_{k+1},\ldots,n^{*}_{k+l}]$. We allow $k=0$ or $l=0$, in these cases we write $[-,n^{*}_{k+1},\ldots,n^{*}_{k+l}]$ and $[n_{1},\ldots,n_{k},-]$, respectively. The canonical form of an element $w\in W$ labelled by $\pi$ can be described as follows.

Let $t=(x_{1},\ldots,x_{n})\in{\mathbf{T}}_{1}$ and set $n^{\prime}:=n_{1}+\cdots+n_{k}$. Then $w$ simply permutes $x_{1},\ldots,x_{n^{\prime}}$ in the way described above for $GL_{n}(\overline{F}_{q})$. (For instance $x_{i}$ ($i\leq n^{\prime}$) goes to the $(i-1)$-th position for $1<i\leq n_{1}$ etc.) Let $n^{\prime}<i$. Then $w$ puts $t_{i}$ on the $(i-1)$-th position provided $i\neq n^{\prime}+n^{*}_{k+1}+\cdots+n^{*}_{k+j}+1$ for some $j<l$. Otherwise, if $i=n^{\prime}+n^{*}_{k+1}+\cdots+n^{*}_{k+j}+1$ for some $j<l$, then $w$ puts $t_{i}^{-1}$ on the $(n^{\prime}+n^{*}_{k+1}+\cdots+n^{*}_{k+j+1})$-th position. (For instance, if the double partition is $[2,3^{*}]$ then $w$ sends $(x_{1},x_{2},x_{3},x_{4},x_{5})$ to $(x_{2},x_{1},x_{4},x_{5},x_{3}^{-1})$.)

It follows that the dual action of $W$ on Hom$({\mathbf{T}}_{1},\overline{F}_{q})$ preserves the set $\{\pm\varepsilon_{1},\ldots,\pm\varepsilon_{n}\}$. In particular, $w$ permutes $\varepsilon_{1},\ldots,\varepsilon_{n^{\prime}}$ by the rule described for the $GL_{n}(\overline{F}_{q})$-case. Let $i>n^{\prime}$ and let $r(j)=n_{1}+\cdots+n_{k}+n^{*}_{k+1}+\cdots+n^{*}_{k+j}$ for some $j\in\{1,\ldots,l\}$. Then $w(\varepsilon_{i})=\varepsilon_{i+1}$ unless $i=r(j)$ for some $j$, and $w(\varepsilon_{r(j)})=-\varepsilon_{r(j-1)+1}$.

Let ${\mathbf{G}}=SO_{2n}(\overline{F}_{q})$, $q$ odd, or $Spin_{2n}(\overline{F}_{q})$, $q$ even. Then the conjugacy classes of $W$ are parameterized by double partitions $[n_{1},\ldots,n_{k},n^{*}_{k+1},\ldots,n^{*}_{k+l}]$ with $l$ is even, except for the cases where $l=0$ and each number $n_{1},\ldots,n_{k}$ is even [6, Prop 25]. In the exceptional cases there are two conjugacy classes corresponding to $[n_{1},\ldots,n_{k},-]$. For a canonical representative of one class we choose the permutation $w$ of $\varepsilon_{1},\ldots,\varepsilon_{n}$ described above for $GL_{n}(\overline{F}_{q})$. A canonical representative $w^{\prime}$ of the second class is chosen as follows. We set $w^{\prime}(\varepsilon_{1})=-w(\varepsilon_{1})$, $w^{\prime}(\varepsilon_{n_{1}})=-w(\varepsilon_{n_{1}})$ and $w^{\prime}(\varepsilon_{i})=w(\varepsilon_{i})$ for $i\neq 1,n_{1}$ ($1\leq i\leq n$).

3.2. Canonical tori

(A) Suppose first that $G=GL_{n}(q)$ and ${\mathbf{T}}_{1}$ is the group of diagonal matrices. Then $F(t)=t^{q}$ for $t\in{\mathbf{T}}_{1}$. Let $w\in W\cong S_{n}$ be a canonical representative of the conjugacy class corresponding to a partition $[n_{1},\ldots,n_{k}]$ of $n$. If $k=1,n_{1}=n$, then $w(i)=i-1$ for $i=2,\ldots,n$, and $w(1)=n$. Then $F(t)=t^{q}=w(t)$ if and only if $t=(b,b^{q},\ldots,b^{q^{n-1}})$ for $b\in\overline{F}_{q}$ with $b^{q^{n}-1}=1.$ This is exactly the canonical $w$-torus $D_{w}$ for this $w$. If one fixes a primitive $(q^{n}-1)$-root of unity $a$, then $D_{w}$ is the group generated by $(a,a^{q},\ldots,a^{q^{n}-1})\in{\mathbf{T}}_{1}$. This special case illustrates for a reader what we do in general.

For an arbitrary partition $\pi=[n_{1},\ldots,n_{k}]$ we consider the group of diagonal matrices $\mathop{\rm diag}\nolimits(D_{1},\ldots,D_{k})$, where $D_{i}$ is generated by a matrix $\mathop{\rm diag}\nolimits(a_{i},a_{i}^{q},\ldots,a_{i}^{q^{n_{i}-1}})$ and $a_{i}$ is an arbitrary primitive $(q^{n_{i}}-1)$-root of unity. One observes that this group coincides with $D_{w}$ for the canonical choice of $w=w(\pi)$ as described above.

(B) The situation with other classical groups is similar but requires adjustments. We start with groups ${\mathbf{G}}=SO_{2n+1}(\overline{F}_{q})$ with $q$ odd, or $Sp_{2n}(\overline{F}_{q})$. Then the conjugacy classes of $W$ are in bijection with the double partitions $\pi=[n_{1},\ldots,n_{k},n_{k+1}^{*},\ldots,n^{*}_{k+l}]$.

Suppose first that $k=0,l=1$. For $t=(x_{1},\ldots,x_{n})\in{\mathbf{T}}_{1}$ we have $w(t)=(x_{2},\ldots,x_{n},x_{1}^{-1})$ and $F(t)=(x^{q}_{1},\ldots,x^{q}_{n})$. The equality $w(t)=F(t)$ implies $x_{i}=x_{i-1}^{q}$ for $i=2,\ldots,n$ and $x_{1}^{-1}=x_{n}^{q}$. So $x_{n}=x_{1}^{q^{n}}=x_{1}^{-1}$, whence $x_{1}^{q^{n}+1}=1$. It follows that the set $\{t\in{\mathbf{T}}_{1}:w(t)=F(t)\}$ coincides with $\{(b,b^{q},\ldots,b^{q^{n_{i}}}):b^{q^{n}+1}=1\}$. In particular, this is a cyclic group of order $q^{n}+1$.

In general, the group $D_{w}=\{t\in{\mathbf{T}}_{1}:w(t)=F(t)\}$ is the direct product of groups $D_{1},\ldots,D_{k+l}$, where $D_{i}$ is a cyclic group of order $q^{n_{i}}-1$ for $i\leq k$ and of order $q^{n_{i}}+1$ for $i>k$. Specifically, if $r=n_{1}+\cdots+n_{i-1}$ then $D_{i}=(1,\ldots,1,b_{i},b_{i}^{q},\ldots,b_{i}^{q^{n_{i}-1}},1,\ldots,1)$, where $b_{i}$ occupies $(r+1)$-th position and $b_{i}^{q^{n_{i}}-1}=1$ for $i\leq k$ and $b_{i}^{q^{n_{i}}+1}=1$ for $i>k$.

(C) Let ${\mathbf{G}}=SO_{2n}(\overline{F}_{q})$, $q$ odd, or $Spin_{2n}(\overline{F}_{q})$, $q$ even , $W=W_{\mathbf{G}}$ and for a moment let $\tilde{W}$ be the Weyl group of $SO_{2n+1}(\overline{F}_{q})$. Then $W$ is a normal subgroup of $\tilde{W}$, and an element of $\tilde{W}$ lies in $W$ if and only if the conjugacy class of it corresponds to the double partition $\pi$ with $l$ even [6, Prop 25]. In particular, the canonical element from such a class lies in $\tilde{W}$. It follows that $D_{w}$ is the same for groups $SO_{2n}(\overline{F}_{q})$ and $SO_{2n+1}(\overline{F}_{q})$ for $w\in W$, as well as for the pair $Spin_{2n}(\overline{F}_{q})$, $q$ even, and $Sp_{2n}(\overline{F}_{q})$, $q$ even. However in the exceptional cases, when $l=0$ and all $n_{1},\ldots,n_{k}$ are even, the Weyl group of the former group has two conjugacy classes corresponding to $\pi=[n_{1},\ldots,n_{k},-]$, so we need to construct a canonical $w$-torus for $w$ from the second class corresponding to $\pi$.

Consider first a special case with $\pi=[n,-]$, $n$ even. If $w$ permutes $\{1,\ldots,n\}$ then the group $\{(b,b^{q},\ldots,b^{q^{n-1}}):b\in F_{q},b^{q^{n}-1}=1\}$ satisfies $w(t)=F(t)=t^{q}$ if $w(1)=n$, $w(i)=i-1$ for $i<n$. A canonical representative $w^{\prime}$ of the other conjugacy class corresponding to $\pi$ is defined to be $w^{\prime}(\varepsilon_{i})=w(\varepsilon_{i})$ for $i\neq 1,n_{1}$, and $w^{\prime}(\varepsilon_{1})=-w(\varepsilon_{1})$, $w^{\prime}(\varepsilon_{n_{1}})=-w(\varepsilon_{n_{1}}).$ So $w(x_{1},\ldots,x_{n})=(x_{2}^{-1},x_{3},\ldots,x_{n-1},x_{1}^{-1})$. Let $t=(b^{-1},b^{q},b^{q^{2}},\ldots,b^{q^{n-1}})\in{\mathbf{T}}_{1}$ with $b\in F_{q}$ and $b^{q^{n}-1}=1$. Then $w^{\prime}(t)=F(t)=t^{q}$, which implies that $t=(b^{-1},b^{q},b^{q^{2}},\ldots,b^{q^{n-1}})$ for some $b\in F_{q}$ with $b^{q^{n}-1}=1$.

Let $k>1$ and $w$ correspond to the double partition $\pi=[n_{1},\ldots,n_{k},-]$ with even parts $n_{1},\ldots,n_{k}$. If $w$ just permutes $\varepsilon_{1},\ldots,\varepsilon_{n}$ then $D_{w}$ is as above, that is, the direct product of subgroups $D_{1},\ldots,D_{k}$, where $D_{i}=(1,\ldots,1,b_{i},b_{i}^{q},\ldots,b_{i}^{q^{n_{i}-1}},1,\ldots,1)$ ($1\leq i\leq k$), $b_{i}$ occupies the $(n_{1}+\cdots+n_{i-1}+1)$-th position and $b_{i}^{q^{n_{i}}-1}=1$. If $w^{\prime}$ is a canonical representative of the other conjugacy class of $W$ corresponding to $\pi$ then $D_{w^{\prime}}$ is the direct product of subgroups $D^{\prime}_{1},D_{2},\ldots,D_{k}$, where $D_{2},\ldots,D_{k}$ are as above, and $D^{\prime}_{1}=\{b_{1}^{-1},b_{1}^{q},\ldots,b_{1}^{q^{n_{1}-1}}\}$ with $b_{1}^{q^{n_{1}}-1}=1.$

(The groups $D_{w},D_{w^{\prime}}$ are known to be conjugate in $O(2n,\overline{F}_{q})$ but not in $SO_{2n}(\overline{F}_{q})$.)

(D) It is well known that $G^{+}=SO^{+}_{2n}(q)$, $q$ odd, can be viewed as a subgroup of $H=SO_{2n+1}(q)$, which agrees with an inclusion ${\mathbf{G}}\subset{\mathbf{H}}=SO_{2n+1}(\overline{F}_{q})$, in the sense that the Frobenius morphism defining $G^{+}$ is the restriction to ${\mathbf{G}}$ of a Frobenius morphism $F$ of ${\mathbf{H}}$ defining $H$. (Similarly, for the pair $Spin_{2n}(\overline{F}_{q})$, $q$ even, and $Sp_{2n}(\overline{F}_{q})$.) Then the maximal reference torus ${\mathbf{T}}_{1}$ of ${\mathbf{H}}$ coincides with ${\mathbf{T}}_{1}$ for ${\mathbf{G}}$, and $F(t)=t^{q}$ for $t\in{\mathbf{T}}_{1}$ for both the groups. This yields an embedding $W_{\mathbf{G}}=N_{{\mathbf{G}}}({\mathbf{T}}_{1})/{\mathbf{T}}_{1}$ into $W_{\mathbf{H}}=N_{{\mathbf{H}}}({\mathbf{T}}_{1})/{\mathbf{T}}_{1}$.

Denote for a moment by $F^{+}$ and $F^{-}$ the Frobenius morphisms of ${\mathbf{G}}$ such that ${\mathbf{G}}^{F^{+}}=G^{+}=SO^{+}_{2n}(q)$ and ${\mathbf{G}}^{F^{-}}=G^{-}=SO^{-}_{2n}(q)$. Then $F^{-}=\sigma\cdot F^{+}$, where $\sigma$ is a graph automorphism of ${\mathbf{G}}$ [16, §11]. Moreover, $\sigma^{2}=1$ and $\sigma$ can be realized as a conjugation by an element $d$ of $N_{{\mathbf{H}}}({\mathbf{T}}_{1})$, that is, if $x\in{\mathbf{G}}$ then $\sigma F^{+}(x)=d(F^{+}(x))d^{-1}$. Let $r$ be the projection of $d$ into $W_{\mathbf{H}}$. As $F$, and hence $F^{+}$, act trivially on $W_{\mathbf{H}}$, the action of $F^{-}$ on $W_{\mathbf{G}}$ coincides with the conjugation by $r$. Clearly, $r\notin W_{\mathbf{G}}$. As $W_{\mathbf{G}}$ has index 2 in $W_{\mathbf{H}}$, we have $W_{\mathbf{H}}=W_{\mathbf{G}}\cup rW_{\mathbf{G}}$, and the coset $rW_{\mathbf{G}}$ is invariant in $W_{\mathbf{H}}$.

We need to describe canonical $w$-tori $D_{w}$ for $G^{-}$. For our purpose it is convenient to describe them in terms of group $H$. To avoid confusion, we use notation $D^{-}_{w}$ for canonicall $w$-tori of $G^{-}$ and keep $D_{w}$ for those of $H$. Note that, for uniformity reason, in Lemma 3.1 and Proposition 3.2 we allow $q$ to be even when dealing with the group $SO_{2n+1}(q)\cong Sp_{2n}(q)$.

Lemma 3.1**.**

$D^{-}_{w}=D_{rw}$, and we use for $D^{-}_{w}$ the canonical $rw$-torus $D_{rw}$ of $H$.

Proof. We have $D^{-}_{w}=\{t\in{\mathbf{T}}_{1}:F^{-}(t)=w(t)\}$. As $F^{-}(t)=rF^{+}(t)=rF(t)$ and $r^{2}=1$, we have $F^{-}(t)=rF^{+}(t)=rF(t),$ so the equality $F^{-}(t)=w(t)$ is equivalent to $F(t)=rw(t)$, whence the result.

We summarize the considerations of this section as follows:

Proposition 3.2**.**

$(1)$* Let ${\mathbf{G}}^{F}=G=Sp_{2n}(q)$ or $SO_{2n+1}(q)$. Let $w\in W_{\mathbf{G}}$ be a canonical element corresponding to the double partition $\pi=[n_{1},\ldots,n_{k},n^{*}_{k+1},\ldots,n^{*}_{k+l}]$. Then ${\mathbf{T}}_{1}^{w^{-1}\circ F}=D_{w}$, where $D_{w}=(D_{1},\ldots,D_{k+l})$, as described above.*

$(2)$* Let ${\mathbf{G}}^{F}=G=SO^{+}_{2n}(q)$ and $w\in W_{\mathbf{G}}$ be a canonical element corresponding to $\pi$. If $\pi$ is non-exceptional then $D_{w}$ is as above. If $\pi$ is exceptional then there is one more canonical element $w^{\prime}$ corresponding to $\pi$ that is not $W_{\mathbf{G}}$-conjugate to w. Then $D_{w}$ is as above, and $D_{w^{\prime}}=(D^{\prime}_{1},\ldots,D_{k})$, where $D_{1}^{\prime}=(b_{1}^{-1},b_{1}^{q},\ldots,b_{1}^{q^{n_{1}-1}})$ with $b_{1}\in\overline{F}_{q}$ and $b_{1}^{q^{n_{1}}-1}=1$.*

$(3)$* Let $G=SO^{-}_{2n}(q)$. Then $D^{-}_{w}$ coincides with the canonical torus $D_{v}$ of ${\mathbf{H}}=SO_{2n+1}(\overline{F}_{q})$ for some $v\in W_{\mathbf{H}}\setminus W_{\mathbf{G}}$.*

3.3. Weights and $q$-characters

Let ${\mathbf{G}}$ be an algebraic group with Frobenius morphism $F$, and $G=G^{F}$. For certain weights $\mu$ and every maximal torus $T$ of $G$ we compute $(s_{\mu}|_{T},1_{T})$, the multiplicity of the trivial character of $T$ in the orbit character $s_{\mu}$. It is well known that $(s_{\mu}|_{T},1_{T})$ is the same for any ${\mathbf{G}}$-conjugate of $T$ in ${\mathbf{G}}$. Therefore, it suffices to do this for the canonical representative $D$ of $T$ in ${\mathbf{T}}_{1}$.

Below ${\mathbf{G}}\in\{GL_{n}(\overline{F}_{q}),SL_{n}(\overline{F}_{q}),$ $Sp_{2n}(\overline{F}_{q})$, $SO_{2n+1}(\overline{F}_{q})$, $q$ odd, ${\mathbf{G}}=SO_{2n}(\overline{F}_{q})$, $q$ odd, $Spin_{2n}(\overline{F}_{q})$, $q$ even$\}$. The weights $\varepsilon_{1},\ldots,\varepsilon_{n}$ and $\omega_{1},\ldots,\omega_{n}$ are defined in the introduction. So $G={\mathbf{G}}^{F}$ is a classical group (except a unitary one, which is not considered). As above, $W$ is the Weyl group of ${\mathbf{G}}$ and $W_{j}=C_{W}(\omega_{j})$. We write $W\omega_{j}$ for the $W$-orbit of $\omega_{j}$.

Definition 3.3**.**

Let $T$ be a maximal torus of $G$, and $\theta\in\mathop{\rm Irr}\nolimits T$. We say that $\theta$ is a $q$-character if $\theta(t)^{q-1}=1$ for all $t\in T$. This also applies to Brauer characters $T\rightarrow\overline{F}_{q}$.

Lemma 3.4**.**

Let $\mu$ be a weight of ${\mathbf{G}}$ and T a maximal torus of G. Then $\mu|_{T}$ is a q-character of T if and only if $((q-1)\mu)|_{T}=1_{T}$.

Proof. If $\mu$ is a $q$-character of $T$ then, obviously, $((q-1)\mu)|_{T}=1_{T}$. Conversely, suppose that $((q-1)\mu)|_{T}=1_{T}$. Let $t\in T$, so a weight $\mu$ is a $q$-character of $T$ if and only if $((q-1)\mu(t))=1$, equivalently, $\mu(t)^{q-1}=1$. As $\mu(t)\in\overline{F}_{q}$, and $x^{q-1}=1$ for $x\in\overline{F}_{q}$ implies $x\in F_{q}$, we have $\mu(t)\in F_{q}$, whence the claim.

Lemma 3.5**.**

††margin:

au1

Let $1\leq k\leq n$, $0<r<q$ and $0\leq l_{1}<\cdots<l_{k}<n$ be integers. Then $r(q-1)(q^{l_{1}}+\cdots+q^{l_{k}})$ is not divisible by $q^{n}-1$, unless $k=n$ and $(l_{1},\ldots,l_{n})=(0,1,\ldots,n-1)$.

Proof. Suppose first that $l_{k}\leq n-2$, so $k\leq n-1$. Then $r(q-1)(q^{l_{1}}+\cdots+q^{l_{k}})\leq r(q-1)(q^{n-k-1}+\cdots+q^{n-3}+q^{n-2})=r(q^{n-1}-q^{n-k-1})$. As $r<q$, we have $r(q^{n-1}-q^{n-k-1})<q^{n}-q^{n-k}=q^{n}-1-(q^{n-k}-1)$, which is less than $q^{n}-1$. So we are left with the case $l_{k}=n-1$.

Let $l_{c+1},\ldots,l_{k-1},l_{k}=n-1$ be the longest string of subsequent natural numbers, that is, $(l_{c+1},\ldots,l_{k})=(n-(k-c),\ldots,n-1)$ and $l_{c}<n-(k-c)-1$. So $c+1\leq k$.

If $c=0$ then $(l_{1},\ldots,l_{k})=(n-k,\ldots,n-1)$ and $r(q-1)(q^{l_{1}}+\cdots+q^{l_{k}})=r(q^{n}-q^{n-k})$, so the lemma follows if $q^{n-k}>1$. If $q^{n-k}=1$ then $n=k$ and $(l_{1},\ldots,l_{k})=(0,\ldots,n-1)$, so we are in the exceptional case of the lemma.) So we assume $c>0$.

Then $l_{j}\leq n-2-(k-j)$ for $1\leq j\leq c$. Therefore, $(q-1)(q^{l_{1}}+\cdots+q^{l_{c}})\leq(q-1)(q^{n-k-1}+q^{n-k-2}+\cdots+q^{n-2-(k-c)})=q^{n-1-(k-c)}-q^{n-k-1}<q^{n-1-(k-c)}$ as $c>0$. We have

[TABLE]

[TABLE]

This is a multiple of $q^{n}-1$ if and only if so is $x:=r(q-1)(q^{l_{1}}+\cdots+q^{l_{c}})-r(q^{n-(k-c)}-1)$. It is easy to observe that $x$ is not divisible by $q^{n}-1$. (Use the inequality $r(q-1)(q^{l_{1}}+\cdots+q^{l_{c}})\leq r(q^{n-1-(k-c)}-q^{n-k-1})<r(q^{n-(k-c)}-1)$.)

(A) ${\mathbf{G}}=GL_{n}(\overline{F}_{q})$ or $SL_{n}(\overline{F}_{q})$.

Let $\pi=[n_{1},\ldots,n_{k}]$ be a partition of $n$ and $w\in W$ the canonical element in the conjugacy class determined by $\pi$. Let $B_{1},\ldots,B_{k}$ be the orbits of $w$ on $\{1,\ldots,n\}$. Then $\{1,\ldots,n\}=B_{1}\cup\cdots\cup B_{k}$, where $B_{1}=\{b:0<b\leq n_{1}\}$ and $B_{i}=\{b:n_{1}+\cdots+n_{i-1}<b\leq n_{1}+\cdots+n_{i-1}\}$ for $i=2,\ldots,k$.

Let $D_{w}=(D_{1},\ldots,D_{k})$ be a canonical $w$-torus in ${\mathbf{T}}_{1}$. Recall that $D_{i}$ $(1\leq i\leq k)$ is generated by an element $(a_{i},a_{i}^{q},,\ldots,a_{i}^{q^{n_{i}-1}})$, where $a_{i}\in\overline{F}_{q}$ is a primitive $(q^{n_{i}}-1)$-root of unity. In this notation we have:

Proposition 3.6**.**

Let $G=GL_{n}(q)$ or $SL_{n}(q)$. Set $\mu=g(\omega_{j})$ for some $g\in W$.

$(1)$* $\mu|_{D_{w}}$ is a $q$-character if and only if $w(\mu)=\mu;$*

$(2)$* The number of distinct q-characters $\mu|_{D_{w}}$ $(\mu\in W\omega_{j})$ equals $1_{W_{j}}^{W}(w)$. In addition, $1_{W_{j}}^{W}(w)=(s_{(q-1)\omega_{j}}|_{T},1_{T})$.*

Proof. (1) The orbit $W\varepsilon_{j}$ consists of all weights $\varepsilon_{m_{1}}+\cdots+\varepsilon_{m_{j}}$ $(1\leq m_{1}<\cdots<m_{j}\leq n)$. Let $\mu=\varepsilon_{m_{1}}+\cdots+\varepsilon_{m_{j}}.$

Suppose first that $k=1$, so $B=B_{1}=\{1,\ldots,n\}$. If $G=GL_{n}(q)$ then $D_{w}=\langle d\rangle$ is a cyclic group of order $q^{n}-1$ with generator $d=(a,a^{q},\ldots,a^{q^{n-1}})$, where $a\in\overline{F}_{q}$ is a primitive $(q^{n}-1)$-root of unity. Then $\mu(d)=a^{q^{m_{1}-1}+\cdots+q^{m_{j}-1}}$. This belongs to $F_{q}$ if and only if $(q-1)(q^{m_{1}-1}+\cdots+q^{m_{j}-1})\equiv 0\pmod{(q^{n}-1)}$. If $G=SL_{n}(q)$ then $D_{w}=\langle d^{q-1}\rangle$ is of order $(q^{n}-1)/(q-1)$. So $(\varepsilon_{m_{1}}+\cdots+\varepsilon_{m_{j}})(d^{q-1})\in F_{q}$ if and only if $(q-1)(\varepsilon_{m_{1}}+\cdots+\varepsilon_{m_{j}})(d)\in F_{q}$, equivalently, $(q-1)^{2}(q^{m_{1}-1}+\cdots+q^{m_{j}-1})\equiv 0\,(q^{n}-1)$. By Lemma 3.5, applied to $\{m_{1}-1,\ldots,m_{j}-1\}$ in place of $\{l_{1},\ldots,l_{k}\}$ with $r=1$ or $q-1$, this is not the case unless $j=n$ and $(m_{1},\ldots,m_{j})=(1,\ldots,n)$.

Let $k>1$. Set $B_{i}^{\prime}=\{b\in B_{i}:b\in\{m_{1},\ldots,m_{j}\}\}$. Let $G=GL_{n}(q)$ and let $d_{i}$ be a generator of the subgroup $D_{i}=(\mathop{\rm Id}\nolimits,\ldots,\mathop{\rm Id}\nolimits,D_{i},\mathop{\rm Id}\nolimits,\ldots,\mathop{\rm Id}\nolimits)$ of $D_{w}$. Then $\mu(d_{i})=\sum_{r\in B_{i}^{\prime}}\varepsilon_{r}(d_{i})$. The above argument shows that this is in $F_{q}$ if and only if $B_{i}^{\prime}=B_{i}$. As this is true for every $i$ with non-empty $B_{i}^{\prime}$, it follows that $\{m_{1},\ldots,m_{j}\}$ is the union of $w$-orbits, whence (1).

Let $G=SL_{n}(q)$. Then $D_{w}$ contains a subgroup which is the direct product of cyclic subgroups $D_{i}^{q-1}$ ($1\leq i\leq k$), where $D_{i}=\langle d_{i}\rangle$ is as above for $GL_{n}(q)$. Suppose that $\mu(D_{w})=1$. Then $\mu(d_{i}^{q-1})=\sum_{r\in B_{i}^{\prime}}\varepsilon_{r}(d_{i}^{q-1})=1$. By the argument for $k=1$, this implies $B_{i}^{\prime}=B_{i}$, so again $\{m_{1},\ldots,m_{j}\}$ is the union of $w$-orbits, as desired.

(2) Note that $g(\omega_{j})=h(\omega_{j})$ ($g,h\in W$) if and only if $gW_{j}=hW_{j}$. Let $\mu=g(\omega_{j})$. Then $w(\mu)=\mu$, or $wg(\omega_{j})=g(\omega_{j})$, is equivalent to $wgW_{j}=gW_{j}$. So the number of distinct $q$-characters $g(\omega_{j})$ is equal to the number of $w$-stable cosets $gW_{j}$. This is well known to be equal to the value at $w$ of the character $1_{W_{j}}^{W}$, as claimed.

Remark. Strictly speaking, the proof of Proposition 3.6 does not require the choice of $w$ to be canonical, which only affects the explicit form of the sets $B_{1},\ldots,B_{k}$.

Corollary 3.7**.**

Let $G=SL_{n}(2)$ and let $V_{j}$ $(j<n)$ be the j-th exterior power of the natural $F_{2}G$-module V. Let $T=T_{w}$ be a maximal torus of G, and $V^{T_{w}}_{j}$ the fixed point subspace of $T_{w}$ on $V_{j}$. Then $\dim V^{T_{w}}_{j}=1_{Y}^{S_{n}}(w)$, where $Y$ is the Young subgroup of $S_{n}$ labeled by $[j,n-j]$.

Proof. Let $\rho$ be the irreducible representation of ${\mathbf{G}}=SL_{n}(\overline{F}_{2})$ with highest weight $\omega_{j}=\varepsilon_{1}+\cdots+\varepsilon_{n-1}$. As $\omega_{j}$ is a miniscule weight [3, §7.3], the weights of $\rho$ are $g(\omega_{j})$ for $g\in W$. As $\dim V^{T_{w}}_{j}$ equals the number of distinct weights $g(\omega_{j})$ such that $g(\omega_{j})(T_{w})=1$, we have $\dim V^{T_{w}}_{j}=1_{W_{j}}^{W}(w)$. Since $W\cong S_{n}$ and $W_{j}\cong Y$, the statement follows.

(B) Let $G\in\{Sp_{2n}(q),SO^{+}_{2n}(q),~{}q~{}odd,~{}SO_{2n+1}(q),~{}q~{}odd,~{}Spin_{2n}^{+}(q),~{}q~{}even\}$, and ${\mathbf{G}}$ the respective algebraic group. We use Proposition 3.2 to compute, for every canonical $w$-tori $D_{w}$, the number of weights $\mu$ in the orbit $W\omega_{j}$ whose restriction to $D_{w}$ yields a $q$-character of $D_{w}$. It is well known that $W\omega_{j}$ consists of all weights $\pm\varepsilon_{1}\pm\cdots\pm\varepsilon_{m_{j}}$, except if ${\mathbf{G}}$ is of type $D_{n}$ and $j=n$, where the orbit consists of all weights $\pm\varepsilon_{1}\pm\cdots\pm\varepsilon_{n}$ with even number of minus signs. So the $W_{\mathbf{G}}$-orbit of $\omega_{j}$ is the same for the above groups ${\mathbf{G}}$, except for the case where $j=n$ and ${\mathbf{G}}$ is of type $D_{n}$.

Below the exceptional $w$-torus $D_{w}$ is one in $SO^{+}_{2n}(q)$ for the exceptional canonical element $w\in W$; the corresponding partition $\pi$ is $[n_{1},\ldots,n_{k},-]$, where all $n_{1},\ldots,n_{k}$ are even. Observe that there are one exceptional and one non-exceptional canonical elements corresponding to this partition but the exceptional canonical $w$-torus exists only in $SO^{+}_{2n}(q)$. Sometimes we denote the exceptional canonical element by $w^{\prime}$ and the corresponding torus by $D_{w^{\prime}}$ to avoid confusion.

Lemma 3.8**.**

††margin:

q-sp

Let $G\in\{Sp_{2n}(q),$ q odd, $SO_{2n+1}(q),SO^{+}_{2n}(q),q~{}odd,Spin^{+}_{2n}(q)$, $q$ even$\}$, and $D_{w}$ the canonical $w$-torus for $G$ corresponding to $w\in W$. Let $\mu\in W\omega_{j}$. Then the following conditions are equivalent:

$(1)$* $\mu|_{D_{w}}$ is a q-character of $D_{w};$*

$(2)$* $w(\mu)=\mu$.*

Proof. Let $\pi=[n_{1},\ldots,n_{k},n^{*}_{k+1},\ldots,n^{*}_{k+l}]$ be a double partition corresponding to $w$. Denote by $B_{i}$ ($1\leq i\leq k+l$) the set of natural numbers $r$ in the range $n_{1}+\cdots+n_{i-1}<r\leq n_{1}+\cdots+n_{i}$. So $|B_{i}|=n_{i}$. We use the canonical form of $D_{w}$ described in Proposition 3.2. So $D_{w}=(D_{1},\ldots,D_{k+l})$ and in the exceptional case (where $G=SO^{+}_{2n}(q)$, $q$ odd, or $Spin^{+}_{2n}(q)$, $q$ even, $l=0$ and all $n_{1},\ldots,n_{k}$ are even) we also consider $D_{w^{\prime}}=(D^{\prime}_{1},D_{2},\ldots,D_{k})$. Note that $D_{i}$ is generated by an element $d_{i}=(1,\ldots,1,a_{i},a_{i}^{q}$ $,\ldots,a_{i}^{q^{n_{i}-1}},1,\ldots,1)$ and $D_{1}^{\prime}$ is generated by $d_{1}^{\prime}=(a_{1}^{-1},a_{1}^{q},\ldots,a_{1}^{q^{n_{1}-1}},1,\ldots,1)$.

$(i)$ Suppose that $\pi=[n,-]$ or $[-,n^{*}]$. Then $D_{w}=\langle d\rangle$ is a cyclic group of order $q^{n}-1$ or $q^{n}+1$, respectively, and $d:=(a,a^{q},a^{q^{2}},\ldots,a^{q^{n-1}})\in{\mathbf{T}}_{1}$ for $a\in\overline{F}_{q}$ with $|a|=|d|$. If $\pi=[n,-]$ with $n$ even and $G=SO^{+}_{2n}(q)$, $q$ odd, or $Spin^{+}_{2n}(q)$, $q$ even, then there is also an exceptional canonical element $w^{\prime}$ for which $D_{w^{\prime}}=\langle d^{\prime}\rangle$, where $d^{\prime}:=(a^{-1},a^{q},a^{q^{2}},\ldots,a^{q^{n-1}})$.

Recall that $\mu=\pm\varepsilon_{m_{1}}\pm\cdots\pm\varepsilon_{m_{j}}$ $(1\leq m_{1}<\cdots<m_{j}\leq n)$ for a certain choice of signs. So $\mu(d)=a^{\pm q^{m_{1}-1}\pm\cdots\pm q^{m_{j}-1}}$, as well as $\mu(d^{\prime})$ in the exceptional case. Note that $a^{\pm q^{m_{1}-1}\pm\cdots\pm q^{m_{j}-1}}\in F_{q}$ if and only if $(q-1)(\pm q^{m_{1}-1}\pm\cdots\pm q^{m_{j}-1})\equiv 0\pmod{|a|}$. The left hand side is not 0 as $m_{1},\ldots,m_{j}$ are distinct. Furthermore, the absolute value of the left hand side is strictly less than $|a|$, unless $\{m_{1},\ldots,m_{j}\}=\{1,\ldots,n\}$, $|a|=q^{n}-1$ and the signs of all $\pm 1,\pm q,\ldots,\pm q^{n-1}$ are the same. In particular, $\mu(d)\notin F_{q}$ if $j<n$ or $j=n$ and $|a|=q^{n}+1$.

Let $j=n$, $|a|=q^{n}-1$. If we compute $\mu(d)$ (respectively, $\mu(d^{\prime})$) then the signs in $\pm 1\ldots,\pm q^{n-1}$ are the same if and only if $\mu=\pm(\varepsilon_{1}+\cdots+\varepsilon_{n})$ (respectively, $\mu=\pm(-\varepsilon_{1}+\varepsilon_{2}+\cdots+\varepsilon_{n}))$.

If ${\mathbf{G}}=SO_{2n}(\overline{F}_{q})$, $q$ odd, or $Spin_{2n}(\overline{F}_{q})$, $q$ even, then the weights $\pm(-\varepsilon_{1}+\varepsilon_{2}+\cdots+\varepsilon_{n})$ are not in $W\omega_{n}$, and this possibility is ruled out. However, we use below the above statement for $\mu(d^{\prime})$.

Thus, if (1) holds then $j=n$ and $w$ permutes $\varepsilon_{1},\ldots,\varepsilon_{n}$, so (2) holds.

Conversely, if $w(\mu)=\mu$ then $w$ is not of type $[-,n^{*}]$ as otherwise $w^{n}(\mu)=-\mu$. So $w$ is of type $[n,-]$, and $|a|=q^{n}-1$, which implies (1) with the above observations.

$(ii)$ The general case.

$(1)\rightarrow(2)$ Suppose that $\mu(D_{w})\subset F_{q}$. Then $\mu(d_{i})\in F_{q}$ for every $i=1,\ldots,k+l$ in the non-exceptional case, otherwise $\mu(d_{1}^{\prime})\in F_{q}$ and $\mu(d_{i})\in F_{q}$ for every $i=2,\ldots,k$. Observe that $\mu(d_{i})=\sum_{r}\pm\varepsilon_{r}(d_{i})$, where $r$ runs over $B_{i}^{\prime}:=\{m_{1},\ldots,m_{j}\}\cap B_{i}$, and similarly for $d_{1}^{\prime}$. Suppose that $B_{i}^{\prime}$ is not empty. Then the same reasoning as above with $a_{i}$ in place of $a$ and $\sum_{j\in B_{i}}\varepsilon_{j}$ in place of $\varepsilon_{1}+\cdots+\varepsilon_{n}$ shows that $i\leq k$ and $B_{i}^{\prime}=B_{i}$ (so $\{m_{1},\ldots,m_{j}\}\subseteq B_{1}\cup\cdots\cup B_{k}$).

Furthermore, the argument in (i) tells us that the signs of $\varepsilon_{r}$ are the same for all $r\in B^{\prime}_{i}$ in the non-exceptional case. In the exceptional case this is only true for $i>1$, whereas for $i=1$ all but one signs of $\varepsilon_{r}$ with $r\in B_{1}^{\prime}$ are the same.

In the non-exceptional case $w$ simply permutes $\varepsilon_{r}$ with $r\in B_{i}$ for $i\leq k$, so $w(\mu)=\mu$ as required.

Consider the exceptional canonical torus $D_{w^{\prime}}=(D_{1}^{\prime},D_{2},\ldots,D_{k})$ for $G=SO^{+}_{2n}(q)$, $q$ odd, and $Spin^{+}_{2n}(q)$, $q$ even, so $l=0$, $|B_{i}|=n_{i}$ are even and $|d_{i}|=q^{n_{i}}-1$. Then $d_{1}^{\prime}=(a_{1}^{-1},a_{1}^{q},\ldots,a_{1}^{q^{n_{1}-1}},1,\ldots,1)$, whereas $D_{i}=\langle d_{i}\rangle$ with $i>1$. If $B^{\prime}_{1}$ is empty then $\{m_{1},\ldots,m_{j}\}$ is the union of $B_{i}$ with $1<i\leq k$ and the signs of $\varepsilon_{r}$ are the same for all $r\in B^{\prime}_{i}$. Suppose that $B_{1}^{\prime}$ is non-empty. Then $B_{1}\subseteq\{m_{1},\ldots,m_{j}\}$, so $n_{1}\leq j$ and $(m_{1},\ldots,m_{n_{1}})=(1,\ldots,n_{1})$. As above, if $\mu(d^{\prime}_{1})\in F_{q}$ then the partial sum of $\mu=\pm\varepsilon_{1}\pm\cdots\pm\varepsilon_{n}$ with indices in $B_{1}$ must be $\pm(-\varepsilon_{1}+\varepsilon_{2}+\cdots+\varepsilon_{n_{1}})$. In addition, the partial sums corresponding to $B_{i}$ with $i>1$ must be $\pm(\sum_{r\in B_{i}}\varepsilon_{r})$. If $j<n$ then $w^{\prime}(\varepsilon_{1})=-\varepsilon_{2}$, $w^{\prime}(\varepsilon_{n_{1}})=-\varepsilon_{1}$, and $w^{\prime}(\varepsilon_{i})=\varepsilon_{i+1}$ for $i=2,\ldots,n_{1}-1$, and also $w^{\prime}(\sum_{r\in B_{i}^{\prime}}\varepsilon_{r})$ for $i>1$. Therefore, $w^{\prime}(\pm(-\varepsilon_{1}+\varepsilon_{2}+\cdots+\varepsilon_{n_{1}}))=\pm(-\varepsilon_{1}+\varepsilon_{2}+\cdots+\varepsilon_{n_{1}})$, and (2) follows.

Let $j=n$. Recall that $|B_{i}|=n_{i}$ is even for every $i$. Therefore, the number of minus signs in the expression $\mu=\pm\varepsilon_{1}\pm\cdots\pm\varepsilon_{n}$ is odd. However, such expression cannot be in the $W$-orbit of $\varepsilon_{1}+\cdots+\varepsilon_{n}$, which is a contradiction. Therefore, this case does not occur, and hence $(1)\rightarrow(2)$ for arbitrary $w\in W$.

$(2)\rightarrow(1)$ Let $w(\mu)=\mu$. Then, obviously, the $w$-orbit of each $\varepsilon_{m_{r}}$ $(r=1,\ldots,j)$ is in the set $\{\pm\varepsilon_{m_{1}},\ldots,\pm\varepsilon_{m_{j}}\}$, so $B_{i}^{\prime}=B_{i}\cap\{m_{1},\ldots,m_{j}\}\neq\emptyset$ implies $B_{i}\subseteq\{m_{1},\ldots,m_{j}\}$. Then $i\leq k$ (as otherwise for a partial sum $\nu:=\sum_{r\in B_{i}}\pm\varepsilon_{m_{r}}$ we have $w^{n_{i}}(\nu)=-\nu$, and hence $w^{n_{i}}(\mu)\neq\mu$). If $w$ is non-exceptional, this means that $w$ permutes $\varepsilon_{r}$ for $r\in B^{\prime}_{i}$ for every $i$ (with $B_{i}^{\prime}\neq\emptyset$), so the signs of these $\varepsilon_{r}$’s are the same. This implies that $\mu(D_{w})\subset F_{q}$. Suppose that $w$ is exceptional. This argument again works if $B_{1}^{\prime}=\emptyset$. Otherwise, we have to pay an additional attention to the case where $w=w^{\prime}$ and $D_{w^{\prime}}=(a_{1}^{-1},a_{1}^{q},\ldots,a_{1}^{q^{n_{1}}-1},a_{2},\ldots,a_{2}^{q^{n_{2}-1}},\ldots,a_{k},\ldots,a_{k}^{q^{n^{k}-1}})$. Then $w^{\prime}(\mu)=\mu$ implies $\mu=\pm(-\varepsilon_{1}+\varepsilon_{2}+\cdots+\varepsilon_{n_{1}})+\sum_{i=2}^{k}\pm(\sum_{r\in B_{i}^{\prime}}\varepsilon_{r})$. If $j=n$ then this $\mu$ is not in the $W$-orbit of $\omega_{n}$. If $j<n$ then $\mu(D_{w})\subset F_{q}$, as required.

Lemma 3.9**.**

$(1)$* Let $G\neq SO^{-}_{2n}(q)$, q odd, $Spin^{-}_{2n}(q)$, $q$ even. For $w\in W$ let $T_{w}$ be a maximal torus of $G$. Let $W_{j}=C_{W}(\omega_{j})$. Then the number of distinct weights $g(\omega_{j})$ $(g\in W)$ that yield q-characters of $T_{w}$ equals the number of the cosets $gW_{j}$ such that $wgW_{j}=gW_{j}$. The latter equals $1_{W_{j}}^{W}(w)$.*

$(2)$* Let $G=SO^{-}_{2n}(q)$, q odd, $H=SO_{2n+1}(q)$, $q$ odd, $\tilde{W}=W_{{\mathbf{H}}}$ and $\tilde{W}_{j}=C_{\tilde{W}}(\omega_{j})$ for $j<n$. Let $r\in\tilde{W}$ be as in Lemma 3.1. Then the number of distinct weights $g(\omega_{j})$ $(g\in W)$ that yield q-characters of $T_{w}$ equals $1_{\tilde{W}_{j}}^{\tilde{W}}(wr)$, where r is as in Lemma 3.1. (Similarly, for $G=Spin^{-}_{2n}(q)$, $Spin_{2n}^{-}(q)$, for $q$ even.)*

Proof. It suffices to proof the lemma for the canonical $w$-torus $D_{w}$ in place of $T_{w}$. (1) Let $\mu=g(\omega_{j})$. By Lemma 3.8, $\mu|_{D_{w}}$ is a $q$-character if and only if $w(\mu)=\mu$, equivalently, $(g^{-1}wg)(\omega_{j})=\omega_{j}$, that is, $g^{-1}wg\in W_{j}$. Let $h\in W$. Then $h(\omega_{j})=\mu$ if and only if $h\in gW_{j}$. It follows that the number of $q$-characters in the set $(g(\mu))|_{D_{w}}$ is equal to the number of cosets $gW_{j}$ such that $wgW_{j}=gW_{j}$. This is exactly $1_{W_{j}}^{W}(w)$.

(2) By Lemma 3.1, $D_{w}=\tilde{D}_{wr}$, where $\tilde{D}_{wr}$ is the canonical $wr$-torus of $SO_{2n+1}(q)$. So $\mu(D_{w})=\mu(\tilde{D}_{wr})$, in particular, $\mu(D_{w})$ is a $q$-character if and only if $\mu|_{\tilde{D}_{wr}}$ is a $q$-character. The latter holds if and only if $wr(\mu)=\mu$ by Lemma 3.8. As in (1), the number of such characters equals $1_{\tilde{W}_{j}}^{\tilde{W}}(wr)$, but a priori $h\omega_{j}$ for $h\in\tilde{W}$ with $wr(h\omega_{j})=h\omega_{j}$ cannot be of the shape $g\omega_{j}$ for $g\in W$. However, this is the case if $j<n$, as then the orbits $W\omega_{j}$ and $\tilde{W}\omega_{j}$ coincide, so the result follows.

Remark. The fundamental weights $\lambda_{j}$ of the root system of type $D_{n}$ are $\varepsilon_{1}+\cdots+\varepsilon_{j}$ for $1=1,\ldots,j$ provided $j<n-1$, and $\lambda_{n-1}=\frac{1}{2}(\varepsilon_{1}+\cdots+\varepsilon_{n-1}-\varepsilon_{n})$, $\lambda_{n}=\frac{1}{2}(\varepsilon_{1}+\cdots+\varepsilon_{n-1}+\varepsilon_{n})$ [2, Planche IV]. Therefore, $\omega_{n}=2\lambda_{n}$ and $\omega_{n-1}=\lambda_{n-1}+\lambda_{n}$. The above results for ${\mathbf{G}}=D_{n}$ can be extended to $\lambda_{n-1}$. If $\mu=g(\lambda_{n-1})$ $(g\in W_{\mathbf{G}})$ and $G=SO^{+}_{2n}(q)$ then $\mu(D_{w})\not\subset F_{q}$, whereas if $G=SO^{-}_{2n}(q)$ then $\mu(D_{w})\subset F_{q}$ if and only if $wg(\mu)=g(\mu)$ for $w\in W_{\mathbf{G}}$. We do not consider this case in detail.

Theorem 3.10**.**

Let $G\in\{GL_{n}(q),SL_{n+1}(q),Sp_{2n}(q),Spin^{\pm}_{2n}(q),$ q even, $SO^{\pm}_{2n}(q)$, q odd, $SO_{2n+1}(q),$ q odd$\}$ and let W be the Weyl group of ${\mathbf{G}}$. Set $\omega_{j}=\varepsilon_{1}+\cdots+\varepsilon_{j}$ for $1\leq j\leq n$, and assume $j<n$ if $G=SO^{-}_{2n}(q)$ or $Spin^{-}_{2n}(q)$. Let ${\mathbf{L}}_{j}$ be a Levi subgroup of $G$ whose Weyl group is $W_{j}=C_{W}(\omega_{j})$, and $L_{j}={\mathbf{L}}_{j}^{F}$. Then $s_{(q-1)\omega_{j}}$ is $L_{j}$-controlled.

Proof. Let $T$ be a maximal torus of $G$ and $\mu=g(\omega_{j})$ for $g\in W$. The weight $(q-1)g(\omega_{j})$ is trivial on $T$ if and only if $g(\omega_{j})|_{T}$ is a $q$-character of $T$ (Lemma 3.4). So $(s_{(q-1)\omega_{j}}|_{T},1_{T})$ is equal to the number of $q$-characters of $T$ in the set $g(\omega_{j})|_{T}$. By Lemma 3.6 for $G=GL_{n}(q)$ and $SL_{n+1}(q)$ and by Lemma 3.9 for other classical groups above but $SO^{-}_{2n}(q)$, $q$ odd and $Spin^{-}_{2n}(q)$, $q$ even, this number is exactly $1_{W_{j}}^{W}(w)$. So the result follows from Lemmas 2.5 in this case. Let $G=SO^{-}_{2n}(q)$, $q$ odd, or $Spin^{-}_{2n}(q)$, $q$ even. Then the number in question equals $1_{\tilde{W}_{j}}^{\tilde{W}}(wr)$ in notation of Lemma 3.9(2). As $F_{1}$ arises from the graph automorphism which fixes the nodes $1,\ldots,n-2$ and permutes $n-1,n$, it follows that $F_{1}\in C_{\tilde{W}}(\omega_{j})$ for $j=1,\ldots,n-1$. (See the remark prior Theorem 3.10.) But then, as $\tilde{W}=W\cdot\langle F_{1}\rangle$, we have $\tilde{W}_{j}=C_{\tilde{W}}(\omega_{j})=C_{W}(\omega_{j})\cdot\langle F_{1}\rangle=W_{j}\cdot\langle F_{1}\rangle$. Now the result follows from Lemma 2.5, where $\tilde{W}_{{\mathbf{L}}}$ is defined as $W_{{\mathbf{L}}}\cdot\langle F_{1}\rangle$ and $W_{{\mathbf{L}}}$ is the Weyl group of a Levi subgroup ${\mathbf{L}}$.

Proof of Theorem 1.2. By Theorem 3.10, the function $s_{(q-1)\omega_{j}}$ is $L_{j}$-controlled. So the result follows from Theorem 2.3.

4. Truncated polynomials and the natural permutation module

4.1. The natural permutation module

Let $G=GL_{n}(q)$, $V$ the natural $F_{q}G$-module and $M$ the permutation module over the complex numbers associated with the action of $G$ on non-zero vectors of $V$. Let $\chi$ be the character of $M$. Let $T$ be a maximal torus of $G$. Observe that $(\chi|_{T},1_{T})$ equals the number of orbits of $T$ on non-zero elements of $V$, see [7, Theorem 32.3]. We write $C_{r}^{j}$ for the $j$-th binomial coefficient (often denoted by ${\tiny\begin{pmatrix}k\cr j\end{pmatrix}})$.

Lemma 4.1**.**

Let $G=GL_{n}(q)$ or $SL_{n}(q)$ and V the natural $F_{q}G$-module. Let T be a maximal torus in $G$ and let k be the composition length of T on V.

$(1)$* Let $G=GL_{n}(q)$. Then the number of T-orbits on the non-zero vectors of $V$ is equal to $\sum_{j=1}^{k}C_{k}^{j}=2^{k}-1$.*

$(2)$* Let $G=SL_{n}(q)$. Then the number of T-orbits on $V\setminus\{0\}$ is equal to $q-3+2^{k}$.*

Proof. (1) Suppose first that $k=1$. Then $|T|=q^{n}-1$. It is well known that $T$ acts transitively on the non-zero vectors of $V$. In general, let $V=V_{1}+\cdots+V_{k}$ be a decomposition of $V$ as a direct sum of irreducible $F_{q}T$-modules. Then $T=T_{1}\times\cdots\times T_{k}$, where $T_{i}$ acts on $V_{i}$ and $|T_{i}|=q^{\dim V_{i}}-1$ ($1\leq i\leq k$). So $T_{i}$ acts transitively on the non-zero vectors of $V_{i}$. For $v\in V$ let $v=v_{1}+\cdots+v_{k}$ with $v_{i}\in V_{i}$ ($1\leq i\leq k$). Then $Tv=T_{1}v_{1}+\cdots+T_{k}v_{k}$. If $v_{i}\neq 0$ then $T_{i}v_{i}$ consists of all non-zero vectors of $V_{i}$. Moreover if $v^{\prime}=v_{1}^{\prime}+\cdots+v_{k}^{\prime}$ with $v_{i}^{\prime}\in V_{i}$ then $Tv=Tv^{\prime}$ if and only if $v_{i}=0$ implies $v_{i}^{\prime}=0$ and conversely. Therefore, the number of $T$-orbits on $V\setminus 0$ equals $\sum_{i=1}^{k}C_{k}^{j}$.

(2) Set $T^{\prime}=T\cap SL_{n}(q)$. Then $V_{1},\ldots,V_{k}$ are irreducible $F_{q}T^{\prime}$-modules so the composition length of $T^{\prime}$ on $V$ again equals $k$. Note that $|T:T^{\prime}|=q-1$. Let $v=v_{1}+\cdots+v_{k}\in V$ as above. If $v_{i}=0$ for some $i$ then, obviously, $T^{\prime}v=Tv$. So the number of $T^{\prime}$-orbits with $v_{i}=0$ for some $i$ is $\sum_{j=1}^{k-1}C_{k}^{j}$.

Suppose that $v_{i}\neq 0$ for all $i$. Let $v^{\prime}=v_{1}^{\prime}+\cdots+v_{k}^{\prime}$ with $0\neq v_{i}^{\prime}\in V_{i}$. Then there is a unique $t\in T$ such that $tv=v^{\prime}$, so the orbit $Tv$ is regular. As $|T:T^{\prime}|=q-1$, the total number of $T^{\prime}$-orbits on the set $Tv$ equals $q-1$. This implies the result.

Corollary 4.2**.**

$(\chi|_{T},1_{T})=2^{k}-1$* and $(\chi|_{T^{\prime}},1_{T^{\prime}})=q-3+2^{k}=q-2+(\chi|_{T},1_{T})$.*

Lemma 4.3**.**

Let $w\in S_{n}$, and let k be the number of cycles in the cycle decomposition of w. Then $2^{k}-1=\sum_{j=1}^{k}C_{k}^{j}=\sum_{i=1}^{n}1_{Y_{i}}^{S_{n}}(w)$, where $Y_{i}=S_{i}\times S_{n-i}$.

Proof. We know (see Proposition 3.6) that $1_{Y_{i}}^{S_{n}}(w)$ equals the number $x_{i}$ of subsets $\{m_{1},\ldots,m_{i}\}$ of $\{1,\ldots,n\}$ such that $w(\{m_{1},\ldots,m_{i}\})=\{m_{1},\ldots,m_{i}\}$. So $\sum_{i=1}^{n}1_{Y_{i}}^{S_{n}}(w)$ is the sum of $x_{i}$, and hence equals the number of subsets $\{m_{1},\ldots,m_{i}\}$ of $\{1,\ldots,n\}$ such that $w(\{m_{1},\ldots,m_{i}\})=\{m_{1},\ldots,m_{i}\}$, where $i$ is in the range $1\leq i\leq n$. One observes that $w(\{m_{1},\ldots,m_{i}\})=\{m_{1},\ldots,m_{i}\}$ is equivalent to saying that $\{m_{1},\ldots,m_{i}\}$ is a union of some cycles of $w.$ This number can be counted in a different way, as the number of sets $\{m_{1},\ldots,m_{i}\}$ for $1\leq i\leq n$ that are obtained as the union of some cycles of $w$. The number of sets $\{m_{1},\ldots,m_{i}\}$ for $1\leq i\leq n$ that consist of a single cycle is $k$, the number of sets $\{m_{1},\ldots,m_{i}\}$ that are unions of two cycles is $C_{k}^{2}$, and the number of sets $\{m_{1},\ldots,m_{i}\}$ that are unions of $j$ cycles is $C_{k}^{j}$. So the result follows.

Lemma 4.4**.**

Let $G=GL_{n}(q)$ and let $T$ be a maximal torus in $G$. Then $(\chi|_{T},1_{T})=\sum_{i=1}^{n}(s_{(q-1)\omega_{i}}|_{T},1_{T})$, and hence $u(\chi\cdot St)=u(\sum_{i=1}^{n}(s_{(q-1)\omega_{i}}\cdot St)$.

Proof. Let $T=T_{w}$ for $w\in W\cong S_{n}$ and let $k$ be the composition length of $T$ on $V$. Then $(\chi|_{T},1_{T})=2^{k}-1$ by Lemma 4.1, and $(s_{(q-1)\omega_{i}}|_{T},1_{T})=1_{Y_{i}}^{S_{n}}(w)$ by Theorem 3.10. So the result follows from Lemma 4.3. The second statement follows from Lemma 2.2.

Lemma 4.5**.**

Let $G=GL_{n}(q)$ and let $\nu$ be a unipotent character of $G$. Then $(\nu,\chi\cdot St)=(\nu,\sum_{i=1}^{n}s_{(q-1)\omega_{i}}\cdot St)=(\nu,\sum_{i=1}^{n}St_{i}^{\#G})$, where $St_{i}$ is the Steinberg character of a Levi subgroup of $G$ isomorphic to $GL(i)\times GL(n-i,q)$.

Proof. The former equality is Lemma 4.4, the latter one follows from Theorem 1.2. Note that the second equality can also be deduced from [11, Theorem 6.2].

Lemma 4.6**.**

Let $G=GL_{n}(q)$, $G^{\prime}=SL_{n}(q)$ and let V be the natural $F_{q}G$-module. Let $\phi=\chi|_{G^{\prime}}$. Let $St$, $St^{\prime}$ be the Steinberg characters of $G,G^{\prime}$, respectively. Then $(St,\chi\cdot St)=n$ and $(St^{\prime},\phi\cdot St^{\prime})=n+q-2$.

Proof. It is well known that $(St,St_{i}^{\#G})=1$. (Indeed, if $\sigma$ denotes the Harish-Chandra restriction of $St$ to $L_{i}$ then $\sigma=St_{i}$ by [10, p.72]. By Harish-Chandra reciprocity [8, Proposition 70.(iii)], $(St,St_{i}^{\#G})=(\sigma,St_{i})=1$.) So Lemma 4.5 yields the result for $G$.

The Weyl groups of ${\mathbf{G}}$ and ${\mathbf{G}}^{\prime}$ coincide. So the mapping ${\mathbf{T}}\rightarrow{\mathbf{T}}^{\prime}={\mathbf{T}}\cap{\mathbf{G}}^{\prime}$ yields a bijection between the $G$-conjugacy classes of $F$-stable maximal tori in ${\mathbf{G}}$ and the $G^{\prime}$-conjugacy classes of $F$-stable maximal tori in ${\mathbf{G}}^{\prime}$. In addition, $W(T)=W(T^{\prime})$ and $(\phi|_{T^{\prime}},1_{T^{\prime}})=q-2+(\chi|_{T},1_{T})$ by Corollary 4.2. So formula (3) yields

[TABLE]

By formula (3) applied to $GL_{n}(q)$, we have $\sum_{{\mathbf{T}}}\frac{(\chi|_{T},1_{T})}{|W(T)|}=(St,\chi\cdot St)$. The latter equals $n$. By formula (4), $1=(St^{\prime},St^{\prime})=\sum_{{\mathbf{T}}^{\prime}}\frac{1}{|W(T^{\prime})|}$, whence the result.

4.2. Truncated polynomials

Let $R_{n}=\overline{F}_{q}[X_{1},\ldots,X_{n}]$ be the polynomial ring with indeterminates $X_{1},\ldots,X_{n}$ over $\overline{F}_{q}$. Let $I$ be the ideal of $R_{n}$ generated by $X_{1}^{p},\ldots,X_{n}^{p}$ and $\overline{R}_{n}=R_{n}/I$. Then $\overline{R}_{n}$ can be viewed as the truncated polynomial ring whose elements are linear combinations of monomials $X_{1}^{c_{1}}\cdots X_{n}^{c_{n}}$ with $0\leq c_{1},\ldots,c_{n}<p$. Let ${\mathbf{G}}=GL_{n}(\overline{F}_{q})$, $G=GL_{n}(q)$. Viewing $X_{1},\ldots,X_{n}$ as a basis of the natural $\overline{F}_{q}{\mathbf{G}}$-module ${\mathbf{V}}_{n}$, one extends the action of ${\mathbf{G}}$ to $\overline{R}_{n}$ in a standard way. Note that homogeneous polynomials of a fixed degree form an $\overline{F}_{q}{\mathbf{G}}$-submodule of $\overline{R}_{n}$. We often view $\overline{R}_{n}$ as an $\overline{F}_{q}G$-module and as an $\overline{F}_{q}{\mathbf{G}}^{\prime}$-module, where ${\mathbf{G}}^{\prime}=SL_{n}(\overline{F}_{q})$.

Let $q=p^{m}$. There is a well known embedding $G=GL_{n}(q)\rightarrow GL_{mn}(p)$ obtained by viewing $F_{q}$ as a vector space over $F_{p}$. So the above constructed $F_{p}GL_{mn}(p)$-module $\overline{R}_{mn}$ is also an $F_{p}GL_{n}(q)$-module, which plays a significant role below.

The action of $G$ on $\overline{R}_{mn}$ extends to ${\mathbf{G}}$ as follows. For $g\in{\mathbf{G}}$ consider the mapping $g\rightarrow\mathop{\rm diag}\nolimits(g,Fr(g),\ldots,Fr^{m-1}(g))$, $Fr(g)$ is obtained from $g$ by raising every matrix entry to the $p$-power and $\mathop{\rm diag}\nolimits(g,Fr(g),\ldots,Fr^{m-1}(g))$ means the block diagonal matrix with diagonal blocks $g,Fr(g)\ldots$. This makes the space ${\mathbf{V}}_{mn}$ to be a ${\mathbf{G}}$-module. One easily observes that the two actions of $G$ on ${\mathbf{V}}_{mn}$ obtained from the embeddings $G\rightarrow GL_{n}(\overline{F}_{q})$ and $G\rightarrow GL_{mn}(p)$ yield equivalent representations of $G$. Note that $\overline{R}_{mn}$ is completely reducible both as $\overline{F}_{q}{\mathbf{G}}$- and $\overline{F}_{q}G$-module (see [18, Proposition 1.6]).

Denote by $M_{q}$ the $\overline{F}_{q}G$-module obtained from the action of $G$ on the vectors of the natural $F_{q}G$-module $V$ (the zero vector is not excluded). The Brauer character of $M_{q}$ coincides with $\chi+1_{G}$ on the $p$-regular elements.

A remarkable fact going back to Bhattacharia [1] states that the Brauer characters of $G$ on $M_{q}$ and on $\overline{R}_{mn}$ are the same. This was exploited in [19] to obtain the decomposition numbers of the irreducible consituents of $M$. The following lemma is essentially [19, Theorem 3.2].

Lemma 4.7**.**

Let $G=GL_{n}(q)$. The Brauer characters of $\overline{R}_{mn}|_{G}$ and $M_{q}$ coincide.

Proof. Let $V$ be the natural $F_{q}G$-module and $V_{m}$ the natural $F_{p}GL_{mn}(p)$-module. Viewing $V$ as a vector space over $F_{p}$, we identify the additive group of $V$ with that of $V_{m}$. Moreover, the regular embedding $F_{q}\rightarrow Mat(m,F_{p})$ yields an embedding $h:GL_{n}(q)\rightarrow GL_{mn}(p)$, and the permutation actions of $G$ on $V$ and $V_{m}$ are isomorphic. Therefore, the permutation characters afforded by these actions coincide. For $m=1$ (that is for $q=p$) the lemma is proved by Bhattacharia [1]. Clearly, this remains true for every subgroup of $GL_{n}(p)$. Applying this to the subgroup $h(G)$ of $GL_{mn}(p)$, we obtain the result for $G$.

Denote $\lambda_{1},\ldots,\lambda_{n-1}$ the fundamental weights of ${\mathbf{G}}^{\prime}=SL_{n}(\overline{F}_{q})$. Weights $a_{1}\lambda_{1}+\cdots+a_{n-1}\lambda_{n-1}$ with $0\leq a_{1},\ldots,a_{n-1}<q$ are called $q$-restricted, and the mapping $\rho_{\lambda}\rightarrow\rho_{\lambda}|_{G^{\prime}}$ sets up a bijection between the irreducible representations of ${\mathbf{G}}$ with $q$-restricted highest weights and irreducible representations of $G^{\prime}=SL_{n}(q)$ over $\overline{F}_{q}G$.

Lemma 4.8**.**

Let $\lambda=a_{1}\lambda_{1}+\cdots+a_{n-1}\lambda_{n-1}$ be a weight of ${\mathbf{G}}^{\prime}=SL_{n}(\overline{F}_{q})$.

$(1)$* If $\lambda$ occurs as a weight of the ${\mathbf{G}}^{\prime}$-module $\overline{R}_{n}$ then $-p<a_{i}<p$ for $i=1,\ldots,n-1$.*

$(2)$* Suppose that $\lambda$ is dominant. Then $\lambda$ is a weight of the ${\mathbf{G}}^{\prime}$-module $\overline{R}_{n}$ if and only if $\lambda$ is strongly $p$-restricted, that is, $a_{1}+\cdots+a_{n-1}<p$.*

Proof. Let ${\mathbf{T}}^{\prime}$ be the group of diagonal matrices in ${\mathbf{G}}^{\prime}$. Then monomial polynomials are weight vectors for ${\mathbf{T}}^{\prime}$ and they form a basis of $\overline{R}_{n}$. Let $f=X_{1}^{c_{1}}\cdots X_{n}^{c_{n}}\in\overline{R}_{n}$, so $0\leq c_{1},\ldots,c_{n}<p$. Then the weight of $f$ for ${\mathbf{G}}^{\prime}$ in terms of $\varepsilon_{1},\ldots,\varepsilon_{n}$ is $c_{1}\varepsilon_{1}+\cdots+c_{n}\varepsilon_{n}$. Note that $\varepsilon_{1}+\cdots+\varepsilon_{n}$ is the zero weight for ${\mathbf{G}}^{\prime}$, so we can write $\varepsilon_{n}=-(\varepsilon_{1}+\cdots+\varepsilon_{n-1})=-\lambda_{n-1}$. As $\varepsilon_{1}=\lambda_{1}$ and $\varepsilon_{i}=\lambda_{i}-\lambda_{i-1}$ for $1<i<n$, the weight of $f$ in terms of $\lambda_{1},\ldots,\lambda_{n-1}$ is $(c_{1}-c_{2})\lambda_{1}+\cdots+(c_{n-1}-c_{n})\lambda_{n-1}$. Set $a_{i}=c_{i}-c_{i+1}$ for $i=1,\ldots,n-1$. As $0\leq c_{i}<p$, (1) follows.

(2) Suppose that $\lambda$ is a dominant weight of $\overline{R}_{n}$. Then $a_{i}=c_{i}-c_{i+1}\geq 0$. This is equivalent to the condition $c_{1}\geq\cdots\geq c_{n}\geq 0$. Furthermore, $a_{1}+\cdots+a_{n-1}=c_{1}-c_{n}<p$. Conversely, suppose that $a_{1}+\cdots+a_{n-1}<p$. Set $c_{i}=a_{i}+\cdots+a_{n-1}$ for $i<n$ and $c_{n}=0$. Then the monomial $X_{1}^{c_{1}}\cdots X_{n-1}^{c_{n-1}}$ has weight $\lambda$ and $c_{i}<p$ for $i=1,\ldots,n$.

Remark. Let $\alpha_{0}$ be the longest root of ${\mathbf{G}}^{\prime}$. Then $a_{1}+\cdots+a_{n-1}=\langle\lambda,\alpha^{\rm v}_{0}\rangle$ in notation of [13, p.16].

Lemma 4.9**.**

$(1)$* Let ${\mathbf{G}}=GL_{n}(\overline{F}_{q})$. There is a ${\mathbf{G}}$-module isomorphism $\overline{R}_{mn}\cong\overline{R}_{n}\otimes Fr(\overline{R}_{n})\otimes\cdots\otimes Fr^{m-1}(\overline{R}_{n})$.*

$(2)$* Let ${\mathbf{G}}^{\prime}=SL_{n}(\overline{F}_{q})$. Then every weight $\mu$ of ${\mathbf{G}}^{\prime}$ on $\overline{R}_{mn}$ can be expressed as $\mu_{0}+\mu_{1}p+\cdots+\mu_{m-1}p^{m-1}$, where $\mu_{0},,\ldots,\mu_{m-1}$ are weights of ${\mathbf{G}}^{\prime}$ on $\overline{R}_{n}$.*

$(3)$* If $\mu=0$ or $(q-1)\lambda_{k}$ $(1\leq k\leq n-1)$ then the expression in $(2)$ is unique.*

Proof. (1) See [18, Proposition 1.6]. Observe that $Fr(\overline{R}_{n})$ is the ${\mathbf{G}}$-module obtaining from $\overline{R}_{n}$ by twisting with morphism $Fr:{\mathbf{G}}\rightarrow{\mathbf{G}}$ defined in the beginning of the section (so $g\in{\mathbf{G}}$ acts via $Fr(g)$). (2) follows from (1).

(3) Let $\mu_{i}=a_{i,1}\lambda_{1}+\cdots+a_{i,n-1}\lambda_{n-1}$ $(0\leq i<m)$ so $\mu=\sum_{i=0}^{m-1}\sum_{j=1}^{n-1}a_{ij}p^{i}\lambda_{j}=\sum_{j=1}^{n-1}(\sum_{i=0}^{m-1}a_{ij}p^{i})\lambda_{j}$. By Lemma 4.8, $-p<a_{i,j}<p$ for $j=1,\ldots,n-1$. Let $\mu=0$. Then $\sum_{i=0}^{m-1}a_{ij}p^{i}=0$ for every $j$. If $\mu_{i}\neq 0$ for some $i$, then $a_{ij}\neq 0$ for some $j$. Let $r=\max\{i:a_{ij}\neq 0\}$. We can assume $a_{rj}>0$. Then $0=\sum_{i=0}^{m-1}a_{ij}p^{i}=a_{rj}p^{r}+\sum_{i=0}^{r-1}a_{ij}p^{i}\geq p^{r}-(p-1)(1+p+\cdots+p^{r-1})=1$, which is a contradiction.

Let $\mu=(q-1)\lambda_{k}$. Suppose that $\mu=\mu_{0}+\mu_{1}p+\cdots+\mu_{m-1}p^{m-1}$, where $\mu_{0},,\ldots,\mu_{m-1}$ are weights of $\overline{R}_{n}$. Then $\sum_{j=1}^{n-1}(\sum_{i=0}^{m-1}a_{ij}p^{i})\lambda_{j}=(q-1)\lambda_{k}$, whence $\sum_{i=0}^{m-1}a_{ij}p^{i}=0$ for $j\neq k$. We have seen in the previous paragraph that this implies $a_{ij}=0$ for $i\neq k$. Then $(q-1)\lambda_{k}=\mu=(a_{k0}+a_{k1}p+\cdots+a_{k,m-1}p^{m-1})\lambda_{k}$, and hence $q-1=a_{k0}+a_{k1}p+\cdots+a_{k,m-1}p^{m-1}\geq(p-1)(1+p+\cdots+p^{m-1})=q-1$. The equality holds if and only if $a_{k0}=a_{k1}=\cdots=a_{k,m-1}=p-1$. So the result follows.

Lemma 4.10**.**

Let $\lambda$ be a strongly $q$-restricted weight of ${\mathbf{G}}^{\prime}$ and $\rho_{\lambda}$ an irreducible representation of ${\mathbf{G}}^{\prime}$ with highest weight $\lambda$. Then all weights of $\rho_{\lambda}$ occur in $\overline{R}_{mn}.$

Proof. Suppose first that $m=1$. By Lemma 4.8, $\lambda$ is a weight of $\overline{R}_{n}.$ Let $\tau$ be an irreducible representation of ${\mathbf{G}}^{\prime}$ on $\overline{R}_{n}$ such that $\lambda$ is a weight of $\tau$. Then the weights of $\rho_{\lambda}$ are weights of $\tau$ (but $\rho_{\lambda}$ is not usually a constituent of $\overline{R}_{n}$.) This follows from a criterion in [2, Ch.VIII, Corollary 2 of Proposition 3] for representations in characteristic 0, and from Suprunenko’s theorem [17] or [13, §3.3] for prime characteristics.

Let $m>1$. Then $\lambda=\nu_{0}+p\nu_{1}+\cdots+p^{m-1}\nu_{m-1}$, where $\nu_{0},\ldots,\nu_{m-1}$ are strongly $p$-restricted weights. By Lemma 4.8, $\nu_{0},\ldots,\nu_{m-1}$ are weights of $\overline{R}_{n}$. By (1), all weights of $\rho_{\nu_{i}}$ $(i=0,\ldots,m-1)$ occur in $\overline{R}_{n}$. As $\rho_{\lambda}=\rho_{\nu_{0}}\otimes Fr(\rho_{\nu_{1}})\otimes\cdots\otimes Fr^{m-1}(\rho_{\nu_{m-1}})$, all weights of $\rho_{\lambda}$ occur in $\overline{R}_{mn}$.

Remark. If $q>p$ then not every weight of $\overline{R}_{mn}$ is strongly $q$-restricted. For instance let $p=n=3$, $q=9$ and $\lambda=2\lambda_{1}+2\lambda_{2}$. As $\lambda=-(\lambda_{1}+\lambda_{2})+3(\lambda_{1}+\lambda_{2})$ and $-(\lambda_{1}+\lambda_{2})$ is a weight of $\rho_{\lambda_{1}+\lambda_{2}}$, it follows that $\lambda$ is a weight of the representation $\rho_{\lambda_{1}+\lambda_{2}}\otimes Fr(\rho_{\lambda_{1}+\lambda_{2}})$, which is a constituent of $\overline{R}_{mn}$ for $m=2,n=3$.

Lemma 4.11**.**

$(1)$* Let $1\leq k\leq p-1$. The polynomial $f_{k}=(X_{1}\cdots X_{n})^{k}\in\overline{R}_{n}$ is a vector of zero weight for ${\mathbf{G}}^{\prime}$.*

$(2)$* Let $f=X_{1}^{a_{1}}\cdots X_{n}^{a_{n}}\in\overline{R}_{n}$, let $e=\max_{1\leq i,j\leq n}(a_{i}-a_{j})$ and let $\nu$ be the ${\mathbf{G}}^{\prime}$-weight of $f$. Then the multiplicity of $\nu$ in $\overline{R}_{n}$ is equal to $p-e$.*

$(3)$* The multiplicity of weight $(q-1)\lambda_{i}$ in $\overline{R}_{mn}$ equals $1$ and that of weight [math] equals $q$.*

Proof. Let ${\mathbf{T}}^{\prime}$ be the group of diagonal matrices of $SL_{n}(\overline{F}_{q})$. Note that monomials $X_{1}^{a_{1}}\cdots X_{n}^{a_{n}}$ form a basis of $\overline{R}_{n}$, and each monomial is a weight vector.

$(1)$ Let $t=\mathop{\rm diag}\nolimits(t_{1},\ldots,t_{n})\in T^{\prime}\subset SL(n,\overline{F}_{q})$. Then $f_{k}(t)=(\det t)^{k}=1$.

$(2)$ If all $a_{1},\ldots,a_{n}$ are non-zero then there is $k>0$ such that $f=f_{k}\cdot X_{1}^{a_{1}-k}\cdots X_{n}^{a_{n}-k}$ and some $a_{i}-k=0$. Clearly, the weight of $f^{\prime}:=X_{1}^{a_{1}-k}\cdots X_{n}^{a_{n}-k}$ is the same as that of $f$, so we can assume that $f=f^{\prime}$, that is, that $a_{j}=0$ for some $j\in\{1,\ldots,n\}$. Let $h=X_{1}^{b_{1}}\cdots X_{n}^{b_{n}}$ be another monomial of weight $\nu$ and $b_{l}=0$ for some $l\in\{1,\ldots,n\}$. Observe that $a_{j}=0$ if and only if $b_{j}=0$. Indeed, suppose the contrary, say, let $a_{j}=0$, $b_{j}\neq 0$, and $b_{l}=0$ for some $l>j$. Let $t=\mathop{\rm diag}\nolimits(1,\ldots,1,t_{j},1,\ldots,1,t_{j}^{-1},1,\ldots,1)\in SL(n,\overline{F}_{q})$, where $0\neq t_{j}\in\overline{F}_{q}$ is arbitrary and $t_{j}^{-1}$ occupies the $l$-th position. Then we have $t\cdot f=t_{j}^{-a_{l}}\cdot f$ and $t\cdot h=t_{j}^{b_{j}}\cdot h$. As $f,h$ are weight vectors of the same weight $\nu$, we have $t_{j}^{-a_{l}}=t_{j}^{b_{j}}$. As $a_{l}\geq 0,b_{j}\neq 0$, this is a contradiction. Similarly, $b_{j}=0$ implies $a_{j}=0$. Finally, fixing $j$ with $a_{j}=b_{j}=0$, let $t=\mathop{\rm diag}\nolimits(t_{1},\ldots,t_{n})$ with $t_{j}=(\Pi_{i\neq j}t_{i})^{-1}$. One easily observes that $t\cdot f=(\Pi_{i\neq j}t_{i}^{a_{i}})f=\nu(t)f$ and $t\cdot h=(\Pi_{i\neq j}t_{i}^{b_{i}})h=\nu(t)h$. As each $t_{i}$ with $i\neq j$ is arbitrary, $a_{i}=b_{i}$ for all $i\neq j$.

It follows that if $f=X_{1}^{a_{1}}\cdots X_{n}^{a_{n}}$ is of weight $\nu$ with $a_{j}=0$ for some $j$ then all other monomials of weight $\nu$ must be $f\cdot f_{k}$ for some $k$. In this case $e=\max_{i}a_{i}$. Therefore, $f\cdot f_{k}\in\overline{R}_{n}$ implies $e+k\leq p-1$, so the result follows.

$(3)$ The monomial $X_{1}^{p-1}\cdots X_{i}^{p-1}\in\overline{R}_{n}$ is a vector of weight $(p-1)\lambda_{i}$ and $f_{1}$ is that of weight 0. So for $m=1$ the lemma follows by applying (2) to these monomials. Since $\overline{R}_{mn}\cong\overline{R}_{n}\otimes Fr(\overline{R}_{n})\otimes\cdots\otimes Fr^{m-1}(\overline{R}_{n})$, the result for arbitrary $m$ follows from Lemma Lemma 4.9(3).

Lemma 4.12**.**

Let $\nu$ be a ${\mathbf{G}}^{\prime}$-weight on $\overline{R}_{mn}$ and T a maximal torus of $G^{\prime}=SL_{n}(q)$. Suppose that $\nu|_{T}=1_{T}$. Then either $\nu=0$ or $\nu$ lies in the $W$-orbit of $(q-1)\lambda_{i}$ for some $i\in\{1\leq i\leq n-1\}$.

Proof. Note that $\overline{R}_{mn}$ is the sum of weight spaces of ${\mathbf{T}}$ on $\overline{R}_{mn}$, so $(\overline{R}_{mn}|_{T},1_{T})=\sum_{\mu}m_{\nu}(s_{\mu}|_{T},1_{T})$, where $\mu$ runs over the dominant weights of ${\mathbf{G}}^{\prime}$ in $\overline{R}_{mn}$ and $m_{\nu}$ is the multiplicity of $\nu$. By Lemma 4.11, the multiplicity of weight 0 of ${\mathbf{G}}^{\prime}$ in $\overline{R}_{mn}$ equals $q$ and that of $(q-1)\lambda_{i}$ is 1 for $i\in\{1,\ldots,n-1\}$. Therefore, $(\overline{R}_{mn}|_{T},1_{T})=q+\sum_{i=1}^{n-1}(s_{(q-1)\lambda_{i}}|_{T},1_{T})+\sum_{\mu}m_{\nu}(s_{\mu}|_{T},1_{T})$, where $\mu$ now runs over the dominant weights of ${\mathbf{G}}^{\prime}$ in $\overline{R}_{mn}$ distinct from 0 and $(q-1)\lambda_{i}$ for $1\leq i<n$. Let $T=T_{w}$ for $w\in W$. By Proposition 3.6 (together with Lemma 3.4), $(s_{(q-1)\lambda_{i}}|_{T_{w}},1_{T_{w}})=1^{W}_{W_{i}}(w)$. By Lemma 4.3, $\sum_{i=1}^{n-1}1^{W}_{W_{i}}(w)=2^{k}-2$, where $k$ is the number of cycles in the cycle decomposition of $w$ (as the term with $i=n$ in Lemma 4.3 equals 1). So $(\overline{R}_{mn}|_{T},1_{T})=q+2^{k}-2+\sum m_{\nu}(s_{\mu}|_{T},1_{T})$, where $\mu\neq 0,(q-1)\lambda_{1},\ldots,(q-1)\lambda_{n-1}$.

Another calculation (Corollary 4.2 and Lemma 4.7) shows that $(\overline{R}_{mn}|_{T},1_{T})=q-2+2^{k}$. It follows that $\sum_{\mu}(s_{\mu}|_{T},1_{T})=0$ for every dominant weight $\mu$ of $\overline{R}_{mn}$ distinct from 0 and $(q-1)\lambda_{i}$ for $i\in\{1,\ldots,n\}$. This implies the lemma.

Remark. At the first sight, Lemma 4.12 cannot be true for $q=2$ since the split torus $T$ has order 1, and hence $\nu|_{T}=1_{T}$ for arbitrary weight $\nu$. However, in fact the Brauer character of $\overline{R}_{n}$ equals $2\cdot 1_{T}+\sum_{i=1}^{n-1}s_{\lambda_{i}}$, which agrees with the statement of the lemma.

Lemma 4.13**.**

Let $j\in\{1,\ldots,n-1\}$, ${\mathbf{G}}=GL_{n}(\overline{F}_{q})$ and ${\mathbf{G}}^{\prime}=SL(n,\overline{F}_{q})$. Then ${\mathbf{G}}^{\prime}$-module $\overline{R}_{mn}$ contains composition factors of highest weights $(q-1)\lambda_{j}$ $(j=1,\ldots,n-1)$, each occurs with multiplicity $1$. If ${\mathbf{G}}=GL_{n}(\overline{F}_{q})$ then the ${\mathbf{G}}$-module $\overline{R}_{mn}$ contains a composition factor of highest weight $(q-1)\omega_{j}$ with multiplicity $1$.

Proof. Suppose first that $m=1$. Then $v_{j}:=X_{1}^{p-1}\cdots X_{j}^{p-1}$ is a vector of weight $(p-1)\lambda_{j}$ for ${\mathbf{G}}^{\prime}$ and of weight $(p-1)\omega_{j}$ for ${\mathbf{G}}$. Moreover, $gv_{j}=v_{j}$ for every upper unitriangular matrix in $GL_{n}(\overline{F}_{q})$. As the space of homogeneous polynomials of fixed degree is an irreducible ${\mathbf{G}}^{\prime}$-module [1, 18], $v_{j}$ is a highest weight vector in this module (both for ${\mathbf{G}}^{\prime}$ and ${\mathbf{G}}$). By Lemma 4.11, the representation afforded by this module occurs in $\overline{R}_{n}$ with multiplicity 1.

As $\overline{R}_{mn}\cong\overline{R}_{n}\otimes Fr(\overline{R}_{n})\otimes Fr^{m-1}(\overline{R}_{mn})$, it follows that $\overline{R}_{mn}$ contains a composition factor in question, and the statement on multiplicity follows again from Lemma 4.11(3).

If $n>2$ then Lemma 4.13 is a special case of [18, Theorem 1.4], which determines the multiplicity of every composition factors of ${\mathbf{G}}^{\prime}$ on $\overline{R}_{mn}$ for $n>2$.

Lemma 4.14**.**

Let $G^{\prime}=SL_{n}(q)$ and $T=T_{w}$ a maximal torus of $G^{\prime}$. Let $\rho_{\nu}$ be an irreducible representation of ${\mathbf{G}}^{\prime}=SL_{n}(\overline{F}_{q})$ with a strongly q-restricted highest weight $\nu\neq 0$, and $d_{0}$ the multiplicity of weight [math] in $\rho_{\nu}$. Let $\beta_{\nu}$ be the Brauer character of $\rho_{\nu}$.

$(1)$* $(\beta_{\nu}|_{T},1_{T})=d_{0}$ unless $\nu=(q-1)\lambda_{i}$ for $i\in\{1,\ldots,n-1\}$ and $w\neq[n]$.*

$(2)$* Let $\nu=(q-1)\lambda_{i}$ for some $i\in\{1,\ldots,n-1\}$. Then $(\beta_{\nu}|_{T},1_{T})$ equals $d_{0}+1_{W_{i}}^{W}(w)$ and $d_{0}\leq 1$, and $d_{0}=1$ if and only if $i(p-1)\equiv 0\pmod{n}$.*

Proof. (1) By Lemma 4.10, every weight of $\rho_{\nu}$ occurs as a weight of $\overline{R}_{mn}$. Note that $\beta_{\nu}|_{T}=d_{0}\cdot 1_{T}+\sum_{\mu}d_{\mu}s_{\mu}|_{T}$, where the sum is over the dominant weights $\mu\neq 0$ of $\rho_{\nu}$ and $d_{\mu}$ denotes the multiplicity of $\mu$ in $\rho_{\nu}$. By Lemma 4.12, either $(s_{\mu}|_{T},1_{T})=0$ or $\mu=(q-1)\lambda_{i}$ for some $i\in\{1,\ldots,n-1\}$. Suppose that the latter holds. Then $\mu=\nu$. Indeed, $(q-1)\lambda_{j}$ occurs as a weight $\overline{R}_{mn}$ with multiplicity 1 (Lemma 4.12(3)), and $\rho_{(q-1)\lambda_{j}}$ is a constituent of $\overline{R}_{mn}$ (Lemma 4.13). Therefore, $(q-1)\lambda_{i}$ is a weight of $\rho_{\nu}$ if and only if $\nu=(q-1)\lambda_{i}$. Thus, if $\nu\neq(q-1)\lambda_{i}$ for some $i$ then $(\beta_{\nu}|_{T},1_{T})=d_{0}$.

Let $\nu=(q-1)\lambda_{i}$ for some $i\in\{1,\ldots,n-1\}$. As above, $\mu\neq(q-1)\lambda_{j}$ for $j\neq i$. By Lemma 3.6, $(s_{(q-1)\lambda_{i}}|_{T},1_{T})=1_{W_{i}}^{W}(w)$. As $1_{W_{i}}^{W}(w)=0$ for $w=[n]$, (1) follows.

(2) We show that $d_{0}\leq 1$ for $\nu=(q-1)\lambda_{i}$. Indeed, $\rho_{\nu}$ is a constituent of $\overline{R}_{mn}$. Every weight of every irreducible constituent of $\overline{R}_{mn}$ has multiplicity 1. This easily follows from the fact that $\chi|_{T_{n}}$ is the character of the regular representation of $T_{n}$, where $T_{n}$ is the maximal torus of $GL_{n}(q)$ of order $q^{n}-1$ (see also the proof of Lemma 4.1). Indeed, this implies, in view of Lemma 4.7, that all weights of ${\mathbf{G}}=GL_{n}(\overline{F}_{q})$ on the space of non-constant polynomials of $\overline{R}_{mn}$ have multiplicity 1. As ${\mathbf{G}}={\bf Z}\cdot{\mathbf{G}}^{\prime}$, where ${\bf Z}$ is the center of ${\mathbf{G}}$, the weight multiplicity of an irreducible representation of ${\mathbf{G}}$ are the same as those of the restriction to ${\mathbf{G}}^{\prime}$.

Furthermore, Lemma 4.9 tells us that $\rho_{(q-1)\lambda_{i}}$ has weight 0 if and only if so is $\rho_{(p-1)\lambda_{i}}$. This happens if and only if $(p-1)\lambda_{i}$ lies in the root lattice. (For Lie algebras over the complex numbers this is well known [3, Ch. VIII, §7, Proposition 5(iv)]. By a result of Suprunenko [17], this remains true for simple algebraic groups of type $A$ and irreducible representations with $p$-restricted highest weights. See also [13, §3.3].) The quotient of the weight lattice over the root lattice is isomorphic to $\mathop{\mathbb{Z}}\nolimits\pmod{n}$, the cyclic group of order $n$, and $i\lambda_{1}-\lambda_{i}$ for $i<n$ is a linear combination of roots [2, Planche I]. It follows that $(p-1)\lambda_{i}$ lies in the root lattice if and only if $i(p-1)\equiv 0\pmod{n}$.

Remark. In general, $\rho_{\nu}$ needs not be a composition factor of $\overline{R}_{mn}$ if $\nu$ is a dominant weight of $\overline{R}_{mn}$. Moreover, if $\rho_{\nu}$ has a weight 0, then the multiplicity of it equals 1 if and only if $\rho_{\nu}$ is a consituent of $\overline{R}_{mn}$.

Corollary 4.15**.**

Let $G^{\prime}$, $\nu$, $\beta_{\nu}$, $d_{0}$ be as in Lemma 4.14.

$(1)$* Let $w=[n]$ and $T=T_{w}$, so $|T|=(q^{n}-1)/(q-1).$ Then $(\beta_{\nu}|_{T},1_{T})=d_{0}$.*

$(2)$* Let $w=[n-1,1]$ and $T=T_{w}$, so $|T|=q^{n-1}-1.$ Then $(\beta_{\nu}|_{T},1_{T})$ equals the multiplicity of weight [math] in $\rho_{\nu}$, unless $\nu=(q-1)\lambda_{1}$ or $(q-1)\lambda_{n-1}$. In these cases $(\beta_{\nu}|_{T},1_{T})=1+d_{0}$, where $d_{0}=1$ if n divides $p-1$, otherwise $d_{0}=0$.*

Proof. (1) In this case $1_{W_{i}}^{W}(w)=0$ as $W_{i}$ contains no conjugate of $w$ for every $i\in\{1,\ldots,n-1\}$. So we are done by Lemma 4.14.

(2) In this case $W_{i}$ contains no conjugate of $w$ if and only if $i=1$ or $n-1$. Then $1_{W_{1}}^{W}$ is the permutation representation of $W\cong S_{n}$ associated with the natural action of $S_{n}$ on $n$ letters. So $1_{W_{1}}^{W}(w)=1$. As $W_{n-1}=W_{1}$, the result follows from Lemma 4.14.

Remark. The special case of Corollary 4.15(2) for $n=2$ and $q$ even is examined in [14, Corollary of Theorem 3]. Note that in this case every $q$-restricted weight $\nu$ of ${\mathbf{G}}$ is strongly $q$-restricted, and $\rho_{\nu}$ has weight zero only if $\nu=0$.

We also mention results of [22, Theorem 8] and [14, Theorem 1] where the authors prove existence of eigenvalue 1 of elements $g$ of cyclic tori $T_{w}$ with $w=[n]$ or $[n-1,1]$ in representations $\rho$ under certain restrictions on $|g|$ and $\rho$.

4.3. Unipotent constituents of certain projective modules

It is well known that the tensor product of any $\overline{F}_{q}G$-module with a projective module is again a projective module. Let $St_{q}$ denote the Steinberg $\overline{F}_{q}G$-module. Then $St_{q}$ is a projective $\overline{F}_{q}G$-module. If $\rho$ is an irreducible representation of $G^{\prime}=SL_{n}(q)$ with Brauer character $\beta$ then $\rho\otimes St_{q}$ is a projective $\overline{F}_{q}G$-module, whose character is $\beta\cdot St$. (If $\beta$ is a generalized Brauer character of $G$ then $\beta\cdot St$ is meant to be the function equal to 0 at the $p$-singular elements of $G$ and $\beta(g)\cdot St(g)$ for $g\in G$ semisimple.)

Theorem 4.16**.**

Let ${\mathbf{G}}^{\prime}=SL_{n}(\overline{F}_{q})$ and $G^{\prime}=SL_{n}(q)$. Let $\rho_{\nu}$ be an irreducible representation of ${\mathbf{G}}^{\prime}$ with highest weight $\nu$, $d_{0}$ the multiplicity of weight [math] of $\rho_{\nu}$ and $\beta_{\nu}$ the Brauer character of $\rho_{\nu}|_{G^{\prime}}$. Suppose that $\nu$ is strongly q-restricted.

$(1)$* $u(\beta_{\nu}\cdot St)=d_{0}\cdot St$, unless $\nu=(q-1)\lambda_{i}$ for $i\in\{1,\ldots,n-1\}$.*

$(2)$* Suppose that $\nu=(q-1)\lambda_{i}$ for $i\in\{1,\ldots,n-1\}$, and let $L_{i}$ be a Levi subgroup of $G$ whose Weyl group is $W_{i}:=C_{W}(\lambda_{i})$. Then $u(\beta_{\nu}\cdot St)=d_{0}\cdot St+St_{L_{i}}^{\#G}$, where $d_{0}\leq 1$.*

Proof. (1) By Lemma 2.2, $\beta_{\nu}\cdot St=\sum_{{\mathbf{T}}}\frac{(\beta_{\nu}|_{T},1_{T})}{|W(T)|}\varepsilon_{{\mathbf{G}}^{\prime}}\varepsilon_{{\mathbf{T}}}R_{{\mathbf{T}},1}$, where the sum is over representatives of $G$-conjugacy classes of $F$-stable maximal tori ${\mathbf{T}}$ of ${\mathbf{G}}^{\prime}$, and $T={\mathbf{T}}^{F}$. By Lemma 4.14, if $\nu\neq(q-1)\lambda_{i}$ for some $i\in\{1,\ldots,n-1\}$ then $(\beta_{\nu}|_{T},1)=d_{0}$. Therefore, $\beta_{\nu}\cdot St=d_{0}\cdot\sum_{{\mathbf{T}}}\frac{1}{|W(T)|}\varepsilon_{{\mathbf{G}}^{\prime}}\varepsilon_{{\mathbf{T}}}R_{{\mathbf{T}},1}=d_{0}\cdot St$ by formula (4).

(2) Let $\nu=(q-1)\lambda_{i}$ and $T=T_{w}$. Then $(\beta_{\nu}|_{T},1)=d_{0}+1_{W_{i}}^{W}(w)$ and $d_{0}\leq 1$ by Lemma 4.14(2). So $\beta_{\nu}\cdot St=d_{0}\cdot\sum_{{\mathbf{T}}}\frac{1}{|W(T)|}\varepsilon_{{\mathbf{G}}^{\prime}}\varepsilon_{{\mathbf{T}}}R_{{\mathbf{T}},1}+\sum_{{\mathbf{T}}}\frac{1_{W_{i}}^{W}(w)}{|W(T)|}\varepsilon_{{\mathbf{G}}^{\prime}}\varepsilon_{{\mathbf{T}}}R_{{\mathbf{T}},1}$. It follows from Lemma 2.5 that the function $\beta_{\nu}-d_{0}$ is $L_{i}$-controlled. The former sum equals $d_{0}\cdot St$ (as above), whereas the latter sum yields $St_{L_{i}}^{\#G^{\prime}}$ by Theorem 2.3.

Remark. The condition for $d_{0}=1$ is given in Lemma 4.14(2).

Proof of Theorem 1.3. The results stated follow from Lemma 4.14 and Theorem 4.16.