Green functions and Glauberman degree-divisibility

TL;DR

This paper proves a key divisibility property for characters in finite groups of Lie type, completing a conjecture related to Green functions and the Glauberman correspondence, which enhances understanding of character degrees in group theory.

Contribution

It provides a general proof that the degree-divisibility property holds universally for characters related by the Glauberman correspondence, confirming a conjecture from 1994.

Findings

Degree-divisibility property holds for all relevant characters

Completes the proof of a conjecture from Hartley and Turull

Confirms the Green functions satisfy the necessary congruence condition

Abstract

The Glauberman correspondence is a fundamental bijection in the character theory of finite groups. In 1994, Hartley and Turull established a degree-divisibility property for characters related by that correspondence, subject to a congruence condition which should hold for the Green functions of finite groups of Lie type, as defined by Deligne and Lusztig. Here, we present a general argument for completing the proof of that congruence condition. Consequently, the degree-divisibility property holds in complete generality.

Equations

$${\mathrm{Irr}}_S(\Gamma )\leftrightarrow \mathrm{Irr}(C_{\Gamma }(S)),\theta \leftrightarrow {\theta }^{\ast },$$

$$Q_{T,F}(u)\equiv Q_{T,F^r}(u)\mathrm{mod}r.$$

$$Q_w:G_{\text{uni}}^F\to {\overline{\mathbb{Q}}}_{\ell },u\mapsto \text{mbox}Tr(u,R_w),$$

$$\sigma_{E}^{d}=\mbox{id}_{E}\qquad\mbox{and}\qquad\mbox{Tr}(\sigma_{E}{\circ}w,E)\in{\mathbb{Z}}\quad\mbox{for all $w\in W$};$$

$$R_{E}(g):=\frac{1}{|W|}\sum_{w\in W}\mbox{Tr}(\sigma_{E}{\circ}w,E)\mbox{Tr}(g,R_{w})\qquad\mbox{for all $g\in G^{F}$}.$$

$$Q_{w}(u)=\sum_{E\in\operatorname{Irr}(W)^{\gamma}}\mbox{Tr}(\sigma_{E}{\circ}w,E)R_{E}(u)\qquad\mbox{for $w\in W$, $u\in G_{\text{uni}}^{F}$}.$$

$$\chi_{A,\phi}(g)=\sum_{i}(-1)^{i}\mbox{Tr}(\phi_{i,g},{\mathscr{H}}_{g}^{i}(A))\qquad\mbox{for $g\in G^{F}$}.$$

$$K\cong \underset{E\in \mathrm{Irr}(W)}{⨁}{V}_E\otimes A_E,$$

$$R_{E}(g)=(-1)^{\dim T_{0}}\chi_{A_{E}}(g)\qquad\mbox{for all $g\in G^{F}$}.$$

$${\iota }_G:\mathrm{Irr}(W)\hookrightarrow {\mathcal{N}}_G$$

$$\mbox{u-supp}(A_{E})\subseteq\overline{C}\qquad\mbox{and}\qquad{\mathscr{H}}^{i}(A_{E})|_{C}\cong\left\{\begin{array}[]{cl}{\mathscr{E}}&\mbox{ if $i=-\dim C-\dim T_{0}$},\\ 0&\mbox{ otherwise}.\end{array}\right.$$

$$\{g\in G_{\text{uni}}\mid{\mathscr{H}}_{g}^{i}(A)\neq\{0\}\mbox{ for some $i$}\}.$$

$$d_E:=(\dim G-\dim C-\dim T_0)/2.$$

$$\chi_{A_{E}}(g)=(-1)^{i}\mbox{Tr}\bigl{(}\phi_{A_{E},i,g},{\mathscr{H}}^{i}_{g}(A_{E})\bigr{)}=(-1)^{i}q^{d_{E}}Y_{E}(g)$$

$$Y_{E}(g)=\left\{\begin{array}[]{cl}\mbox{Tr}(\psi_{g},{\mathscr{E}}_{g})&\quad\mbox{if $g\in C^{F}$},\\ 0&\quad\mbox{otherwise};\end{array}\right.$$

$$R_{E}(g)=(-1)^{\dim T_{0}}\chi_{A_{E}}(g)=q^{d_{E}}Y_{E}(g)\qquad\mbox{for all $g\in C^{F}$}.$$

$$Y_{E}(g)\in{\mathbb{Z}}\qquad\mbox{for all $E\in\operatorname{Irr}(W)^{\gamma}$ and $g\in G_{\text{uni}}^{F}$}.$$

$$R_{E}|_{G_{\text{uni}}^{F}}=\sum_{E^{\prime}\in\operatorname{Irr}(W)^{\gamma}}q^{d_{E}}\,p_{E^{\prime},E}Y_{E^{\prime}}\qquad\mbox{for all $E\in\operatorname{Irr}(W)^{\gamma}$},$$

$$p_{E^{\prime},E}\in{\mathbb{Z}}\qquad\mbox{for all $E,E^{\prime}\in\operatorname{Irr}(W)^{\gamma}$}.$$

$$p_{E,E}=1\qquad\mbox{and}\qquad p_{E^{\prime},E}=0\quad\mbox{if $E^{\prime}\neq E$ and $d_{E^{\prime}}\geqslant d_{E}$}.$$

$$Q_w(u)=\sum _{E^{\prime },E\in \mathrm{Irr}(W{)}^{\gamma }}\mathrm{Tr}({\sigma }_E\circ w,E)q^{d_E}{p}_{E^{\prime },E}{Y}_{E^{\prime }}(u),$$

$$F=\tilde{\gamma }\circ F_p^m=F_p^m\circ \tilde{\gamma }(m⩾1)$$

$$\delta_{E}=\left\{\begin{array}[]{rl}1&\quad\mbox{if $q\equiv 1\bmod 3$},\\ -1&\quad\mbox{if $q\equiv-1\bmod 3$}.\end{array}\right.$$

$$\mathcal{M}:=\{r\in {\mathbb{Z}}_{⩾1}\mid r\equiv 1\mathrm{mod}d\}.$$

$$R_E(u_0)=q^{d_E}{\delta }_E\dim {\mathcal{E}}_{u_0}\text{mbox}(see\S \text{ref}sub25).$$

$$R_E^{(r)}(u_0)=q^{d_{E}r}{\delta }_E^{(r)}\dim {\mathcal{E}}_{u_0}.$$

$$(F^{*})^{j-1}(\phi)\colon(F^{*})^{j}A_{E}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(F^{*})^{j-1}A_{E}\qquad\mbox{for $1\leqslant j\leqslant r$}.$$

$${\tilde{\varphi }}^{(r)}:=\varphi \circ F^{\ast }(\varphi )\circ \dots \circ (F^{\ast }{)}^{r-1}(\varphi ):(F^{\ast }{)}^r{A}_E\stackrel{\sim }{\to }{A}_E.$$

$${\varphi }^{(r)}:(F^r{)}^{\ast }{A}_E\stackrel{\sim }{\to }{A}_E\text{mbox}(see\text{cite}[cite][\text{@@bibref}S2,1.1]).$$

$$\chi^{(r)}(g)=\sum_{i}(-1)^{i}\mbox{Tr}\bigl{(}\phi_{i,g}^{(r)},{\mathscr{H}}_{g}^{i}(A_{E})\bigr{)}\qquad\mbox{for $g\in G^{F^{r}}$}.$$

Full text

1. Introduction

This paper is mainly about representations of finite groups of Lie type, but the motivation comes from the general character theory of finite groups. Let $\Gamma$, $S$ be finite groups of coprime order such that $S$ is solvable and acts by automorphisms on $\Gamma$. Then the Glauberman correspondence [10] is a certain canonical bijection

[TABLE]

where $\operatorname{Irr}_{S}(\Gamma)$ is the set of $S$-invariant irreducible characters of $\Gamma$ and $C_{\Gamma}(S)$ the subgroup of $\Gamma$ fixed by all elements of $S$. It is of fundamental importance in various current trends of research; see, e.g., Navarro [26, §2.5]. Using the classification of finite simple groups, and subject to a certain congruence condition on Green functions of finite groups of Lie type, Hartley–Turull [11, Theorem A] showed the following result, which gives a positive answer to a problem described as perhaps one of the deepest in character theory by Navarro [26, §1].

Glauberman degree-divisibility. Assume that $\theta\in\operatorname{Irr}_{S}(\Gamma)$ and $\theta^{*}\in\operatorname{Irr}(C_{\Gamma}(S))$ correspond to each other as above. Then $\theta^{*}(1)$ divides $\theta(1)$.

That congruence condition on Green functions has been explicitly verified in [11, Prop. 7.5] for groups of type $A_{n}$ and in [11, Prop. 7.7] for groups of type ${{}^{3}\!D}_{4}$, ${{}^{2}\!B}_{2}$, ${{}^{2}\!G}_{2}$, ${{}^{2}\!F}_{4}$; by the comments on [11, p. 204] it is also known in a large number of further cases. In this paper, we present a general argument which completes the proof of that condition. Hence, the Glauberman degree-divisibility property will hold unconditionally and in complete generality.

We shall use the full power of the geometric representation theory of finite groups of Lie type, as developed by Lusztig [14], [16]–[20], [21], [23]; an essential role will also be played by the results of Shoji [28], [29] concerning the relation between irreducible representations and character sheaves.

Let us now explain that congruence condition on Green functions. We consider a connected reductive algebraic group $G$ (over an algebraic closure of ${\mathbb{F}}_{p}$ where $p$ is a prime) and an endomorphism $F\colon G\rightarrow G$ such that some power of $F$ is a Frobenius map. The Green function $Q_{T}$ corresponding to an $F$-stable maximal torus $T\subseteq G$ is introduced by Deligne–Lusztig [4] (see also Carter [3]). It is a function defined on the set of unipotent elements of $G^{F}$, with values in ${\mathbb{Z}}$. The construction involves the theory of $\ell$-adic cohomology applied to certain algebraic varieties on which the finite group $G^{F}$ acts. In order to indicate the dependence on $F$, we shall write $Q_{T,F}$ instead of just $Q_{T}$. Now we can state:

Congruence Condition (Hartley–Turull [11, Condition 6.9, 6.10]). Let $T\subseteq G$ be an $F$-stable maximal torus and $u\in G^{F}$ be unipotent. Let $r\in{\mathbb{N}}$ be a prime such that $r$ does not divide the order of $|G^{F^{r}}|$. Then

[TABLE]

A quick informal argument to establish this condition goes as follows. Since Suzuki and Ree groups have already been dealt with, we can assume that $F$ is a Frobenius map defining an ${\mathbb{F}}_{q}$-rational structure on $G$, where $q$ is a power of $p$. It is expected that $Q_{T,F}(u)$ is given by a well-defined polynomial in $q$ with integer coefficients, such that $Q_{T,F^{r}}(u)$ is given by evaluating that same polynomial at $q^{r}$. Then it simply remains to use Fermat’s Little Theorem. For example, this works perfectly well if $G$ is of type $A_{n}$, as already noted in [11, Prop. 7.5], and in many further cases; see Shoji [27, §6]. However, the required information is not yet available for all groups over fields of small characteristic. And even when it is known, then some additional care is needed since there are cases where the Green functions are only “PORC” (in the sense of Higman), that is, polynomial on residue classes of $q$; see Beynon–Spaltenstein [2]. Thus, it seems desirable to find a general argument, uniformly for all characteristics $p$ and appropriately dealing with the “PORC” phenomenon — and this is what we will do in this paper.

It was first shown by Lusztig [21] (with some mild restrictions on $q$) and then by Shoji [28], [29] (in complete generality) that the original Green functions of [4] can be identified with another type of Green functions defined in terms of Lusztig’s character sheaves [17]. This provides new, extremely powerful tools.

In Section 2, we review the general plan for determining the Green functions, taking into account the above developments. The main result of Section 3 is Theorem 3.7, which is inspired by [7] and implies a crucial “PORC” property. In Section 4 we recall the basic ingredients of the Lusztig–Shoji algorithm which reduces the computation of the Green functions to the determination of certain signs. Our Theorem 3.7 does not determine these signs, but it ensures that the signs behave well with respect to replacing $F$ by $F^{r}$. In Section 5 we put all these pieces together to complete the proof of the Congruence Condition.

We only mention here that Theorem 3.7 is also useful in another direction, for computational purposes: in [8] it is the main theoretical tool to complete the computation of Green functions for several cases where these have not been previously known (e.g., type $E_{7}$ in characteristic $2,3$).

We assume some general familiarity with the theory of finite groups of Lie type and the character theory of these groups; see, e.g., [3], [13], [6].

Acknowledgements. The author is indebted to Gunter Malle and Jay Taylor for pointing out the open question about Green functions in the article of Hartley–Turull [11] at the Oberwolfach workshop “Representations of finite groups” in March 2019. I am also grateful to Gabriel Navarro for helpful comments, and to Jay Taylor for a careful reading of the paper. This work is a contribution to the SFB-TRR 195 “Symbolic Tools in Mathematics and their Application” of the German Research Foundation (DFG).

2. Green functions and character sheaves

Let $p$ be a prime and $k=\overline{{\mathbb{F}}}_{p}$ be an algebraic closure of the field with $p$ elements. Let $G$ be a connected reductive algebraic group over $k$ and assume that $G$ is defined over the finite subfield ${\mathbb{F}}_{q}\subseteq k$, where $q=p^{m}$ for some $m\geqslant 1$. Let $F\colon G\rightarrow G$ be the corresponding Frobenius map. Let $B_{0}\subseteq G$ be an $F$-stable Borel subgroup and $T_{0}\subseteq B_{0}$ be an $F$-stable maximal torus. Let $W=N_{G}(T_{0})/T_{0}$ be the corresponding Weyl group. For each $w\in W$, let $R_{w}$ be the virtual representation of the finite group $G^{F}$ defined by Deligne–Lusztig [4, §1]. (In the setting of [3, §7.2], we have $\mbox{Tr}(g,R_{w})=R_{T_{w},1}(g)$ for $g\in G^{F}$, where $T_{w}\subseteq G$ is an $F$-stable maximal torus obtained from $T_{0}$ by twisting with $w$, and $1$ stands for the trivial character of $T^{F}$.) This construction is carried out over $\overline{{\mathbb{Q}}}_{\ell}$, an algebraic closure of the $\ell$-adic numbers where $\ell$ is a prime not equal to $p$. The corresponding Green function is defined by

[TABLE]

where $G_{\text{uni}}$ denotes the set of unipotent elements of $G$. It is known that $Q_{w}(u)\in{\mathbb{Z}}$ for all $u\in G_{\text{uni}}^{F}$; see [3, §7.6]. So the character formula [3, 7.2.8] shows that we also have $\mbox{Tr}(g,R_{w})\in{\mathbb{Z}}$ for all $g\in G^{F}$.

The general plan for determining the values of $Q_{w}$ is explained in Lusztig [20, Chap. 24] and Shoji [27, §5], [30, 1.1–1.3] (even for generalised Green functions, which we will not consider here). We will have to go through some of the steps of that plan.

2.1. Almost characters

The Frobenius map $F$ induces an automorphism of $W$ which we denote by $\gamma\colon W\rightarrow W$. Let $\operatorname{Irr}(W)$ be the set of irreducible representations of $W$ over $\overline{{\mathbb{Q}}}_{\ell}$ (up to isomorphism). Let $\operatorname{Irr}(W)^{\gamma}$ be the set of all those $E\in\operatorname{Irr}(W)$ which are $\gamma$-invariant, that is, there exists a bijective linear map $\sigma_{E}\colon E\rightarrow E$ such that $\sigma_{E}{\circ}w=\gamma(w){\circ}\sigma_{E}\colon E\rightarrow E$ for all $w\in W$. Note that $\sigma_{E}$ is only unique up to scalar multiples but, if $\gamma$ has order $d\geqslant 1$, then one can always find some $\sigma_{E}$ such that

[TABLE]

see [14, 3.2]. In what follows, we assume that a fixed choice of $\sigma_{E}$ satisfying the above conditions has been made for each $E\in\operatorname{Irr}(W)^{\gamma}$. (For example, one could take the “preferred” choice for $\sigma_{E}$ specified by Lusztig [19, 17.2].) For $E\in\operatorname{Irr}(W)^{\gamma}$, the corresponding almost character is the class function $R_{E}\colon G^{F}\rightarrow\overline{{\mathbb{Q}}}_{\ell}$ defined by

[TABLE]

We have $R_{E}(g)\in{\mathbb{Q}}$ for all $g\in G^{F}$. By [14, 3.9], the functions $R_{E}$ are orthonormal with respect to the standard inner product on class functions of $G^{F}$. Furthermore, by [13, 3.19], we have

[TABLE]

Hence, knowing the values of all Green functions $Q_{w}$ is equivalent to knowing the values of all $R_{E}$ on $G_{\text{uni}}^{F}$.

2.2. Constructible $\overline{{\mathbb{Q}}}_{\ell}$-sheaves

Let ${\mathscr{D}}G$ be the bounded derived category of constructible $\overline{{\mathbb{Q}}}_{\ell}$-sheaves on $G$ (in the sense of Beilinson, Bernstein, Deligne [1]), which are equivariant for the action of $G$ on itself by conjugation. The “character sheaves” on $G$, defined by Lusztig [16], all belong to ${\mathscr{D}}G$. Consider any object $A\in\mathscr{D}G$ and suppose that its inverse image $F^{*}A$ under the Frobenius map is isomorphic to $A$ in $\mathscr{D}G$. Let $\phi\colon F^{*}A\stackrel{{\scriptstyle\sim}}{{\rightarrow}}A$ be an isomorphism. Then $\phi$ induces a linear map $\phi_{i,g}\colon{\mathscr{H}}_{F(g)}^{i}(A)\rightarrow{\mathscr{H}}_{g}^{i}(A)$ for each $i$ and $g\in G$, where ${\mathscr{H}}^{i}(A)$ denotes the $i$-th cohomology sheaf of $A$ and ${\mathscr{H}}_{g}^{i}(A)$ the stalk at $g\in G$. By [17, 8.4], this gives rise to a class function $\chi_{A,\phi}\colon G^{F}\rightarrow\overline{{\mathbb{Q}}}_{\ell}$, called a “characteristic function” of $A$, defined by

[TABLE]

Note that, by a version of Schur’s Lemma, $\phi$ is unique up to a non-zero scalar; hence, $\chi_{A,\phi}$ is unique up to a non-zero scalar. If $F^{*}A\cong A$, then one can choose an isomorphism $\phi_{A}\colon F^{*}A\stackrel{{\scriptstyle\sim}}{{\rightarrow}}A$ such that the values of $\chi_{A,\phi_{A}}$ are cyclotomic integers and the standard inner product of $\chi_{A,\phi_{A}}$ with itself is equal to $1$. The precise conditions which guarantee these properties are formulated in [18, 13.8], [20, 25.1]; note that these conditions specify $\phi_{A}$ up to multiplication by a root of unity.

2.3. The complexes $A_{E}$

Lusztig [17, §8.1] describes a geometric induction process by which one obtains objects in ${\mathscr{D}}G$ from objects in ${\mathscr{D}}L$ where $L$ is a Levi subgroup of some parabolic subgroup of $G$. Applying this to $L=T_{0}$ and the constant local system $\overline{{\mathbb{Q}}}_{\ell}$ on $T_{0}$, we obtain a well-defined complex $K\in{\mathscr{D}}G$ together with a canonical isomorphism $\varphi\colon F^{*}K\stackrel{{\scriptstyle\sim}}{{\rightarrow}}K$. The restriction of the corresponding characteristic function $\chi_{K,\varphi}\colon G^{F}\rightarrow\overline{{\mathbb{Q}}}_{\ell}$ to $G_{\text{uni}}^{F}$ is an example of a “generalised Green function”, as defined in [17, §8.3]; see also [28, 1.7]. We have $\mbox{End}(K)\cong\overline{{\mathbb{Q}}}_{\ell}[W]$ (see [17, 24.2]) and $K$ has a canonical decomposition

[TABLE]

where $A_{E}\in{\mathscr{D}}G$ is a character sheaf and $V_{E}=\mbox{Hom}(A_{E},K)$ is an irreducible $W$-module isomorphic to $E\in\operatorname{Irr}(W)$; see [30, 1.2]. Now let $E\in\operatorname{Irr}(W)^{\gamma}$. Then we also have $F^{*}A_{E}\cong A_{E}$ and, using our choice of $\sigma_{E}\colon E\rightarrow E$ in §2.1, we can single out a particular isomorphism $\phi_{A_{E}}\colon F^{*}A_{E}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}A_{E}$ as in §2.2. Since this will be important later, let us briefly indicate how this is done, following [20, 24.2] or [30, 1.3]. We start with any isomorphism $\phi_{A_{E}}\colon F^{*}A_{E}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}A_{E}$. Then there is a unique linear map $\psi_{E}\colon V_{E}\rightarrow V_{E}$ such that $\phi_{A_{E}}\otimes\psi_{E}$ corresponds to $\varphi\colon F^{*}K\stackrel{{\scriptstyle\sim}}{{\rightarrow}}K$ under the above direct sum decomposition; see [17, 10.4]. Furthermore, by [20, 24.2], $\psi_{E}$ corresponds under a $W$-module isomorphism $V_{E}\cong E$ to a non-zero scalar $\zeta\in\overline{{\mathbb{Q}}}_{\ell}$ times the map $\sigma_{E}\colon E\rightarrow E$. Hence, replacing $\phi_{A_{E}}$ by a scalar multiple, we can achieve that $\zeta=1$. Having fixed this choice of $\phi_{A_{E}}\colon F^{*}A_{E}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}A_{E}$, let $\chi_{A_{E}}\colon G^{F}\rightarrow\overline{{\mathbb{Q}}}_{\ell}$ be the corresponding characteristic function. Then, by the main result of Lusztig [21] and by Shoji [29, Theorem 5.5] (see also the argument in [28, 2.17, 2.18]), we have

[TABLE]

(In [28, 2.18], it is assumed that $q$ is a sufficiently large power of $p$, but this condition is later removed thanks to [29, Theorem 5.5].) The above identity is a special case of a more general conjecture about the relation between almost characters and characteristic functions of character sheaves; see [17, p. 226], [28], [29].

2.4. The Springer correspondence

Let ${\mathscr{N}}_{G}$ be the set of all pairs $(C,{\mathscr{E}})$ where $C$ is a unipotent class in $G$ and ${\mathscr{E}}$ is a $G$-equivariant irreducible $\overline{{\mathbb{Q}}}_{\ell}$-local system on $C$ (up to isomorphism). The Springer correspondence defines an injective map

[TABLE]

such that, if $E\in\operatorname{Irr}(W)$ and $\iota_{G}(E)=(C,{\mathscr{E}})$, then we have

[TABLE]

See Lusztig [15], [20, Chap 24], and the references there. Here, $\mbox{u-supp}(A)$ for any $A\in{\mathscr{D}}G$ is defined as the Zariski closure of

[TABLE]

Given $E\in\operatorname{Irr}(W)$ and $\iota_{G}(E)=(C,{\mathscr{E}})$, we define

[TABLE]

Note that $\dim C_{G}(g)\geqslant\dim T_{0}$ for $g\in G$. Furthermore, $d_{E}\in{\mathbb{Z}}_{\geqslant 0}$ since $\dim G-\dim T_{0}$ is always even and so is $\dim C$; see [3, §5.10] and the references there.

2.5. The $Y$-functions

Let $E\in\operatorname{Irr}(W)^{\gamma}$ and $\iota_{G}(E)=(C,{\mathscr{E}})$. Then $F(C)=C$ and $F^{*}{\mathscr{E}}\cong{\mathscr{E}}$. Since ${\mathscr{E}}\cong{\mathscr{H}}^{i}(A_{E})|_{C}$ for $i=-\dim C-\dim T_{0}$, the isomorphism $\phi_{A_{E}}\colon F^{*}A_{E}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}A_{E}$ induces a map $F^{*}{\mathscr{E}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}{\mathscr{E}}$ which we can write as $q^{d_{E}}\psi$ where $\psi\colon F^{*}{\mathscr{E}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}{\mathscr{E}}$ is an isomorphism. With this normalisation, $\psi$ induces an automorphism of finite order $\psi_{g}\colon{\mathscr{E}}_{g}\rightarrow{\mathscr{E}}_{g}$ at each stalk ${\mathscr{E}}_{g}$ where $g\in C^{F}$; see [20, 24.2.4]. For $g\in C^{F}$ and $i=-\dim C-\dim T_{0}$, we have

[TABLE]

where the class function $Y_{E}\colon G_{\text{uni}}^{F}\rightarrow\overline{{\mathbb{Q}}}_{\ell}$ is defined by

[TABLE]

see [20, 24.2.3]. In particular, the values of $Y_{E}$ are algebraic integers. Since $\dim C$ is an even number, we obtain that

[TABLE]

Since the values of $R_{E}$ are rational numbers (see §2.1), we conclude that

[TABLE]

The $Y$-functions $\{Y_{E}\mid E\in\operatorname{Irr}(W^{\gamma})\}$ are linearly independent by [20, 24.2.7].

2.6. The coefficients $p_{E^{\prime},E}$

Having established the above framework, Lusztig [20, Theorem 24.4] shows that we have unique equations

[TABLE]

where the coefficients $p_{E^{\prime},E}\in\overline{{\mathbb{Q}}}_{\ell}$ are determined by a purely combinatorial algorithm which we will consider in more detail in Section 4. Note that the hypotheses of [20, Theorem 24.4] (“cleanness”) are always satisfied by the main result of [23]. (Since we are only dealing with Green functions of $G^{F}$, and not with generalised Green functions, it would actually be sufficient to refer to [5, §3] instead of [23].) By [20, 24.5.2], we have

[TABLE]

Furthermore, by [20, 24.2.10, 24.2.11], we have

[TABLE]

Consequently, for a suitable ordering of $\operatorname{Irr}(W)^{\gamma}$, the matrix of coefficients $(p_{E^{\prime},E})$ will be triangular with $1$ along the diagonal (see also Section 4).

Thus, the whole problem of computing the Green functions $Q_{w}$ of $G^{F}$ is reduced to the determination of the functions $\{Y_{E}\mid E\in\operatorname{Irr}(W)^{\gamma}\}$ (cf. Shoji [30, 1.3]).

As in [30, 1.3], the above discussion also applies, with additional technical complications, to the generalized Green functions defined in [17, §8.3], but in this article we restrict ourselves to the “ordinary” Green functions $Q_{w}$.

3. Evaluating the $Y$-functions

Combining and summarizing the formulae in Section 2, we can state the following result about the values of the Green functions of $G^{F}$.

Proposition 3.1**.**

Let $w\in W$ and $u\in G^{F}$ be unipotent. Then

[TABLE]

where $\sigma_{E},d_{E},Y_{E^{\prime}},p_{E^{\prime},E}$ are defined in §2.1, §2.4, §2.5, §2.6, respectively.

Proof.

By §2.1, we can express $Q_{w}(u)$ as a linear combination of $R_{E}(u)$, for various $E\in\operatorname{Irr}(W)^{\gamma}$. By §2.6, we can express each term $R_{E}(u)$ as a linear combination of $Y_{E^{\prime}}(u)$, for various $E^{\prime}\in\operatorname{Irr}(W)^{\gamma}$. ∎

In this section, we address the further evaluation of the terms $Y_{E^{\prime}}(u)$. As we will use results from Shoji [28], [29] we will assume from now on that the Frobenius map $F\colon G\rightarrow G$ is given by

[TABLE]

where $\tilde{\gamma}\colon G\rightarrow G$ is an automorphism of finite order leaving $T_{0},B_{0}$ invariant, and $F_{p}\colon G\rightarrow G$ is a Frobenius map corresponding to a split ${\mathbb{F}}_{p}$-rational structure, such that $F_{p}(t)=t^{p}$ for all $t\in T_{0}$. Note that $F_{p}$ acts trivially on $W$ and that $\tilde{\gamma}$ induces an automorphism of $W$ which is just the automorphism $\gamma\colon W\rightarrow W$ induced by $F$ considered earlier. Thus, if $G$ is semisimple, then $G^{F}$ is an untwisted or twisted Chevalley group, as in Steinberg [32, §11.6].

Remark 3.2**.**

It is known that all unipotent classes of $G$ are $F_{p}$-stable (since, in each case, representatives of the classes are known which lie in $G^{F_{p}}=G({\mathbb{F}}_{p})$; see, e.g., Liebeck–Seitz [12]). Let $C$ be an $F$-stable unipotent class. We shall also make the following assumption.

• ($\clubsuit$)

There exists an element $u_{0}\in C^{F}$ such that $F$ acts trivially on the finite group of components $A(u_{0}):=C_{G}(u_{0})/C_{G}^{\circ}(u_{0})$.

If ($\clubsuit$) holds, then there is a bijective correspondence between the conjugacy classes of $A(u_{0})$ and the conjugacy classes of $G^{F}$ that are contained in the set $C^{F}$ (see, e.g., [12, Lemma 2.12]). For $a\in A(u_{0})$, an element in the corresponding $G^{F}$-conjugacy class is given by $u_{a}=hu_{0}h^{-1}$ where $h\in G$ is such that $h^{-1}F(h)\in C_{G}(u_{0})$ maps to $a$ under the natural homomorphism $C_{G}(u_{0})\rightarrow A(u_{0})$. (The existence of $h$ is guaranteed by Lang’s Theorem; note that $h$ is not unique but $u_{a}=hu_{0}h^{-1}$ is well-defined up to $G^{F}$-conjugacy.)

Let $E\in\operatorname{Irr}(W)^{\gamma}$ and $\iota_{G}(E)=(C,{\mathscr{E}})\in{\mathscr{N}}_{G}$. As in §2.5, we have $F(C)=C$ and $F^{*}{\mathscr{E}}\cong{\mathscr{E}}$. Furthermore, there is a certain isomorphism $\psi\colon F^{*}{\mathscr{E}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}{\mathscr{E}}$ which induces a map of finite order $\psi_{g}\colon{\mathscr{E}}_{g}\rightarrow{\mathscr{E}}_{g}$ for each $g\in C^{F}$. Now let us fix an element $u_{0}\in C^{F}$ as in ($\clubsuit$), such that $F$ acts trivially on $A(u_{0})$.

Lemma 3.3** (Cf. Lusztig [22, 19.7]).**

In the above setting, let $u_{0}\in C^{F}$ be such that ($\clubsuit$) holds. There is a natural $A(u_{0})$-module structure on the stalk ${\mathscr{E}}_{u_{0}}$. We have ${\mathscr{E}}_{u_{0}}\in\operatorname{Irr}(A(u_{0}))$ and the map $\psi_{u_{0}}\colon{\mathscr{E}}_{u_{0}}\rightarrow{\mathscr{E}}_{u_{0}}$ is given by scalar mulplication with a sign $\delta_{E}=\pm 1$. Furthermore, $Y_{E}(u_{a})=\delta_{E}\operatorname{Tr}(a,{\mathscr{E}}_{u_{0}})$ for all $a\in A(u_{0})$.

Note that $\mbox{Tr}(a,{\mathscr{E}}_{u_{0}})$ is just an entry in the ordinary character table of $A(u_{0})$. In particular, if $a=1$, then $u_{1}$ is $G^{F}$-conjugate to $u_{0}$ and so $Y_{E}(u_{0})=\delta_{E}\dim{\mathscr{E}}_{u_{0}}$.

Proof.

The $A(u_{0})$-module structure on ${\mathscr{E}}_{u_{0}}$ is explained in the proof of [22, 19.7]; this also shows that $\psi_{u_{0}}\circ a=a\circ\psi_{u_{0}}\colon{\mathscr{E}}_{u_{0}}\rightarrow{\mathscr{E}}_{u_{0}}$ for all $a\in A(u_{0})$. (Recall from ($\clubsuit$) that $F$ acts trivially on $A(u_{0})$.) Since ${\mathscr{E}}_{u_{0}}\in\operatorname{Irr}(A(u_{0}))$, we conclude that $\psi_{u_{0}}$ acts as a scalar times the identity on ${\mathscr{E}}_{u_{0}}$; let us denote this scalar by $\delta_{E}\in\overline{{\mathbb{Q}}}_{\ell}$. Then, for any $g\in C^{F}$, we have $Y_{E}(g)=\delta_{E}\mbox{Tr}(a,{\mathscr{E}}_{u_{0}})$ where $a\in A(u_{0})$ is such that $g$ is $G^{F}$-conjugate to $u_{a}$; see [22, 19.7]. Since $\psi_{u_{0}}\colon{\mathscr{E}}_{u_{0}}\rightarrow{\mathscr{E}}_{u_{0}}$ is a map of finite order, the scalar $\delta_{E}$ is a root of unity. Since the values of the almost characters $R_{E}$ are in ${\mathbb{Q}}$ (see §2.1), and since $R_{E}(u_{1})=q^{d_{E}}Y_{E}(u_{1})=q^{d_{E}}\delta_{E}\mbox{Tr}(1,{\mathscr{E}}_{u_{0}})$ (see §2.5), we conclude that we also have $\delta_{E}\in{\mathbb{Q}}$. Hence, we finally see that $\delta_{E}=\pm 1$. ∎

Thus, the problem of computing the Green functions $Q_{w}$ is further reduced to the determination of the signs $\delta_{E}=\pm 1$ for $E\in\operatorname{Irr}(W)^{\gamma}$ (cf. Shoji [30, 1.3, p. 161]).

Example 3.4**.**

Let $G$ be of type $E_{8}$ and $p>5$. Let $C$ be the unipotent class denoted by $D_{8}(a_{3})$ in Mizuno [24], or by $E_{8}(b_{6})$ in Carter [3, p. 407]. We have $\dim C=28$ and, up to conjugation within $G^{F}$, there is a unique $u_{0}\in C^{F}$ such that $|C_{G}(u_{0})^{F}|=6q^{28}$; see [24, p. 455]. We have $A(u_{0})\cong{\mathfrak{S}}_{3}$ and $F$ acts trivially on $A(u_{0})$. Let $E\in\operatorname{Irr}(W)$ be the irreducible representation denoted by $\phi_{840,13}$ in [3, §13.2], or by $840_{z}$ in [14, 4.13.1]. Then $\iota_{G}(E)=(C,{\mathscr{E}})$ where the irreducible $A(u_{0})$-module ${\mathscr{E}}_{u_{0}}$ corresponds to the sign representation of ${\mathfrak{S}}_{3}$; see [3, p. 432]. It is shown by Beynon–Spaltensetin [2, §3, Case 5] that

[TABLE]

In particular, there do exist cases in which $\delta_{E}=-1$.

Returning to the general setting, the following corollary interprets the signs $\delta_{E}=\pm 1$ somewhat more directly in terms of the character sheaves $A_{E}$ in §2.3.

Corollary 3.5**.**

Let $E\in\operatorname{Irr}(W)^{\gamma}$ and $\iota_{G}(E)=(C,{\mathscr{E}})\in{\mathscr{N}}_{G}$. Let $u_{0}\in C^{F}$ be as in ($\clubsuit$). Then the isomorphism $\phi_{A_{E}}\colon F^{*}A_{E}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}A_{E}$ in §2.3 induces the scalar multiplication by $\delta_{E}\,q^{d_{E}}$ on the stalk ${\mathscr{H}}_{u_{0}}^{i}(A_{E})$, where $i=-\dim C-\dim T_{0}$.

Proof.

This is just a reformulation of Lemma 3.3, noting that ${\mathscr{H}}^{i}(A_{E})|_{C}\cong{\mathscr{E}}$ and ${\mathscr{H}}^{j}(A_{E})|_{C}=0$ for $j\neq i$; see §2.4. ∎

Remark 3.6**.**

Let $d\geqslant 1$ be the order of the automorphism $\gamma\colon W\rightarrow W$; we set

[TABLE]

Let $E\in\operatorname{Irr}(W)^{\gamma}$ and $\iota_{G}(E)=(C,{\mathscr{E}})$ where $C$ is $F$-stable and $F^{*}{\mathscr{E}}\cong{\mathscr{E}}$. Let $u_{0}\in C^{F}$ be such that $F$ acts trivially on $A(u_{0})$; by Lemma 3.3, we have a corresponding sign $\delta_{E}=\pm 1$ such that

[TABLE]

Now let $r\in{\mathscr{M}}$ and replace $F$ by $F^{r}$. Thus, since $r\equiv 1\bmod d$, the automorphism of $W$ induced by $F^{r}$ is again given by $\gamma$. Hence, we can use the chosen map $\sigma_{E}\colon E\rightarrow E$ (see §2.1) to define a corresponding almost character of $G^{F^{r}}$, which we denote by $R_{E}^{(r)}$. Finally, we still have $u_{0}\in C^{F^{r}}$ and $F^{r}$ acts trivially on $A(u_{0})$. Hence, we also have a corresponding sign $\delta_{E}^{(r)}=\pm 1$ as in Lemma 3.3, such that

[TABLE]

The following result relates $\delta_{E}$ and $\delta_{E}^{(r)}$.

Theorem 3.7**.**

With the above notation, we have $\delta_{E}^{(r)}=(\delta_{E})^{r}$ for all $r\in{\mathscr{M}}$.

Proof.

We use the interpretation of $\delta_{E}$ in Corollary 3.5. Starting with the isomorphism $\phi:=\phi_{A_{E}}\colon F^{*}A_{E}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}A_{E}$ in §2.3, we obtain natural isomorphisms

[TABLE]

These give rise to an isomorphism

[TABLE]

We also have a canonical isomorphism $(F^{*})^{r}A_{E}\cong(F^{r})^{*}A_{E}$ which, finally, induces an isomorphism

[TABLE]

We denote the corresponding characteristic function of $A_{E}$ by $\chi^{(r)}\colon G^{F^{r}}\rightarrow\overline{{\mathbb{Q}}}_{\ell}$. Thus, we have

[TABLE]

Now assume that $g$ is an element in $G^{F}$, and not just in $G^{F^{r}}$. Then we have

[TABLE]

see [28, 1.1]. If we take $g=u_{0}$ (the chosen element in $C^{F}\subseteq C^{F^{r}}$) and let $i=-\dim C-\dim T_{0}$, then $\phi_{i,u_{0}}\colon{\mathscr{H}}_{u_{0}}^{i}(A_{E})\rightarrow{\mathscr{H}}_{u_{0}}^{i}(A_{E})$ is given by scalar multiplication with $\delta_{E}\,q^{d_{E}}$; see Corollary 3.5. So we conclude that

[TABLE]

Hence, again in view of Corollary 3.5, it remains to show that the isomorphism $\phi^{(r)}\colon(F^{r})^{*}A_{E}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}A_{E}$ constructed above is the particular isomorphism singled out in the discussion in §2.3. But this has already been checked essentially by Shoji in [28, 2.18.1]. For this purpose, we have to consider once more the decomposition

[TABLE]

As above, starting with the isomorphism $\varphi\colon F^{*}K\stackrel{{\scriptstyle\sim}}{{\rightarrow}}K$, we obtain a natural isomorphism $\varphi^{(r)}\colon(F^{r})^{*}K\stackrel{{\scriptstyle\sim}}{{\rightarrow}}K$. Then we have a unique linear map $\psi_{E}^{(r)}\colon V_{E}\rightarrow V_{E}$ such that $\phi^{(r)}\otimes\psi_{E}^{(r)}$ corresponds to $\varphi^{(r)}$ under the above direct sum decomposition. By an argument completely analogous to that in [7, 3.7], we see that $\psi_{E}^{(r)}=(\psi_{E})^{r}$, where $\psi_{E}\colon V_{E}\rightarrow V_{E}$ is determined by $\varphi$ and $\phi_{A_{E}}$ as in §2.3. Since $r\equiv 1\bmod d$ and $\psi_{E}$ has order dividing $d$, we conclude that $\psi_{E}^{r}=\psi_{E}$. Thus, $\phi^{(r)}$ also satisfies the requirements in §2.3. ∎

If $\tilde{\gamma}=\mbox{id}_{G}$ and $F=F_{p}^{m}$, then Theorem 3.7 shows that the determination of $\delta_{E}$ is reduced to the case where $m=1$. This is exploited in [8] to compute the values of Green functions for groups of exceptional type in small characteristic.

Remark 3.8**.**

Assume that $F=\tilde{\gamma}\circ F_{0}=F_{0}\circ\tilde{\gamma}$ where $F_{0}:=F_{p}^{m}$ as above and $\tilde{\gamma}\colon G\rightarrow G$ is non-trivial. Let $E\in\operatorname{Irr}(W)^{\gamma}$ and $\iota_{G}(E)=(C,{\mathscr{E}})\in{\mathscr{N}}_{G}$ where $C$ is $F$-stable and $F^{*}{\mathscr{E}}\cong{\mathscr{E}}$. Assume that there exists an element $u_{0}\in C$ such that

[TABLE]

Then, by Lemma 3.3, we have signs $\delta_{E}=\pm 1$ (with respect to $F$) and $\delta_{E}^{\circ}=\pm 1$ (with respect to $F_{0}=F_{p}^{m}$). We claim that

[TABLE]

(This remark is used, for example, in the determination of Green functions for groups of type ${{}^{2}\!E}_{6}$ in [8, §7].) A proof is as follows. As noted in [28, 2.17], we have $\tilde{\gamma}^{*}A_{E}\cong A_{E}$. Let $\nu_{A_{E}}\colon\tilde{\gamma}^{*}A_{E}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}A_{E}$ be an isomorphism. Then $\nu_{A_{E}}$ induces linear maps $\nu_{A_{E},i,u}\colon{\mathscr{H}}_{u_{0}}^{i}(A_{E})\rightarrow{\mathscr{H}}_{u_{0}}^{i}(A_{E})$ for all $i$. Let $\phi_{A_{E}}^{\circ}\colon F_{0}^{*}A_{E}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}A_{E}$ be an isomorphism as in §2.3. Since $F_{0}^{*}(\tilde{\gamma}^{*}A)\cong(F_{0}\circ\tilde{\gamma})^{*}A=F^{*}A$, we obtain

[TABLE]

where $F_{0}^{*}(\nu_{A_{E}})\colon F_{0}^{*}(\tilde{\gamma}^{*}A_{E})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}F_{0}^{*}A_{E}$ is induced by $\tilde{\gamma}^{*}A_{E}\cong A_{E}$. Replacing $\nu_{A_{E}}$ by a scalar multiple if necessary, we can assume that the above isomorphism satisfies the requirements in §2.3 (see also [28, 2.1, 2.17]). Now consider stalks at $u_{0}\in C$. Since $F_{p}(u_{0})=u_{0}=\tilde{\gamma}(u_{0})$, the above map $\nu_{A_{E},i,u_{0}}$ agrees with the map induced by $(F_{p}^{m})^{*}(\nu_{A_{E}})$ on ${\mathscr{H}}_{u_{0}}^{i}(A_{E})$. Hence, we obtain that

[TABLE]

Now let $i=-\dim C-\dim T_{0}$. By Corollary 3.5, $\phi_{A_{E},i,u_{0}}$ is given by scalar multiplication with $\delta_{E}\,q^{d_{E}}$; similarly, $\phi_{A_{E},i,u_{0}}^{\circ}$ is given by scalar multiplication with $\delta_{E}^{\circ}\,q^{d_{E}}$. Hence, $\nu_{A_{E},i,u_{0}}$ must also be given by scalar multiplication with a sign $\delta_{E}^{\tilde{\gamma}}=\pm 1$, such that $\delta_{E}=\delta_{E}^{\tilde{\gamma}}\,\delta_{E}^{\circ}$. Note that $\delta_{E}^{\tilde{\gamma}}$ only depends on $\tilde{\gamma}$.

4. Determining the coefficients $p_{E^{\prime},E}$

We keep the notation from the previous section. Taking into account the information in §2.6 and the orthogonality relations for Green functions, Lusztig [20, §24.4] has described a purely combinatorial algorithm for determining the coefficients $p_{E^{\prime},E}$, which modifies and simplifies an earlier algorithm of Shoji [27, §5].

We say that $w,w^{\prime}\in W$ are $\gamma$-conjugate, and write $w\sim_{\gamma}w^{\prime}$, if there exists some $x\in W$ such that $w^{\prime}=x^{-1}w\gamma(x)$. This defines an equivalence relation on $W$; the equivalence classes are called $\gamma$-conjugacy classes. For $w\in W$, we set

[TABLE]

Then the size of the $\gamma$-conjugacy class of $w$ is given by the index of $C_{W}^{\gamma}(w)$ in $W$. By [3, Prop. 3.3.6] and [3, Prop. 7.6.2], the Green functions of $G^{F}$ satisfy the following orthogonality relations, for any $w,w^{\prime}\in W$:

[TABLE]

We define the matrix $\tilde{\Omega}=(\tilde{\omega}_{E^{\prime},E})_{E^{\prime},E\in\operatorname{Irr}(W)^{\gamma}}$ where

[TABLE]

here, $T_{w}\subseteq G$ denotes an $F$-stable maximal torus obtained from $T_{0}$ by twisting with $w$ and the maps $\sigma_{E}\colon E\rightarrow E$, $\sigma_{E^{\prime}}\colon E^{\prime}\rightarrow E^{\prime}$ are as in §2.1. As in [13, 3.19], we have the formula

[TABLE]

Note also that, for $E\in\operatorname{Irr}(W)^{\gamma}$, the function $w\mapsto\mbox{Tr}(\sigma_{E}{\circ}w,E)$ is constant on $\gamma$-conjugacy classes (by the definition of $\sigma_{E}$). Combining this with the formulae in §2.1, the above orthogonality relations can be restated as follows:

[TABLE]

Lemma 4.1**.**

We have $\det(\tilde{\Omega})=\prod_{w}[G^{F}:T_{w}^{F}]$ where $w$ runs over a set of representatives of the $\gamma$-conjugacy classes of $W$.

Proof.

Let $w_{1},\ldots,w_{n}\in W$ be representatives of the $\gamma$-conjugacy classes of $W$. It is known that then $|\operatorname{Irr}(W)^{\gamma}|=n$; see, e.g., [9, Lemma 7.3]. Let us write $\operatorname{Irr}(W)^{\gamma}=\{E_{1},\ldots,E_{n}\}$. We define a matrix

[TABLE]

Then, by §2.1, we have $Q_{w_{i}}(g)=\sum_{1\leqslant i^{\prime}\leqslant n}x_{ii^{\prime}}R_{E_{i^{\prime}}}(g)$ for all $g\in G_{\text{uni}}^{F}$ and so

[TABLE]

On the other hand, the left hand side equals $[G^{F}:T_{w_{i}}^{F}]|C_{W}^{\gamma}(w_{i})|$ if $i=j$, and [math] otherwise. Hence, we find that

[TABLE]

It remains to show that $\det(X\cdot X^{\text{tr}})=\prod_{1\leqslant i\leqslant n}|C_{W}^{\gamma}(w_{i})|$. But this immediately follows from the identity

[TABLE]

which holds by the above-mentioned formula from [13, 3.19]. ∎

Following [20, 24.3.4], we define three matrices

[TABLE]

where, in each case, the indices run over all $E^{\prime},E\in\operatorname{Irr}(W)^{\gamma}$. Here, $p_{E^{\prime},E}$ are the coefficients in §2.6; furthermore,

[TABLE]

(The integers $d_{E}$ are defined in §2.4.)

Proposition 4.2** (Lusztig [20, 24.4]).**

We have $P^{\operatorname{tr}}\cdot\Lambda\cdot P=\Omega$. Furthermore, for all $E^{\prime},E\in\operatorname{Irr}(W)^{\gamma}$, we have $p_{E^{\prime},E}\in{\mathbb{Z}}$, $\lambda_{E^{\prime},E}\in{\mathbb{Z}}$, and $\omega_{E^{\prime},E}\in{\mathbb{Z}}$.

Proof.

Recall from §2.6 the following relations:

[TABLE]

This immediately implies the above matrix identity. The fact that $p_{E^{\prime},E}\in{\mathbb{Z}}$ was already mentioned in §2.6. The fact that $\lambda_{E^{\prime},E}\in{\mathbb{Z}}$ follows from the fact that the $Y$-functions are integer-valued; see §2.5. Finally, the above matrix identity implies that we also have $\omega_{E^{\prime},E}\in{\mathbb{Z}}$. ∎

We obtain further information about the matrices $P$ and $\Lambda$ by taking into account the additional information on $p_{E^{\prime},E}$ in §2.6. Let $E,E^{\prime}\in\operatorname{Irr}(W)^{\gamma}$ and $\iota_{G}(E)=(C,{\mathscr{E}})\in{\mathscr{N}}_{G}$, $\iota_{G}(E^{\prime})=(C^{\prime},{\mathscr{E}}^{\prime})\in{\mathscr{N}}_{G}$ where $C,C^{\prime}$ are $F$-stable and $F^{*}{\mathscr{E}}\cong{\mathscr{E}}$, $F^{*}{\mathscr{E}}^{\prime}\cong{\mathscr{E}}^{\prime}$. As in [20, 24.1], we write $E\sim E^{\prime}$ if $C=C^{\prime}$. This gives rise to a partition

[TABLE]

where ${\mathscr{I}}_{1},\ldots,{\mathscr{I}}_{h}$ are the equivalence classes for the relation $\sim$. Note that $d_{E}=d_{E^{\prime}}$ if $E\sim E^{\prime}$. Thus, we can define $d_{i}:=d_{E}$ where $E\in{\mathscr{I}}_{i}$.

Remark 4.3**.**

We fix a labelling of the equivalence classes ${\mathscr{I}}_{1},\ldots,{\mathscr{I}}_{h}$ such that

[TABLE]

and enumerate $\operatorname{Irr}(W)^{\sigma}$ in a way which is compatible with the above partition of $\operatorname{Irr}(W)^{\sigma}$. Then it is clear that $\Lambda$ has a block diagonal shape, where the blocks correspond to the sets ${\mathscr{I}}_{1},\ldots,{\mathscr{I}}_{h}$ (see [20, 24.3.2]). Furthermore, $P$ has an upper block triangular shape with identity matrices on the diagonal (see [20, 24.2.10, 24.2.11]). More precisely, we can write:

[TABLE]

here, $e_{j}=|{\mathscr{I}}_{j}|$ and $I_{e_{j}}$ denotes the identity matrix of size $e_{j}$. For $1\leqslant i<j\leqslant h$, the block $P_{i,j}$ has size $e_{i}\times e_{j}$ and entries $p_{E^{\prime},E}$ for $E^{\prime}\in{\mathscr{I}}_{i}$ and $E\in{\mathscr{I}}_{j}$; similarly, the block $\Lambda_{j}$ has size $e_{j}\times e_{j}$ and entries $\lambda_{E^{\prime},E}$ for $E^{\prime},E\in{\mathscr{I}}_{j}$.

With the additional requirement that $P$ and $\Lambda$ have block shapes as above, it easily follows that $P,\Lambda$ are uniquely determined by $\Omega$ and the equation $P^{\text{tr}}\cdot\Lambda\cdot P=\Omega$; see [20, 24.4], [27, §5] (or the proof of Lemma 4.4 below).

With these preparations, we can now establish a first step towards the proof of the Congruence Condition in Section 1. Let $d\geqslant 1$ be the order of $\gamma\colon W\rightarrow W$ and

[TABLE]

as in Remark 3.6. Let $r\in{\mathscr{M}}$ and replace $F$ by $F^{r}$. We obtain analogous matrices as above, which we now denote by

[TABLE]

and where the indices run again over all $E^{\prime},E\in\operatorname{Irr}(W)^{\gamma}$. (Note that $\gamma$ is also the automorphism of $W$ induced by $F^{r}$, since $r\in{\mathscr{M}}$.)

Lemma 4.4**.**

Let $r\in{\mathscr{M}}$ be a prime such that

[TABLE]

Then $p^{(r)}_{E^{\prime},E}\equiv p_{E^{\prime},E}\bmod r$ and $\lambda^{(r)}_{E^{\prime},E}\equiv\lambda_{E^{\prime},E}\bmod r$ for all $E^{\prime},E\in\operatorname{Irr}(W)^{\gamma}$.

Proof.

For $a\in{\mathbb{Z}}$, we denote by $\bar{a}\in{\mathbb{F}}_{r}$ the reduction modulo $r$. If $A=(a_{ij})$ is a matrix with entries in ${\mathbb{Z}}$, then we denote $\bar{A}=(\bar{a}_{ij})$. With this notation, we must show that $\bar{P}=\bar{P}^{(r)}$ and $\bar{\Lambda}=\bar{\Lambda}^{(r)}$ under the given assumptions, which mean that

[TABLE]

Now note that the Springer correspondence $\iota_{G}\colon\operatorname{Irr}(W)\rightarrow{\mathscr{N}}_{G}$ does not depend on any Frobenius map. Hence, the partition $\operatorname{Irr}(W)^{\gamma}={\mathscr{I}}_{1}\sqcup\ldots\sqcup{\mathscr{I}}_{h}$ and the block structure of $P$, $\Lambda$ in Remark 4.3 remain the same for $P^{(r)}$, $\Lambda^{(r)}$. Thus, $\Lambda$ and $\Lambda^{(r)}$ are block diagonal matrices; furthermore, $P$ and $P^{(r)}$ are upper block triangular matrices with identity blocks along the diagonal and so $\det(\bar{P})=\det(\bar{P}^{(r)})=1$. Consequently, we have $\det(\bar{\Lambda})=\det(\bar{\Omega})\neq 0$ and $\det(\bar{\Lambda}^{(r)})=\det(\bar{\Omega}^{(r)})=\det(\bar{\Omega})\neq 0$. So all of the above matrices are invertible. Hence, from the identities

[TABLE]

we can deduce the identiy

[TABLE]

Since the block shape remains the same when passing from $F$ to $F^{r}$, the left hand side of the above identity is a block lower triangular matrix with identity blocks along the diagonal, while the right hand side is a block upper triangular matrix. Hence, we conclude that $\bar{P}=\bar{P}^{(r)}$ and $\bar{\Lambda}=\bar{\Lambda}^{(r)}$, as desired. ∎

Remark 4.5**.**

Arguing as in the first part of the proof of [20, Theorem 24.8], one sees that there are well-defined polynomials $\pi_{E^{\prime},E}\in{\mathbb{Q}}[{\boldsymbol{q}}]$ (where ${\boldsymbol{q}}$ is an indeterminate) such that $p_{E^{\prime},E}^{(r)}=\pi_{E^{\prime},E}(q^{r})$ for all $r\in{\mathscr{M}}$. With a little extra work, one can show that $\pi_{E^{\prime},E}\in{\mathbb{Z}}[{\boldsymbol{q}}]$. This would, of course, also imply the conclusion of Lemma 4.4, as far as the $p_{E^{\prime},E}$ and $p_{E^{\prime},E}^{(r)}$ are concerned.

5. Proof of the Congruence Condition

As remarked in [11, p. 202], it is sufficient to prove the Congruence Condition in Section 1 in the case where $G$ is simple of adjoint type. We assume for the rest of this section that this is the case. Let us begin with an endomorphism $F^{\prime}\colon G\rightarrow G$ such that some power of $F^{\prime}$ is a Frobenius map. Then $G^{F^{\prime}}$ is one of the groups considered by Steinberg [32, §11.6]. It has been already shown in [11, Prop. 7.7] that the Congruence Condition holds for $(G,F^{\prime})$ of type ${{}^{3}\!D}_{4}$, ${{}^{2}\!B}_{2}$, ${{}^{2}\!G}_{2}$, ${{}^{2}\!F}_{4}$. So it remains to consider the case where $F^{\prime}=F$ is a Frobenius map. Assume now that this is the case. Thus, we have

[TABLE]

as in Section 3, where $\tilde{\gamma}\colon G\rightarrow G$ is a graph automorphism of order $d\in\{1,2,3\}$ leaving $T_{0},B_{0}$ invariant, and $F_{p}\colon G\rightarrow G$ is a Frobenius map corresponding to a split ${\mathbb{F}}_{p}$-rational structure, such that $F_{p}(t)=t^{p}$ for all $t\in T_{0}$. Then the map $\gamma\colon W\rightarrow W$ induced by $\tilde{\gamma}$ also has order $d$. Since groups of type ${{}^{3}\!D}_{4}$ have already been dealt with, we may actually assume that $d\in\{1,2\}$. We set

[TABLE]

as in Remark 3.6. (Thus, ${\mathscr{M}}$ consists of all positive integers if $d=1$, and of all odd positive integers if $d=2$.) Let $r\in{\mathscr{M}}$. Working with both $F$ and $F^{r}$, we will have to consider two sets of matrices:

[TABLE]

defined as in the previous section. For each $E\in\operatorname{Irr}(W)^{\gamma}$, we have an almost character $R_{E}$ and a $Y$-function $Y_{E}$ for $G^{F}$; similarly, we have an almost character and a $Y$-function for $G^{F^{r}}$, which we denote by $R_{E}^{(r)}$ and $Y_{E}^{(r)}$, respectively. This notation will be used throughout this section.

Lemma 5.1**.**

Let $r\in{\mathscr{M}}$ be a prime. Then we have $|G^{F^{r}}|\equiv|G^{F}|\bmod r$ and $|T_{w}^{F^{r}}|\equiv|T_{w}^{F}|\bmod r$ for any $w\in W$.

Proof.

Let $X^{\vee}(T_{0})=\mbox{Hom}(k^{\times},T_{0})$ be the co-character group of $T_{0}$ (a free abelian group of rank equal to $\dim T_{0}$). Then $\tilde{\gamma}$ induces an automorphism $\tilde{\gamma}^{*}\in\mbox{Aut}(X^{\vee})$ such that $\tilde{\gamma}^{*}(\nu)(x)=\tilde{\gamma}(\nu(x))$ for all $x\in k^{\times}$ and $\nu\in X^{\vee}$. We can also naturally regard $W$ as a subgroup of $\mbox{Aut}(X^{\vee})$; see [3, §1.9]. Let ${\boldsymbol{q}}$ be an indeterminate over ${\mathbb{Z}}$ and define

[TABLE]

Then we have $|T_{w}^{F}|=f_{w}(q)$ where $q=p^{m}$; see [3, Prop. 3.3.5]. Furthermore, we have $|G^{F}|=f(q)$, where we define

[TABLE]

see [3, §2.9]. Now let us replace $F$ by $F^{r}$, where $r\in{\mathscr{M}}$. Then $F^{r}=\tilde{\gamma}\circ F_{p}^{rm}=F_{p}^{rm}\circ\tilde{\gamma}$. Consequently, we obtain that $|T_{w}^{F^{r}}|=f_{w}(q^{r})$ and $|G^{F^{r}}|=f(q^{r})$. If $r\in{\mathscr{M}}$ is a prime, then Fermat’s Little Theorem yields the desired congruences. ∎

Lemma 5.2**.**

Let $r\in{\mathbb{N}}$ be a prime such that $r\nmid|G^{F^{r}}|$. Then $\omega_{E^{\prime},E}^{(r)}\equiv\omega_{E^{\prime},E}\bmod r$, $p_{E^{\prime},E}^{(r)}\equiv p_{E^{\prime},E}\bmod r$ and $\lambda_{E^{\prime},E}^{(r)}\equiv\lambda_{E^{\prime},E}\bmod r$ for all $E^{\prime},E\in\operatorname{Irr}(W)^{\gamma}$.

Proof.

Since $G$ is non-trivial, the order of $|G^{F^{r}}|$ is even and so $r>2$. Hence, we automatically have $r\in{\mathscr{M}}$. Let $w_{1},\ldots,w_{n}\in W$ be a set of representatives of the $\gamma$-conjugacy classes of $W$. First we consider the matrix $\tilde{\Omega}$. We have

[TABLE]

Since $G^{F}\subseteq G^{F^{r}}$, we also have $r\nmid|G^{F}|$. Hence, using Lemma 5.1, we obtain that

[TABLE]

By [3, Prop. 3.3.6], we have $N_{G}(T_{w}^{F})/T_{w}^{F}\cong C_{W}^{\gamma}(w)$. Since $N_{G}(T_{w})^{F}\subseteq G^{F}$, we conclude that $r\nmid|C_{W}^{\gamma}(w)|$. Consequently, we have $\tilde{\omega}_{E^{\prime},E}^{(r)}\equiv\tilde{\omega}_{E^{\prime},E}\bmod r$. Now

[TABLE]

Using $r\nmid q$ and Fermat’s Little Theorem, we conclude that we also have ${\omega}_{E^{\prime},E}^{(r)}\equiv{\omega}_{E^{\prime},E}\bmod r$. Finally, by Lemma 4.1, we have $r\nmid\det(\tilde{\Omega})$ and, hence, also $r\nmid\det(\Omega)$ (again, since $r\nmid q$). So we can apply Lemma 4.4. ∎

Lemma 5.3**.**

Let $u\in G^{F}$ be unipotent and $E\in\operatorname{Irr}(W)^{\gamma}$. Let $r\in{\mathbb{N}}$ be a prime such that $r\nmid|G^{F^{r}}|$. Then $Y_{E}^{(r)}(u)=Y_{E}(u)$.

Proof.

As in the previous proof, we have $r>2$ and $r\in{\mathscr{M}}$. Let $\iota_{G}(E)=(C,{\mathscr{E}})$ where $C$ is $F$-stable and $F^{*}\cong{\mathscr{E}}$. If $u\not\in C$, then $Y_{E}(u)=Y_{E}^{(r)}(u)=0$. So now let $u\in C$. Since we are assuming that $G$ is simple of adjoint type, it is known that there exists an element $u_{0}\in C^{F}$ such that $(\clubsuit)$ in Remark 3.2 holds. For $G$ of classical type, such a representative $u_{0}\in C$ is explicitly described by Shoji [31, §2]; see, e.g., [31, 2.10] for type ${{}^{2}\!D}_{n}$ in arbitrary characteristic. If $G$ is of exceptional type, the existence of $u_{0}$ is guaranteed by [12, Lemma 20.16], the proof of which involves a certain amount of case–by–case considerations. For example, for type $E_{7},E_{8}$ one can simply look through the list of class representatives determined by Mizuno [24]. See also the discussion in [33, §2] (in good characteristic). In any case, let us now fix an element $u_{0}\in C^{F}$ such that $F$ acts trivially on $A(u_{0})$. Then $u=gu_{0}g^{-1}$ for some $g\in G$, and we have $x:=g^{-1}F(g)\in C_{G}(u_{0})$; we denote by $a$ the image of $x$ in $A(u)$. Consequently, we have

[TABLE]

Note that, since the values of the $Y$-functions are integers, the same is true for $\mbox{Tr}(a,{\mathscr{E}}_{u_{0}})$. Let us now replace $F$ by $F^{r}$. We still have $x^{\prime}:=g^{-1}F^{r}(g)\in C_{G}(u_{0})$; consequently, if we denote by $a^{\prime}$ the image of $x^{\prime}$ in $A(u_{0})$, then

[TABLE]

(and, again, these values are integers). By Theorem 3.7, we have $\delta_{E}^{(r)}=(\delta_{E})^{r}$. Since $r$ is odd, we conclude that $\delta_{E}^{(r)}=\delta_{E}$. Hence, it remains to show that

[TABLE]

Now, since $F$ acts trivially on $A(u_{0})$, we have $A(u_{0})\cong C_{G}(u_{0})^{F}/C_{G}^{\circ}(u_{0})^{F}$ (see [3, p. 33]). Hence, since $r\nmid|G^{F}|$, we also have $r\nmid|A(u_{0})|$. But then a well-known result in the character theory of finite groups implies that $\mbox{Tr}(a,{\mathscr{E}}_{u_{0}})=\mbox{Tr}(a^{r},{\mathscr{E}}_{u_{0}})$. So it will be enough to show that $a^{r}=a^{\prime}$. This is seen as follows. We have $F(g)=gx$ and $F^{r}(g)=gx^{\prime}$, with $x,x^{\prime}\in C_{G}(u_{0})$, which implies that

[TABLE]

Since $F$ acts trivially on $C_{G}(u_{0})/C_{G}^{\circ}(u_{0})$, we have $x^{-1}F^{i}(x)\in C_{G}^{\circ}(u_{0})$ for all $i\geqslant 1$. This yields $x^{\prime}=x^{r}c$ for some $c\in C_{G}^{\circ}(u_{0})$ and, hence, $a^{\prime}=a^{r}$, as desired. ∎

We can now complete the proof of the Congruence Condition, as follows. Let $u\in G^{F}$ be unipotent and $T\subseteq G$ be an $F$-stable maximal torus. Let $w\in W$ be such that $T$ is obtained from $T_{0}$ by twisting with $w$ (relative to $F$). Thus, we have $Q_{T,F}=Q_{w}$. Let $r\in{\mathbb{N}}$ be a prime such that $r\nmid|G^{F^{r}}|$; then $r>2$ and $r\in{\mathscr{M}}$ (see the above proofs). As pointed out in [11, p. 204], we also have that $T$ is obtained by twisting with $w$ relative to $F^{r}$. Thus, we have $Q_{T,F^{r}}(u)=Q_{w}^{(r)}(u)$ where, as usual, we indicate by the superscript “$(r)$” that we mean the Green function for $G^{F^{r}}$. Now, by Proposition 3.1, we have the following formulae.

[TABLE]

By Lemma 5.2, we have $p_{E^{\prime},E}^{(r)}\equiv p_{E^{\prime},E}\bmod r$; furthermore, by Lemma 5.3, we have $Y_{E}^{(r)}(u)=Y_{E}(u)$. Finally, by Fermat’s Little Theorem, we have $q^{d_{E}}\equiv q^{d_{E}r}\bmod r$. Hence, we conclude that $Q_{w}(u)\equiv Q_{w}^{(r)}(u)\bmod r$. Thus, the Congruence Condition in Section 1 is proved. ∎