2.2.1

For any $n\in \mathbb {Z}_{>0}$ , we denote by $\mathfrak {S}_n$ the group of permutations of the set

$$ \begin{align*}\mathbb{I}(n):=\{1,2,\ldots,n\}.\end{align*} $$

Each permutation $\tau \in \mathfrak {S}_n$ can be decomposed into a product of cycles $c_{I_1}\cdots c_{I_l}$ , where the disjoint subsets $I_r\subset \mathbb {I}(n)$ form a partition of $\mathbb {I}(n)$ and $c_{I_r}$ is a circular permutation on $I_r$ . For each $1\le r\le l$ , put $\tau _r=|c_{I_r}|$ , the size of $I_r$ , and then the conjugacy class of $\tau $ is determined by the partition $(\tau _1,\ldots ,\tau _l)$ .

Let $n\in \mathbb {Z}_{\ge 0}$ . A partition of n will often be written as a nonincreasing sequence of nonnegative integers $\lambda =(\lambda _1\ge \lambda _2\ge \cdots )$ such that $n=\sum _k\lambda _k$ , and $0$ only has the empty partition $(0,0,\ldots )$ . The length of $\lambda $ will be denoted by $l(\lambda )$ , and n is called the size of $\lambda $ . For each partition $\lambda $ , we define the integer $n(\lambda ):=\sum _i(i-1)\lambda _i$ . The dual of a partition $\lambda $ will be denoted by $\lambda ^{\ast }$ . The set of partitions of n will be denoted by $\mathcal {P}(n)$ . Write $\mathcal {P}=\sqcup _{n\in \mathbb {Z}_{\ge 0}}\mathcal {P}(n)$ . The irreducible characters and the conjugacy classes of $\mathfrak {S}_n$ are both parametrised by $\mathcal {P}(n)$ . For any $\lambda \in \mathcal {P}(n)$ , we will denote by $\chi ^{\lambda }$ the corresponding irreducible character, and by $\chi ^{\lambda }_{\mu }$ the value of this character on the conjugacy class corresponding to $\mu \in \mathcal {P}(n)$ .

Write $\mathcal {P}^2=\mathcal {P}\times \mathcal {P}$ . The elements of $\mathcal {P}^2$ will be called 2-partitions. For any $\boldsymbol{\lambda} =(\lambda ^{(0)},\lambda ^{(1)})\in \mathcal {P}^2$ , its size is defined by $|\boldsymbol{\lambda} |:=|\lambda ^{(0)}|+|\lambda ^{(1)}|$ , and its length is defined by $l(\boldsymbol{\lambda} ):=l(\lambda ^{(0)})+l(\lambda ^{(1)})$ . The dual of a 2-partition $\boldsymbol{\lambda} =(\lambda ^{(0)},\lambda ^{(1)})$ is defined by $\boldsymbol{\lambda} ^{\ast }:=(\lambda ^{(1)\ast },\lambda ^{(0)\ast })$ . The set of 2-partitions of size n will be denoted by $\mathcal {P}^2(n)$ .

2.2.2

Given a partition $\lambda =(\lambda _1,\lambda _2,\ldots )$ of size n, we take $r\ge l(\lambda )$ , and we put

$$ \begin{align*}\delta_r=(r-1,r-2,\ldots,1,0).\end{align*} $$

Let $(2y_1>\cdots >2y_{l_0})$ and $(2y_1'+1>\cdots >2y^{\prime }_{l_1}+1)$ be the even parts and the odd parts of $\lambda +\delta _r$ , where the sum is taken term by term. Denote by $\lambda '$ the partition that has as its parts the numbers $2s+t$ , $0\le s\le l_t-1$ , $t=0,1$ . We have $l(\lambda ')=l(\lambda )$ . The 2-core of $\lambda $ is the partition defined by $(\lambda ^{\prime }_k-l(\lambda )+k)_{1\le k\le l(\lambda )}$ . It is independent of r and necessarily of the form $(d,d-1,\ldots ,2,1,0)$ , for some $d\in \mathbb {Z}_{\ge 0}$ . Denote by $\smash {\lambda ^{(0)}}$ the partition defined by $\smash {\lambda ^{(0)}_k}=y_k-l_0+k$ and denote by $\smash {\lambda ^{(1)}}$ the partition defined by $\smash {\lambda ^{(1)}_k}=y^{\prime }_k-l_1+k$ . Then $(\lambda ^{(0)},\lambda ^{(1)})_r$ is a 2-partition that depends on the parity of r, which we call the 2-quotient of $\lambda $ . Changing the parity of r will permute $\lambda ^{(0)}$ and $\lambda ^{(1)}$ . We make the convention that $r\equiv 1\quad\mod 2$ if the 2-core is $(0)$ and $r\equiv 0\quad\mod 2$ if the 2-core is $(1)$ .

The above constructions give a bijection

(2.2.2.1) $$ \begin{align} \left\{\begin{array}{c}\text{Partitions of } n\\ \text{with 2-core } (d,d-1,\ldots,1,0)\end{array}\right\} \longleftrightarrow\left\{\text{2-partitions of } \frac{1}{2}(n-\frac{d(d+1)}{2})\right\}. \end{align} $$

We call this bijection the quotient-core decomposition of partitions.

2.3.1

Given a finite group $\Gamma $ and a positive integer m, the symmetric group $\mathfrak {S}_m$ acts on the direct product $\Gamma ^m$ of m copies of $\Gamma $ by permuting the factors. This defines a semi-direct product $\Gamma ^m\rtimes \mathfrak {S}_m$ , called a wreath product. We will denote by $\mathfrak {W}_m$ the wreath product defined by $\Gamma =\mathbb {Z}/2\mathbb {Z}$ . It is the Weyl group of $\operatorname {\mathrm {Sp}}_{2m}$ and $\operatorname {\mathrm {SO}}_{2m+1}$ . More general wreath products $(\mathbb {Z}/r\mathbb {Z})^m\rtimes \mathfrak {S}_m$ for $r>2$ cannot be realised as the Weyl groups of any algebraic groups. We will always identify $\mathbb {Z}/2\mathbb {Z}$ with $\boldsymbol{\mu} _2$ and write its elements in the multiplicative form.

2.3.2

Denote by $w_0$ the permutation $(1,-1)(2,-2)\cdots (m,-m)$ of the set

$$ \begin{align*}\bar{\mathbb{I}}(m):=\{1,\ldots,m,-m,\ldots,-1\}.\end{align*} $$

We can identify $\mathfrak {W}_m$ as the subgroup of the group of permutations on $\bar {\mathbb {I}}(m)$ consisting of elements commuting with $w_0$ . Then each element $\tau \in \mathfrak {S}_m\subset \mathfrak {W}_m$ is identified with the permutation

$$ \begin{align*}i\mapsto \tau(i),\quad-i\mapsto\tau(-i)=-\tau(i),\end{align*} $$

and for any $i\in \{1,\ldots ,m\}$ , the element $(e_1,\ldots ,e_m)\in (\mathbb {Z}/2\mathbb {Z})^m\subset \mathfrak {W}_m$ , with $e_i=-1$ and $e_j=1$ if $j\ne i$ , is identified with the permutation $(i,-i)$ .

2.3.3

Let $w=((e_1,\ldots ,e_m),\tau )\in (\mathbb {Z}/2\mathbb {Z})^m\rtimes \mathfrak {S}_m$ . Then $\tau $ can be written as $\tau =c_{I_1}\cdots c_{I_l}$ as in §2.2.1. For each $1\le r\le l$ , put $\bar {e}_r=\prod _{k\in I_r}e_k$ . For any $1\le r\le l$ , we will call $((e_r)_{r\in I_r},c_{I_r})$ a positive (resp. negative) cycle in w if $\bar {e}_r=1$ (resp. $\bar {e}_r=-1$ ), so that w is a product of signed cycles. The size of a signed cycle in w is defined as the size of the corresponding cycle in $\tau $ . Define the permutations

(2.3.3.1) $$ \begin{align} \tau^{(0)}=\prod_{\bar{\epsilon}_r=1}c_{I_r},\quad \tau^{(1)}=\prod_{\bar{\epsilon}_r=-1}c_{I_r}, \end{align} $$

so that $\tau =\tau ^{(0)}\tau ^{(1)}$ . Also denote by $\tau ^{(0)}$ and $\tau ^{(1)}$ the associated partitions by abuse of notation. We then have a 2-partition $(\tau ^{(0)},\tau ^{(1)})$ , which is sometimes called a signed partition of m. This 2-partition determines the conjugacy class of w. The conjugacy classes and irreducible characters of $\mathfrak {W}_m$ are both parametrised by $\mathcal {P}^2(m)$ . For any $\boldsymbol{\lambda} \in \mathcal {P}^2(m)$ , we will denote by $\chi ^{\boldsymbol{\lambda} }$ the corresponding irreducible character, and by $\chi ^{\boldsymbol{\lambda} }_{\boldsymbol{\mu} }$ the value of this character on the conjugacy class corresponding to $\boldsymbol{\mu} \in \mathcal {P}^2(m)$ .

Regarded as a permutation on the set $\bar {\mathbb {I}}(m)$ , an element w such that $\tau =\tau ^{(0)}=(1,2,\ldots ,m)$ is typically of the form

(2.3.3.2)

and an element w such that $\tau =\tau ^{(1)}=(1,2,\ldots ,m)$ is typically of the form

(2.3.3.3)

2.4.1

Let G be a connected reductive group and let W be the Weyl group of G defined by a maximal torus. Denote by $\mathscr {N}$ the set consisting of pairs $(C,\phi )$ , where $C\subset G$ is a unipotent conjugacy class and $\phi $ is an irreducible character of $C_G(u)/C_G(u)^{\circ }$ for some $u\in C$ . The Springer correspondence is an injection ([Reference ShojiSho87, §4])

(2.4.1.1) $$ \begin{align} \operatorname{\mathrm{Irr}}(W) \hookrightarrow \mathscr{N}. \end{align} $$

There is an equivalence relation on $\mathscr {N}$ that identifies $(C_1,\phi _1)$ and $(C_2,\phi _2)$ whenever $C_1=C_2$ , in which case we say $(C_1,\phi _1)$ and $(C_2,\phi _2)$ are similar. Such an equivalence class is called a similarity class.

Symbols are certain combinatorial data that are in bijection with the image of (2.4.1.1). If $G=\operatorname {\mathrm {Sp}}_{2m}$ or $\operatorname {\mathrm {SO}}_{2m+1}$ , then $\operatorname {\mathrm {Irr}}(W)$ is in bijection with $\mathcal {P}^2(m)$ . Therefore, the symbols are also in bijection with $\mathcal {P}^2(m)$ . Similarity classes in $\mathcal {P}^2(m)$ or in the set of symbols are induced from $\mathscr {N}$ . If $\boldsymbol{\alpha} $ is a 2-partition, we will denote by $\boldsymbol{\Lambda} (\boldsymbol{\alpha} )$ the corresponding symbol. In this article, we do not need the notion of symbols in an essential way. We use them only to match the notations in [Reference ShojiSho01].

2.4.2

For any symbol $\boldsymbol{\Lambda }$ , let $a(\boldsymbol{\Lambda })$ be the integer defined by [Reference ShojiSho01, (1.2.2)]. This is the analogue of $n(\lambda )$ for partitions. Its value is constant on the similarity classes of symbols. If we denote by u an element of the unipotent conjugacy class corresponding to $\boldsymbol{\Lambda }$ , then

(2.4.2.1) $$ \begin{align} a(\boldsymbol{\Lambda})=\dim\mathcal{B}_u, \end{align} $$

where $\mathcal {B}_u$ is the Springer fiber over u. For any 2-partition $\boldsymbol{\alpha} $ , we define $a(\boldsymbol{\alpha }):=a(\boldsymbol{\Lambda }(\boldsymbol{\alpha }))$ . We will fix once and for all a total order $\prec $ on the set of symbols in such a way that if $C_1$ (resp. $C_2$ ) is the unipotent class corresponding to $\boldsymbol{\Lambda} _1$ (resp. $\boldsymbol{\Lambda} _2$ ), then $\boldsymbol{\Lambda} _2\prec \boldsymbol{\Lambda} _1$ if $C_2\subset \bar {C}_1$ , and each similarity class forms an interval. We will write $\boldsymbol{\Lambda} _1\sim \boldsymbol{\Lambda} _2$ if $\boldsymbol{\Lambda} _1$ and $\boldsymbol{\Lambda} _2$ are similar. Note that $a(\boldsymbol{\Lambda} _1)\le a(\boldsymbol{\Lambda} _2)$ whenever $\boldsymbol{\Lambda} _2\prec \boldsymbol{\Lambda} _1$ . The set of 2-partitions acquires a total order via the bijection with symbols, so that $a(\boldsymbol{\alpha })\le a(\boldsymbol{\beta })$ whenever $\boldsymbol{\beta }\prec \boldsymbol{\alpha }$ . In particular, the element $\boldsymbol{\alpha }_0:=(\varnothing ,(1^n))$ corresponding to the the identity of the finite classical group is minimal, and it is alone in its similarity class.

2.5.1

Let G be a not necessarily connected linear algebraic group over an algebraically closed field . We say that G is reductive if $G^{\circ }$ is reductive. Let $B\subset G^{\circ }$ be a Borel subgroup and let $T\subset B$ be a maximal torus. Then $\bar {T}:=N_G(T,B)$ has nonempty intersection with every connected component of G, since the conjugation by any element of G sends T and B to some other maximal torus and Borel subgroup, which are conjugate to T and B by an element of $G^{\circ }$ . An element of G is called quasi-semi-simple if it lies in $\bar {T}$ for some $T\subset B$ . It is known that semi-simple elements are always quasi-semi-simple ([Reference SteinbergSte68, Theorem 7.5]). We will always assume that . Under this assumption, all unipotent elements of G are contained in $G^{\circ }$ and all quasi-semi-simple elements are semi-simple ([Reference Digne and MichelDM94, Remarque 2.7]).

2.5.2

Let $G^1$ be a connected component of G and let $\sigma \in G^1$ be a semi-simple element. Let $T\subset G^{\circ }$ be a maximal torus that is normalised by $\sigma $ . Denote by $W=W_{G^{\circ }}(T)$ the Weyl group of $G^{\circ }$ defined by T. Then $\sigma $ induces an action on W. Denote by $W^{\sigma }$ the subgroup of $\sigma $ -fixed points. The subtori $(T^{\sigma })^{\circ }$ and $[T,\sigma ]$ are preserved by the action of $W^{\sigma }$ on T.

A semi-simple element $\sigma $ is quasi-central in $G^1$ if there is no element $g\in G^{\circ }$ such that $\dim C_{G^{\circ }}(\sigma )^{\circ }<\dim C_{G^{\circ }}(g\sigma )^{\circ }.$

2.5.3

A closed subgroup of G is a parabolic subgroup if $G/P$ is proper. According to [Reference SpringerSpr98, Lemma 6.2.4], a subgroup $P\subset G$ is parabolic if and only if $P^{\circ }$ is parabolic in $G^{\circ }$ . For example, if P is a parabolic subgroup of $G^{\circ }$ , then P and $N_G(P)$ are both parabolic subgroups of G. It is obvious that the intersection of $N_G(P)$ with any connected component of G is isomorphic to P if it is nonempty. We want to determine which connected components of G have nonempty intersection with $N_G(P)$ .

There is a well-defined action of $G/G^{\circ }$ on the set of $G^{\circ }$ -conjugacy classes of parabolic subgroups of $G^{\circ }$ , induced by the conjugation action of G. Let $G^1\subset G$ be a connected component and $P\subset G^{\circ }$ a parabolic subgroup. Then $N_G(P)$ meets $G^1$ if and only if the $G^{\circ }$ -conjugacy class of P is $G^1$ -stable. Let $L\subset P$ be a Levi factor. If $N_G(P)$ meets a connected component $G^1$ of G, then $N_G(L,P)$ also meets $G^1$ , since any two Levi factors of P are conjugate under P.

Fix a semi-simple element $\sigma \in G^1$ . Then by [Reference Digne and MichelDM94, Proposition 1.6] and the above discussions, $N_G(P)$ meets $G^1$ if and only if there exists a $G^{\circ }$ -conjugate of P that is $\sigma $ -stable, and $N_G(L,P)$ meets $G^1$ if and only if there exists a $G^{\circ }$ -conjugate of the pair $(L,P)$ that is $\sigma $ -stable. The following propositions describe the set of $\sigma $ -stable parabolic subgroups and the set of $\sigma $ -stable Levi factors of $\sigma $ -stable parabolic subgroups.

2.5.4

The following proposition will be used via its corollaries.

With the notion of isolated elements, the following two corollaries could be integrated into one uniform result. But we prefer not to introduce more notions.

Corollary 2.5.6. Let $\sigma \in G$ be quasi-central and let L be a $\sigma $ -stable Levi factor of a $\sigma $ -stable parabolic subgroup of $G^{\circ }$ . Write

$$ \begin{align*}W_{(G^{\sigma})^{\circ}}((L^{\sigma})^{\circ}):=N_{(G^{\sigma})^{\circ}}((L^{\sigma})^{\circ})/(L^{\sigma})^{\circ},\quad W_{G^{\circ}}(L):=N_{G^{\circ}}(L)/L.\end{align*} $$

Then there is a natural injective map

$$ \begin{align*}W_{(G^{\sigma})^{\circ}}((L^{\sigma})^{\circ})\hookrightarrow W_{G^{\circ}}(L)^{\sigma}.\end{align*} $$

Corollary 2.5.7. Let $\sigma \in G$ be quasi-central. Let $T\subset G^{\circ }$ be a $\sigma $ -stable maximal torus and let $s\in T$ . Write

$$ \begin{align*}W_{(G^{s\sigma})^{\circ}}((T^{s\sigma})^{\circ}):=N_{(G^{s\sigma})^{\circ}}((T^{s\sigma})^{\circ})/(T^{s\sigma})^{\circ},\quad W_{G^{\circ}}(T):=N_{G^{\circ}}(T)/T.\end{align*} $$

Then there is a natural injective map

$$ \begin{align*}W_{(G^{s\sigma})^{\circ}}((T^{s\sigma})^{\circ})\hookrightarrow W_{G^{\circ}}(T)^{\sigma}.\end{align*} $$

2.6.1

An algebraic group G over is defined over $\mathbb {F}_q$ if there is an algebraic group $G_0$ over $\mathbb {F}_q$ such that . Given $G_0$ , the geometric Frobenius endomorphism on G is denoted by F. The set of fixed points of F, denoted by $G^F$ , is a finite group, which we will sometimes write as $G(q)$ . If $X\subset G$ is a subvariety, then we say that X is F-stable if $F(X)=X$ .

2.6.2

Let G be a connected reductive group defined over $\mathbb {F}_q$ . Let $T\subset G$ be an F-stable maximal torus. Then F-acts naturally on $W_G(T)$ , and so we can talk about the F-conjugacy classes in $W_G(T)$ as defined in §2.1. The $G^F$ -conjugacy classes of F-stable maximal tori are in bijection with the F-conjugacy classes of $W_G(T)$ . This bijection is described as follows. Given $w\in W_G(T)$ , let $\dot {w}\in G$ be a representative of w. By Lang-Steinberg theorem, there exists $g\in G$ such that $g^{-1}F(g)=\dot {w}$ . Then $gTg^{-1}$ is again an F-stable maximal torus, and its $G^F$ -conjugacy class only depends on the F-conjugacy class of w. Conversely, if $gTg^{-1}$ is an F-stable maximal torus for some $g\in G$ , then $g^{-1}F(g)$ normalises T and so defines an element of $W_G(T)$ . The two constructions are inverse to each other.

Fix an F-stable maximal torus T and a Borel subgroup B containing T. Let $\Delta =\Delta (T,B)$ be the set of simple roots defined by T and B. The parabolic subgroups containing B, called standard parabolic subgroups, are parametrised by the set of the subsets of $\Delta $ . For any $I\subset \Delta $ , denote by $P_I$ the corresponding standard parabolic subgroup. Its unique Levi factor containing T is called a standard Levi subgroup, denoted by $L_I$ . Every Levi subgroup of G is conjugate to a standard Levi subgroup. Fix $I\subset \Delta $ . The $G^F$ -conjugacy classes of the G-conjugates of $L_I$ are in bijection with the F-conjugacy classes of $N_G(L_I)/L_I$ . The construction of this correspondence resembles the case of maximal tori.

2.6.3

Suppose that G is a reductive group defined over $\mathbb {F}_q$ and that $G/G^{\circ }$ is a cyclic group. Let $\sigma \in G$ be an F-stable quasi-central element such that $G/G^{\circ }$ is generated by the component of $\sigma $ .

We give a more concrete description of this correspondence. Fix an F-stable and $\sigma $ -stable maximal torus T and a $\sigma $ -stable Borel subgroup $B\subset G^{\circ }$ that contains T. Let I be a subset of the set of simple roots defined by $(T,B)$ . Let $L_I$ be the standard Levi subgroup with respect to $(T,B)$ . Suppose that I is $\sigma $ -stable and so $(L_I^{\sigma })^{\circ }$ is a standard Levi subgroup of $(G^{\sigma })^{\circ }$ with respect to $((T^{\sigma })^{\circ },(B^{\sigma })^{\circ })$ . Then the $((G^{\sigma })^{\circ })^F$ -conjugacy classes of the $(G^{\sigma })^{\circ }$ -conjugates of $(L_I^{\sigma })^{\circ }$ are in bijection with the F-conjugacy classes in $W_{(G^{\sigma })^{\circ }}((L_I^{\sigma })^{\circ })$ . For any $w\in W_{(G^{\sigma })^{\circ }}((L_I^{\sigma })^{\circ })$ , the procedure of §2.6.2 gives an F-stable Levi factor $L^{\prime }_{I,w}$ of some parabolic subgroup $P^{\prime }_{I,w}\subset (G^{\sigma })^{\circ }$ , which is $(G^{\sigma })^{\circ }$ -conjugate to $(L_I^{\sigma })^{\circ }$ and whose $((G^{\sigma })^{\circ })^F$ -conjugacy class corresponds to the F-conjugacy class of w. By Proposition 2.5.4, the pair $(L^{\prime }_{I,w},P^{\prime }_{I,w})$ determines an F-stable and $\sigma $ -stable Levi factor $L_{I,w}$ of a $\sigma $ -stable parabolic subgroup $P_{I,w}$ . In fact, if $\dot {w}$ , $g\in (G^{\sigma })^{\circ }$ are as in §2.6.2, then $L_{I,w}=gL_Ig^{-1}$ . Put

$$ \begin{align*}\bar{L}_{I,w}:=N_G(L_{I,w},P_{I,w})=L_{I,w}\sqcup L_{I,w}\sigma.\end{align*} $$

It is F-stable and meets the connected component $G^{\circ }\sigma $ . By Proposition 2.6.1, such groups for varying I and w exploit all $G^F$ -conjugacy classes of F-stable subgroups of the form $\bar {L}$ that meet $G^{\circ }\sigma $ . This implies, in particular, that the $G^F$ -conjugacy classes of the F-stable subgroups $N_G(T',B')$ , defined by an F-stable maximal torus $T'$ contained in a Borel subgroup $B'\subset G^{\circ }$ , are parametrised by the F-conjugacy classes of $W_{G^{\circ }}(T)^{\sigma }$ (see Remark 2.5.8).

Define another Frobenius $F_w:=\operatorname {\mathrm {ad}}\dot {w}\circ F$ on G. Then there is a commutative diagram

(2.6.3.1)

We will simply say that $(L_I,F_w)$ is isomorphic to $(L_{I,w},F)$ via $\operatorname {\mathrm {ad}} g$ .

2.7.1

Let H be a finite group and let $H_0\lhd H$ be a normal subgroup of index 2. Let $\sigma $ be an element of $H\setminus H_0$ . Then $\sigma $ defines an involution on $\operatorname {\mathrm {Irr}}(H_0)$ by composing a character with the conjugation by $\sigma $ . This involution does not depend on the choice of $\sigma $ in $H\setminus H_0$ . Denote by $\operatorname {\mathrm {Irr}}(H_0)^{\sigma }$ the subset of $\sigma $ -fixed elements. By Clifford theory, each element $\chi \in \operatorname {\mathrm {Irr}}(H_0)^{\sigma }$ admits an extension to H (i.e., some $\chi _H\in \operatorname {\mathrm {Irr}}(H)$ such that $(\chi _H)|_{H_0}=\chi $ ). Moreover, there are precisely two such extensions, and they differ by $-1$ on $H\setminus H_0$ . For every $\chi \in \operatorname {\mathrm {Irr}}(H_0)^{\sigma }$ , we choose a preferred extension, denoted by $\tilde {\chi }$ . If the restriction of $\chi _H\in \operatorname {\mathrm {Irr}}(H)$ to $H_0$ is not irreducible (or rather, $\chi _H$ is not an extension of any $\chi \in \operatorname {\mathrm {Irr}}(H_0)$ ), then $\chi _H$ must vanish on $H\setminus H_0$ .

Proposition 2.7.1. Let $\mathcal {C}=(C_j)_{1\le j\le 2k}$ be an arbitrary $2k$ -tuple of $H_0$ -conjugacy classes contained in $H\setminus H_0$ . Then we have the following counting formula:

(2.7.1.1) $$ \begin{align} \begin{aligned} &\#\left\{((A_i,B_i)_{1\le i\le g},(X_j)_{1\le j\le 2k})\in H_0^{2g}\times \prod_{j=1}^{2k}C_j\Big\arrowvert\prod_{i=1}^{g}[A_i,B_i]\prod_{j=1}^{2k}X_j=1\right\}\\ =\ &|H_0|\sum_{\chi\in\operatorname{\mathrm{Irr}}(H_0)^{\sigma}}\Big(\frac{|H_0|}{\chi(1)}\Big)^{2g-2}\prod_{i=1}^{2k} \frac{|C_i|\tilde{\chi}(C_i)}{\chi(1)}. \end{aligned} \end{align} $$

Note that the value of the counting formula is independent of the choices of $\tilde {\chi }$ due to the product of $2k$ terms. The proof is given below.

2.7.2

Let us prepare some notations of finite groups.

Denote by $\mathcal {C}(H)$ the vector space of complex valued class functions on H. Put $\hat {H}=\operatorname {\mathrm {Irr}}(H)$ . Denote by $\mathcal {C}(\hat {H})$ the vector space of linear combinations $\sum _{\chi \in \hat {H}}a_{\chi }\chi $ of irreducible characters, which is the same as $\mathcal {C}(H)$ but will be equipped with a different product operation.

The convolution product $\ast $ in $\mathcal {C}(H)$ is defined by

(2.7.2.1) $$ \begin{align} (f_1\ast f_2)(x)=\sum_{yz=x}f_1(y)f_2(z),\quad f_1,f_2\in\mathcal{C}(H). \end{align} $$

The dot product $\cdot $ in $\mathcal {C}(\hat {H})$ is defined by

(2.7.2.2) $$ \begin{align} F_1\cdot F_2(\chi)=F_1(\chi)F_2(\chi),\quad F_1,F_2\in\mathcal{C}(\hat{H}), \end{align} $$

(i.e., the coefficient of $\chi $ in the product is the product of the coefficients of $\chi $ ).

The Fourier transform $\mathcal {F}:\mathcal {C}(H)\rightarrow \mathcal {C}(\hat {H})$ is defined by

(2.7.2.3) $$ \begin{align} \mathcal{F}(f)(\chi)=\sum_{h\in H}\frac{f(h)\chi(h)}{\chi(1)},\quad f\in\mathcal{C}(H). \end{align} $$

We also have the transform $\mathbb {F}:\mathcal {C}(\hat {H})\rightarrow \mathcal {C}(H)$ defined by

(2.7.2.4) $$ \begin{align} \mathbb{F}(F)(h)=\sum_{\chi\in\hat{H}}F(\chi)\chi(1)\overline{\chi(h)}. \end{align} $$

We have

(2.7.2.5) $$ \begin{align} \mathbb{F}\circ\mathcal{F}=|H|\cdot \operatorname{\mathrm{Id}}_{\mathcal{C}(H)},\quad \mathcal{F}\circ\mathbb{F}=|H|\cdot\operatorname{\mathrm{Id}}_{\mathcal{C}(\hat{H})}. \end{align} $$

They are compatible with the product operations

(2.7.2.6) $$ \begin{align} \mathcal{F}(f_1)\cdot\mathcal{F}(f_2)=\mathcal{F}(f_1\ast f_2),\quad \mathbb{F}(F_1)\ast\mathbb{F}(F_2)=|H|\cdot\mathbb{F}(F_1\cdot F_2). \end{align} $$

From the equality $\mathbb {F}\circ \mathcal {F}=|H|\cdot \operatorname {\mathrm {Id}}_{\mathcal {C}(H)}$ and the definition of the transforms, we deduce that

(2.7.2.7) $$ \begin{align} f(1)=\frac{1}{|H|}\sum_{\chi\in\hat{H}}\chi(1)^2\mathcal{F}(f)(\chi) \end{align} $$

for any class function f.

2.7.3

We define the function $\mathfrak {n}^g:H\rightarrow \mathbb {C}$ by

(2.7.3.1) $$ \begin{align} \mathfrak{n}^g(x)=\#\left\{(A_1,B_1,\cdots,A_g,B_g)\in H_0^{2g}\Big\arrowvert\prod_{i=1}^g[A_i,B_i]=x\right\}. \end{align} $$

We find that $\mathfrak {n}^g\in \mathcal {C}(H)$ (identically zero on $H\setminus H_0$ ) and that $\mathfrak {n}^g=\mathfrak {n}^1\ast \cdots \ast \mathfrak {n}^1$ . Denote by $1_{C_i}$ the characteristic function of the class $C_i$ . Then

(2.7.3.2) $$ \begin{align} \begin{aligned} &\#\left\{((A_i,B_i)_{1\le i\le g},(X_j)_{1\le j\le 2k})\in H_0^{2g}\times \prod_{j=1}^{2k}C_j\Big\arrowvert\prod_{i=1}^{g}[A_i,B_i]\prod_{j=1}^{2k}X_j=1\right\}\\ =\ &(\mathfrak{n}^g\ast 1_{C_1}\ast\cdots\ast 1_{C_{2k}})(1). \end{aligned} \end{align} $$

By (2.7.2.6) and (2.7.2.7), we have

(2.7.3.3) $$ \begin{align} \begin{aligned} &(\mathfrak{n}^g\ast 1_{C_1}\ast\cdots\ast 1_{C_{2k}})(1)\\ =\ &\frac{1}{|H|}\sum_{\chi\in\operatorname{\mathrm{Irr}}(H)}\chi(1)^2(\mathcal{F}(\mathfrak{n}^1)(\chi))^g\prod_{i=1}^{2k}\frac{|C_i|\chi(C_i)}{\chi(1)}. \end{aligned} \end{align} $$

It is known ([Reference Hausel, Letellier and Rodriguez-VillegasHLRV11, Lemma 3.1.3]) that

(2.7.3.4) $$ \begin{align} \mathcal{F}(\mathfrak{n}^1)(\chi)= \big(\frac{|H_0|}{\chi(1)}\big)^2. \end{align} $$

Therefore,

(2.7.3.5) $$ \begin{align} \begin{aligned} &\frac{1}{|H|}\sum_{\chi_H\in\operatorname{\mathrm{Irr}}(H)}\chi_H(1)^2(\mathcal{F}(\mathfrak{n}^1)(\chi_H))^g\prod_{i=1}^{2k}\frac{|C_i|\chi_H(C_i)}{\chi_H(1)}\\ =\ &|H_0|\sum_{\chi\in\operatorname{\mathrm{Irr}}(H_0)^{\sigma}}\left(\frac{|H_0|}{\chi(1)}\right)^{2g-2}\prod_{i=1}^{2k}\frac{|C_i|\tilde{\chi}(C_i)}{\chi(1)}, \end{aligned} \end{align} $$

where we have used the fact that $\chi _H$ vanishes on $H\setminus H_0$ if it is not an extension of any element of $\operatorname {\mathrm {Irr}}(H_0)$ . This completes the proof of Proposition 2.7.1.

3.1.1

Denote by $J_n$ the matrix

If n is even, put $t_0=\operatorname {\mathrm {diag}}(a_1,\ldots ,a_n)$ with $a_i=1$ if $i\le N$ and $a_i=-1$ otherwise. Put $J^{\prime }_n=t_0J_n$ . Write

(3.1.1.1) $$ \begin{align} \mathscr{J}_n:= \begin{cases} J^{\prime}_n & \text{if } n \text{ is even}\\ J_n & \text{if } n \text{ is odd} \end{cases}, \quad \mathscr{J}^{\prime}_n:=J_n \text{ if } n \text{ is even}. \end{align} $$

Define $\sigma \in \operatorname {\mathrm {Aut}} G$ as sending g to $\mathscr {J}_ng^{-t}\mathscr {J}^{-1}_n$ , where $g^{-t}$ is the transpose-inverse of g, and define $\sigma '\in \operatorname {\mathrm {Aut}} G$ by replacing $\mathscr {J}_n$ with $\mathscr {J}^{\prime }_n$ in the definition of $\sigma $ . They are exterior involutions of G. Their fixed points in G are described as follows:

(3.1.1.2) $$ \begin{align} \text{ if } n=2N, \text{ then } \begin{cases} (G^{\sigma})^{\circ}\cong\operatorname{\mathrm{Sp}}_{2N}\\ (G^{\sigma'})^{\circ}\cong\operatorname{\mathrm{SO}}_{2N} \end{cases}\!\!\!\!\!\!\!,\quad \text{ if } n=2N+1, \text{ then } (G^{\sigma})^{\circ}\cong\operatorname{\mathrm{SO}}_{2N+1}. \end{align} $$

We say that an automorphism is of symplectic type or orthogonal type according to the type of its centraliser.

3.1.2

An involution $\tau $ of G defines a semi-direct product $G\rtimes \!<\!\tau \!>\!$ . When n is even, there are two $\operatorname {\mathrm {PGL}}_n$ -conjugacy classes of exterior involutions in $\operatorname {\mathrm {Aut}} G$ , represented by $\sigma $ and $\sigma '$ , respectively ([Reference Liebeck and SeitzLS12, Lemma 2.9]). We will write and , in order to indicate that the type of the defining automorphism is symplectic or orthogonal. These two semi-direct products are not isomorphic over any field of characteristic different from $2$ (This follows from [Reference ShuShu23, Theorem 1.12]). It is convenient to regard $\sigma $ also as an element of , identified with $t_0\sigma '$ , since $\sigma $ and $t_0\sigma '$ induce the same automorphism of G. Note, however, that in , we have $\sigma ^2=-1$ . Therefore, we write $G\!<\!\sigma \!>\!$ instead of $G\rtimes \!<\!\sigma '\!>\!$ . We will write $\bar {G}=G\!<\!\sigma \!>\!$ if n is odd or if there is no need to distinguish and .

Since $\sigma $ normalises T and B, it is semi-simple as an element of $\bar {G}$ . Moreover, according to [Reference Digne and MichelDM94, Proposition 1.22], $\sigma $ is a quasi-central element of $\bar {G}$ . It is easy to see that the action of $\sigma $ on G commutes with F, and so F can be extended to $\bar {G}$ in such a way that $\sigma $ is an F-stable element.

3.1.3

Now suppose . For the moment, we do not distinguish and . We apply Proposition 2.5.1 to the connected component $G^1=G\sigma $ .

The subtorus $(T^{\sigma })^{\circ }$ consists of the matrices

(3.1.3.1) $$ \begin{align} \operatorname{\mathrm{diag}}(a_1,a_2,\ldots,a_N,a_N^{-1},\ldots,a_2^{-1},a_1^{-1}), &\text{ if } n=2N, \end{align} $$

(3.1.3.2) $$ \begin{align} \operatorname{\mathrm{diag}}(a_1,a_2,\ldots,a_N,1,a_N^{-1},\ldots,a_2^{-1},a_1^{-1}), &\text{ if } n=2N+1, \end{align} $$

with for all i, and the commutator $[T,\sigma ]$ consists of the matrices

(3.1.3.3) $$ \begin{align} \operatorname{\mathrm{diag}}(b_1,b_2,\ldots,b_N,b_N,\ldots,b_2,b_1),&\text{ if } n=2N, \end{align} $$

(3.1.3.4) $$ \begin{align} \operatorname{\mathrm{diag}}(b_1,b_2,\ldots,b_N,b_{N+1},b_N,\ldots,b_2,b_1),&\text{ if } n=2N+1, \end{align} $$

with for all i. So the elements of $[T,\sigma ]\cap (T^{\sigma })^{\circ }$ are the matrices

(3.1.3.5) $$ \begin{align} \operatorname{\mathrm{diag}}(e_1,e_2,\ldots,e_N,e_N,\ldots,e_2,e_1),&\text{ if } n=2N, \end{align} $$

(3.1.3.6) $$ \begin{align} \operatorname{\mathrm{diag}}(e_1,e_2,\ldots,e_N,1,e_N,\ldots,e_2,e_1),&\text{ if } n=2N+1, \end{align} $$

with $e_i=\pm 1$ for all i.

The parametrisation of semi-simple conjugacy classes in $G\sigma $ is then given as follows. Denote by the set of $(\boldsymbol{\mu} _2\times \boldsymbol{\mu} _2)$ -orbits in for the action

(3.1.3.7) $$ \begin{align} (-1,1):a\mapsto a^{-1},\quad (1,-1):a\mapsto -a. \end{align} $$

Then the semi-simple conjugacy classes in $G\sigma $ are parametrised by the $\mathfrak {S}_N$ -orbits in , or rather, the N-tuples $(\hat {a}_1,\ldots ,\hat {a}_N)$ , , up to permutation, whether $n=2N$ or $n=2N+1$ . We thus regard the elements of as the eigenvalues of a semi-simple element. For example, we may say that $\sigma $ has eigenvalues $(\hat {1},\ldots ,\hat {1})$ . Beware that the eigenvalues depend on the choice of $\sigma $ in Proposition 2.5.1. For example, if n is even and if we had used $\sigma '$ in Proposition 2.5.1, then $\sigma $ would have eigenvalues $(\hat {\mathfrak {i}},\ldots ,\hat {\mathfrak {i}})$ . In the rest of the article, we will always use $\sigma \in G^1$ when we specify a semi-simple element in terms of elements of .

3.1.4

Let $P\subset G$ be a parabolic subgroup and let $L\subset P$ be a Levi factor. We will be interested in those $L\subset P$ such that $N_{\bar {G}}(L,P)$ meets the connected component $G\sigma $ . According to §2.5.3, this is to consider the $\sigma $ -stable Levi factors of $\sigma $ -stable parabolic subgroups. If P is a $\sigma $ -stable standard parabolic subgroup with respect to B, then its unique Levi factor containing T is of the form

(3.1.4.1)

, where $L_i\cong \operatorname {\mathrm {GL}}_{n_i}$ , for some integers $n_i$ , $1\le i\le s$ . Therefore, up to the conjugation by G, we only need to consider these Levi subgroups. Note that for these groups, $N_{\bar {G}}(L,P)=L\sqcup L\sigma $ .

3.2.1

By the theorem of Lang-Steinberg, a G-conjugacy class in $G\sigma $ contains an element of $G^F\sigma $ if and only if it is F-stable. Let $C\subset G\sigma $ be a semi-simple conjugacy class with a representative $t\sigma \in (T^{\sigma })^{\circ }\sigma $ . According to the parametrisation of semi-simple conjugacy classes, C is F-stable if and only if there exist $w\in W^{\sigma }$ and $s\in [T,\sigma ]\cap (T^{\sigma })^{\circ }$ such that $F(t)=wtw^{-1}s$ (see §3.1.3). We will, however, only be concerned with those semi-simple conjugacy classes that have representatives in $(T^{\sigma })^{\circ F}\sigma $ ; that is, $w=1$ and $s=1$ in the above equation. Now let C be such a class with representative $t\sigma \in (T^{\sigma })^{\circ F}\sigma $ . If we represent C by the N-tuple $(\hat {a}_1,\ldots ,\hat {a}_N)$ , , then $a_i\in \mathbb {F}_q^{\ast }$ for all $1\le i\le N$ .

Lemma 3.2.1 [Reference ShuShu22, Lemma 6.2.1]

Denote by $N_+$ (resp. $N_-$ ) the multiplicity of $\hat {1}$ (resp. $\hat {\mathfrak {i}}$ ) in $(\hat {a}_1,\ldots ,\hat {a}_N)$ . Then the centraliser of $t\sigma $ in G is isomorphic to

$$ \begin{align*} \operatorname{\mathrm{Sp}}_{2N_+}\times\operatorname{\mathrm{O}}_{2N_-}\times\prod_i\operatorname{\mathrm{GL}}_{n_i} &\text{ if } n \text{ is even and,}\\ \operatorname{\mathrm{O}}_{2N_++1}\times\operatorname{\mathrm{Sp}}_{2N_-}\times\prod_i\operatorname{\mathrm{GL}}_{n_i} &\text{ if } n \text{ is odd,} \end{align*} $$

for some integers $n_i$ .

In general, $C^F$ is not a single $G^F$ -conjugacy class. By [Reference Digne and MichelDM20, Proposition 4.2.14], the number of $G^F$ -conjugacy classes contained in $C^F$ is equal to the number of connected components of $C_G(t\sigma )$ . If n is odd, then $C_G(t\sigma )$ always has two connected components. The $G^F$ -class of $t\sigma $ has a representative of the form

(3.2.1.1) $$ \begin{align} \operatorname{\mathrm{diag}}(a_1,a_2,\ldots,a_N,1,a_N^{-1},\ldots,a_2^{-1},a_1^{-1})\sigma, \end{align} $$

with every $a_i\in \mathbb {F}_q$ , while the other $G^F$ -class in $C^F$ is represented by

(3.2.1.2) $$ \begin{align} \operatorname{\mathrm{diag}}(a_1,a_2,\ldots,a_N,b,a_N^{-1},\ldots,a_2^{-1},a_1^{-1})\sigma, \end{align} $$

with $b\in \mathbb {F}_q^{\ast }\setminus (\mathbb {F}_q^{\ast })^2$ .

3.2.2

Let I be a subset of $\Delta (T,B)$ and let $L_I$ be the corresponding standard Levi subgroup. There are positive integers $\{n_i\mid i\in \Gamma _I\}$ indexed by a finite set $\Gamma _I$ , such that $L_I\cong \prod _{i\in \Gamma _I}\operatorname {\mathrm {GL}}_{n_i}$ . Then $Z_{L_I}\cong \mathbb {G}_m^{\Gamma _I}$ (i.e., direct product of copies of $\mathbb {G}_m$ indexed by $\Gamma _I$ ). For any $r\in \mathbb {Z}_{>0}$ , put $\Gamma _{I,r}=\{i\in \Gamma _I\mid n_i=r\}$ and put $N_r=|\Gamma _{I,r}|$ . Then the G-conjugacy class of $L_I$ is uniquely determined by the sequence $(N_r)_{r\in \mathbb {Z}_{>0}}$ , and $W_G(L_I)\cong \prod _r\mathfrak {S}_{N_r}$ .

Suppose that I is $\sigma $ -stable, so that the associated $\sigma $ -stable standard Levi subgroup $L_I$ is of the form (3.1.4.1). Now the action of $\sigma $ on $Z_{L_I}$ induces an involution on $\Gamma _I$ . There is at most one element of $\Gamma _I$ fixed by $\sigma $ , and we denote it by $0$ so that it corresponds to the direct factor $\operatorname {\mathrm {GL}}_{n_0}$ in (3.1.4.1). In case no element of $\Gamma _I$ is fixed by $\sigma $ , we define $n_0=0$ . For any $r\in \mathbb {Z}_{>0}$ , put $N_r'=N_r/2$ if $n_0\ne r$ and $N_r'=(N_r-1)/2$ if $n_0=r$ . Then $W_{G}(L_I)^{\sigma }\cong \prod _r\mathfrak {W}_{N_r'}$ .

The action of an element of $W_{G}(L_I)^{\sigma }$ on $L_I$ (up to an inner automorphism of $L_I$ ) can be easily visualised. Let $r\in \mathbb {Z}_{>0}$ and let $\tau $ be a signed cycle of size d in $\mathfrak {W}_{N_r'}$ . Then it acts on a subgroup of $L_I$ that is isomorphic to $(\operatorname {\mathrm {GL}}_r\times \operatorname {\mathrm {GL}}_r)^d$ . Depending on whether $\tau $ is a positive cycle or a negative cycle (see §2.3.3), its action can be schematically described as

(3.2.2.1)

, respectively. If $r=n_0$ , then $\tau $ simply ignores the component $\operatorname {\mathrm {GL}}_{n_0}$ .

For each $w\in W_{G}(L_I)^{\sigma }$ , there exists $\dot {w}\in (G^{\sigma })^{\circ }$ that represents w. For example, it is easy to see that in (3.2.2.1), one can find explicit matrices in $\operatorname {\mathrm {GL}}_{2rd}$ fixed by $\sigma $ representing the corresponding cycles in $\mathfrak {W}_{N_r'}$ . This implies that the inclusion

$$ \begin{align*}W_{(G^{\sigma})^{\circ}}((L_I^{\sigma})^{\circ})\hookrightarrow W_{G}(L_I)^{\sigma}\end{align*} $$

is, in fact, an isomorphism. Now the groups $L_{I,w}$ defined for $w\in W_{(G^{\sigma })^{\circ }}((L_I^{\sigma })^{\circ })$ in §2.6.3 make sense if we regard w as an element of $W_{G}(L_I)^{\sigma }$ . The discussions in §2.6.3 give the following:

3.3.1

Let H be a connected reductive group defined over $\mathbb {F}_q$ equipped with a geometric Frobenius endomorphism F. Let $L\subset H$ be an F-stable Levi subgroup. The Deligne-Lusztig induction $R^H_L$ ([Reference Digne and MichelDM20, §9.1]) is a $\bar {\mathbb {Q}}_{\ell }$ -linear map from the set of $L^F$ -invariant functions on $L^F$ to the set of $H^F$ -invariant functions on $H^F$ (invariant for the conjugation actions).

Let $T_H$ be an F-stable maximal torus of H. Denote by $W_H$ the Weyl group of H defined by $T_H$ . By §2.6.2, to each $w\in W_H$ , we can associate an F-stable maximal torus $T_w\subset H$ in such a way that $T_H$ is associated to $1\in W_H$ . For any $w\in W_H$ , we denote by $\mathbf {1}$ the trivial character of $T_w^F$ . Then the Green function is an $H^F$ -invariant function on the subset $H^F_u\subset H^F$ of unipotent elements, defined by

(3.3.1.1) $$ \begin{align} Q_{T_w}^H(u):=R_{T_w}^H\mathbf{1}|_{H^F_u}. \end{align} $$

Such a function only depends on the $H^F$ -conjugacy class of $T_w$ , or rather, the F-conjugacy class of w.

Let $\tau $ be an automorphism of finite order of H that commutes with F and let $L\subset H$ be an F-stable and $\tau $ -stable Levi factor of a $\tau $ -stable parabolic subgroup $P\subset H$ . Then $N_{H\!<\!\tau \!>\!}(L,P)=L\!<\!\tau \!>\!$ . The generalised Deligne-Lusztig induction $R^{H\tau }_{L\tau }$ ([Reference Digne and MichelDM94, §2]) is a $\bar {\mathbb {Q}}_{\ell }$ -linear map from the set of $L^F$ -invariant functions on $L^F\tau $ to the set of $H^F$ -invariant functions on $H^F\tau $ .

3.3.2

By §2.6.3 and Remark 2.5.8, to each $w\in W^{\sigma }$ , we can associate an F-stable and $\sigma $ -stable maximal torus $T_w$ contained in a $\sigma $ -stable Borel subgroup of G, in such a way that T is associated to $1\in W^{\sigma }$ . If $B_w\subset G$ is some $\sigma $ -stable Borel subgroup containing $T_w$ , then $\bar {T}_w:=N_{\bar {G}}(T_w,B_w)=T_w\sqcup T_w\sigma $ and the $G^F$ -conjugacy class of $\bar {T}_w$ is determined by the conjugacy class of $w\in W^{\sigma }$ . Let $\theta \in \operatorname {\mathrm {Irr}}(T_w^F)^{\sigma }$ and denote by $\tilde {\theta }\in \operatorname {\mathrm {Irr}}(T_w^F\!<\!\sigma \!>\!)$ an extension of $\theta $ . By abuse of notation, we may also denote by $\tilde {\theta }$ its restriction on $T_w^F\sigma $ . Computation of Deligne-Lusztig inductions is reduced to Green functions according to the character formula below.

Proposition 3.3.1. [Reference Digne and MichelDM94, Proposition 2.6]

Let $s\sigma \in G^F\sigma $ be semi-simple and let $u\in C_G(s\sigma )^{\circ F}$ . Then

(3.3.2.1) $$ \begin{align} R^{G\sigma}_{T_w\sigma}\tilde{\theta}(us\sigma)=\frac{|(T_w^{\sigma})^{\circ F}|}{|T_w^F|\cdot|C_G(s\sigma)^{\circ F}|}\sum_{\{h\in G^{F}\mid hs\sigma h^{-1}\in T_w\sigma\}}Q^{C_G(s\sigma)^{\circ}}_{C_{h^{-1}T_wh}(s\sigma)^{\circ}}(u)\tilde{\theta}(hs\sigma h^{-1}), \end{align} $$

where $Q^{C_G(s\sigma )^{\circ }}_{C_{h^{-1}T_wh}(s\sigma )^{\circ }}(u)$ is the Green function associated to the connected reductive group $C_G(s\sigma )^{\circ }$ and its F-stable maximal torus $C_{h^{-1}T_wh}(s\sigma )^{\circ }$ .

An invariant function on $G^F\sigma $ is called uniform if it is a linear combination of $R_{T_w\sigma }^{G\sigma }\tilde {\theta }$ for various w and $\tilde {\theta }$ .

3.3.3

Fix semi-simple element $s\sigma \in G^F\sigma $ and write $L':=C_G(s\sigma )^{\circ }$ . Let $\tau \in W^{\sigma }$ . Write

$$ \begin{align*}A_{\tau}=\{h\in G\mid hs\sigma h^{-1}\in T_{\tau}\sigma\},\quad A_{\tau}^F=A_{\tau}\cap G^F.\end{align*} $$

If $A^F_{\tau }$ is nonempty, we may assume that $s\sigma \in T_{\tau }\sigma $ , since the problem we will consider is not affected by conjugation by $G^F$ . So, in particular, $A^F_{\tau }$ contains the unit $1$ . For any $h\in A_{\tau }^F$ , $C_{h^{-1}T_{\tau }h}(s\sigma )^{\circ }$ is an F-stable maximal torus of $L'$ by Proposition 2.5.3. In particular, $T^{\prime }_{\tau }:=C_{T_{\tau }}(s\sigma )^{\circ }$ is an F-stable maximal torus of $L'$ . The $L^{\prime F}$ -conjugacy classes of the F-stable maximal tori of $L'$ are parametrised by the F-conjugacy classes of $W':=W_{L'}(T_{\tau }')$ . Then the map $h\mapsto C_{h^{-1}T_{\tau }h}(s\sigma )^{\circ }$ induces a map from $A_{\tau }^F$ to the set of F-conjugacy classes of $W'$ . Denote by $B_{\tau }$ the image of this map so that we have a surjection $A_{\tau }^F\rightarrow B_{\tau }$ . Let $\nu \in W'$ represent an F-conjugacy class of $W'$ and denote by $A^F_{\tau ,\nu }$ the inverse image in $A_{\tau }^F$ of this F-conjugacy class.

We will fix $g_{\tau }$ such that $T_{\tau }=g_{\tau }Tg^{-1}_{\tau }$ . Write $\dot {\tau }=g_{\tau }^{-1}F(g_{\tau })$ and $s_0\sigma =g_{\tau }^{-1}s\sigma g_{\tau }$ and $L^{\prime }_0=C_G(s_0\sigma )^{\circ }$ . Let $W^{\prime }_0$ denote the Weyl group of $L^{\prime }_0$ defined by $C_T(s_0\sigma )^{\circ }$ . Then the F-conjugacy classes of $W'$ are in natural bijection with the $\tau $ -conjugacy classes of $W^{\prime }_0$ . We will therefore regard $B_{\tau }$ as a set of $\tau $ -conjugacy classes of $W^{\prime }_0$ . By Corollary 2.5.7, $W^{\prime }_0$ is naturally a subgroup of $W^{\sigma }$ .

3.3.4

We will need the following results.

Lemma 3.3.4. There is a natural isomorphism:

$$ \begin{align*}N_G(T\sigma)/T\smash{\begin{array}{c}\sim\\[-1em] \longrightarrow\end{array}} W^{\sigma}.\end{align*} $$

Lemma 3.3.5. We have

$$ \begin{align*}\{x\in G\mid xs_0\sigma x^{-1}\in T\sigma\}=N_{G}(T\sigma)L_0'.\end{align*} $$

Proof. Since $xs_0\sigma x^{-1}$ lies in $T\sigma $ , it normalises T and besides,

$$ \begin{align*}C_T(xs_0\sigma x^{-1})^{\circ}=C_T(s_0\sigma)^{\circ}=T':=C_T(\sigma)^{\circ}.\end{align*} $$

By Proposition 2.5.3, $T'$ is a maximal torus of $L'$ . Now $s\sigma $ normalises $x^{-1}Tx$ and so $C_{x^{-1}Tx}(s_0\sigma )^{\circ }$ is also a maximal torus of $L'$ . There exists $l\in L'$ such that $C_{x^{-1}Tx}(s_0\sigma )^{\circ }=lT'l^{-1}$ . Note that

$$ \begin{align*}C_{x^{-1}Tx}(s_0\sigma)^{\circ}=x^{-1}C_T(xs_0\sigma x^{-1})^{\circ}x=x^{-1}T'x.\end{align*} $$

We deduce that $xl$ normalises $T'$ and thus normalises T by Proposition 2.5.4. Write $\xi =xl$ . From the relation $xs_0\sigma x^{-1}\in T\sigma $ , we see that $\sigma (\xi )\in \xi T$ . By the proof of Lemma 3.3.4, we have $\xi \in N_G(T\sigma )$ . Therefore, $x\in N_G(T\sigma )L'$ . Conversely, it is easy to see that every element $x\in N_G(T\sigma )L'$ satisfies $xs_0\sigma x^{-1}\in T\sigma $ .

Proposition 3.3.6. [Reference ShuShu22, Proposition 10.2.13 (ii)]

We have

$$ \begin{align*}|A^F_{\tau,\nu}|=|T_{\tau}^F|\cdot|L^{\prime F}|\cdot|(T_{\tau}^{\sigma})^{F}|^{-1}\mathfrak{z}_{\tau}\mathfrak{z}_{\nu}^{-1},\end{align*} $$

where $\mathfrak {z}_{\tau }$ is the cardinality of the centraliser of $\tau $ in $W^{\sigma }$ and $\mathfrak {z}_{\nu }$ is the cardinality of the stabiliser of an element of the $\tau $ -conjugacy class $\nu $ under the $\tau $ -twisted action.

3.4.1

Let $H=\prod _i\operatorname {\mathrm {GL}}_{n_i}$ for some positive integers $n_i$ and suppose that H is defined over $\mathbb {F}_q$ equipped with a Frobenius F. Fix an F-stable maximal torus $T_H\subset H$ and denote by $W_H$ the Weyl group of H defined by $T_H$ . By §2.6.2, to each $w\in W_H$ , we can associate an F-stable maximal torus $T_w$ of H in such a way that $T_H$ is associated to $1\in W$ . The $H^F$ -conjugacy class of $T_w$ is determined by the F-conjugacy class of $w\in W_H$ . The natural action of F on $W_H$ induces an action on $\operatorname {\mathrm {Irr}}(W_H)$ . Denote by $\operatorname {\mathrm {Irr}}(W_H)^F$ the set of F-stable irreducible characters of $W_H$ . Let $\varphi \in \operatorname {\mathrm {Irr}}(W_H)^F$ . Then it can be extended to $\tilde {\varphi }\in \operatorname {\mathrm {Irr}}(W_H\rtimes \!<\! F\!>\!)$ . Such an extension is not unique. We choose such an extension and put

$$ \begin{align*}R^{H}_{\varphi}\mathbf{1}:=|W_H|^{-1}\sum_{w\in W_H}\tilde{\varphi}(wF)R_{T_w}^{H}\mathbf{1},\end{align*} $$

where $\mathbf {1}$ is the trivial character of $T_w^F$ .

Let $\theta \in \operatorname {\mathrm {Hom}}(H^F,\bar {\mathbb {Q}}_{\ell }^{\ast })$ (i.e., a linear character of $H^F$ ). Then $\theta \otimes R^H_{\varphi }\mathbf {1}$ is also an irreducible character of $H^F$ . Denote by $\theta _{T_w}$ the restriction of $\theta $ to $T_w^F$ for any $w\in W_H$ . Then

$$ \begin{align*}\theta\otimes R^H_{\varphi}\mathbf{1}=R^H_{\varphi}\theta:=|W_H|^{-1}\sum_{w\in W_H}\tilde{\varphi}(wF)R_{T_w}^{H}\theta_{T_w}.\end{align*} $$

3.4.2

Let $M\subset G$ be an F-stable Levi subgroup of G. Fix an F-stable maximal torus $T_M\subset M$ and let $T_w$ , $w\in W_M$ be defined as in the previous paragraph. Denote by $\operatorname {\mathrm {Irr}}_{reg}(M^F)$ the set of regular linear characters of $M^F$ (see [Reference Lusztig and SrinivasanLS77, §3.1 (a), (b)]). Concretely, if we choose an isomorphism $M^F\cong \prod _i\operatorname {\mathrm {GL}}_{n_i}(q^{d_i})$ for some positive intergers $n_i$ , $d_i$ and write a linear character $\theta \in \operatorname {\mathrm {Irr}}(M^F)$ as $(\theta _i)_i$ for some $\theta _i\in \operatorname {\mathrm {Hom}}(\operatorname {\mathrm {GL}}_{n_i}(q^{d_i}),\bar {\mathbb {Q}}_{\ell }^{\ast })$ , then $\theta $ is regular if and only if $\theta _i\ne \theta _j$ whenever $i\ne j$ and $\theta _i^{q^r}\ne \theta _i$ for any $1\le r< d_i$ and any i.

For any connected reductive group H defined over $\mathbb {F}_q$ , define $\epsilon _H:=(-1)^{rk_H}$ where $rk_H$ is the $\mathbb {F}_q$ -rank of H. For any $\varphi \in \operatorname {\mathrm {Irr}}(W_M)^F$ and $\theta \in \operatorname {\mathrm {Hom}}(M^F,\bar {\mathbb {Q}}_{\ell }^{\ast })$ , choose an extension $\tilde {\varphi }\in \operatorname {\mathrm {Irr}}(W_M\rtimes \!<\! F\!>\!)$ and put

$$ \begin{align*}R^{G}_{\varphi}\theta:=\epsilon_G\epsilon_M|W_M|^{-1}\sum_{w\in W_M}\tilde{\varphi}(wF)R_{T_w}^{G}\theta_{T_w}.\end{align*} $$

Note that $R_{\varphi }^G\theta =\epsilon _G\epsilon _MR^G_M(R^M_{\varphi }\theta )$ ([Reference Digne and MichelDM20, Proposition 9.1.8]).

3.4.3

Let $n_0\in \mathbb {Z}_{>0}$ , and let $n_+$ , $n_-\in \mathbb {Z}_{\ge 0}$ be such that $n_++n_-=n_0$ . Let F be the Frobenius of $\operatorname {\mathrm {GL}}_{n_0}$ that sends each entry of a matrix to its q-th power. Let $M_{00}\subset \operatorname {\mathrm {GL}}_{n_0}$ be a standard Levi subgroup isomorphic to $\operatorname {\mathrm {GL}}_{n_+}\times \operatorname {\mathrm {GL}}_{n_-}$ ; that is,

(3.4.3.1)

With respect to the isomorphism $M_{00}^F\cong \operatorname {\mathrm {GL}}_{n_+}(q)\times \operatorname {\mathrm {GL}}_{n_-}(q)$ , we define a linear character $\theta \in \operatorname {\mathrm {Irr}}(M_{00}^F)$ to be $(\mathbf {1}\circ \det ,\eta \circ \det )$ , where $\mathbf {1}$ is the trivial character of $\mathbb {F}_q^{\ast }$ and $\eta $ is the order 2 irreducible character of $\mathbb {F}_q^{\ast }$ . Now $W_{M_{00}}$ is isomorphic to $\mathfrak {S}_{n_+}\times \mathfrak {S}_{n_-}$ and so $\operatorname {\mathrm {Irr}}(W_{M_{00}})$ is in bijection with $\mathcal {P}(n_+)\times \mathcal {P}(n_-)$ . By Theorem 3.4.2, each $(\mu _+,\mu _-)\in \mathcal {P}(n_+)\times \mathcal {P}(n_-)$ defines an irreducible character of $\operatorname {\mathrm {GL}}_{n_0}(q)$ , called a quadratic-unipotent character. By [Reference ShuShu22, Lemma 5.2.1], this character is $\sigma $ -stable (Here, $\sigma $ is the automorphism of $\operatorname {\mathrm {GL}}_{n_0}$ defined in the same way as it is defined for $\operatorname {\mathrm {GL}}_n$ ).

3.4.4

Let $M\subset G$ be a Levi subgroup of the form $L_{I,w}$ as defined in §2.6.3, which is an F-stable and $\sigma $ -stable Levi factor of some $\sigma $ -stable parabolic subgroup $P\subset G$ . It can be written as $M_0\times M_1$ with $M_0\cong \operatorname {\mathrm {GL}}_{n_0}$ and $M_1\cong \prod _i(\operatorname {\mathrm {GL}}_{n_i}\times \operatorname {\mathrm {GL}}_{n_i})$ following the notations of §3.1.4. The actions of F and $\sigma $ respect the isomorphism $M\cong M_0\times M_1$ , and we will also denote by F and $\sigma $ their restrictions to $M_0$ or $M_1$ .

Let $T_M\subset M$ be an F-stable and $\sigma $ -stable maximal torus. Then we can write $T_M=T_0\times T_1$ with $T_0\subset M_0$ and $T_1\subset M_1$ . Denote by $W_0$ (resp. $W_1$ ) the Weyl group of $M_0$ (resp. $M_1$ ) defined by $T_0$ (resp. $T_1$ ). Then $\sigma $ induces an action on $W_1$ and so an action on $\operatorname {\mathrm {Irr}}(W_1)$ . Let $\varphi \in \operatorname {\mathrm {Irr}}(W_1)^F\cap \operatorname {\mathrm {Irr}}(W_1)^{\sigma }$ and denote by $\tilde {\varphi }$ an extension of $\varphi $ to $W_1\rtimes \!<\! F\!>\!$ . Let $\theta $ be a linear character of $M_1^F$ that is $\sigma $ -stable. Then $\chi _1:=R^{M_1}_{\varphi }\theta $ as defined in §3.4.1 is a $\sigma $ -stable irreducible character of $M_1^F$ . Let $\chi _0$ be a quadratic-unipotent character of $M_0^F$ . It is induced from a Levi subgroup $M_{00}$ of $M_0$ as in §3.4.3.

Let $(\mu _+,\mu _-)$ be the 2-partition defining the quadratic-unipotent character $\chi _0$ . Let $m_+$ and $m_-$ be the nonnegative integers such that $(m_+,m_+-1,\ldots ,1,0)$ and $(m_-,m_--1,\ldots ,1,0)$ are the 2-cores of $\mu _+$ and $\mu _-$ , respectively. Write

$$ \begin{align*}m=m_+(m_++1)/2+m_-(m_-+1)/2.\end{align*} $$

3.4.5

In the rest of this section, we will restrict ourselves to the special case where $M=L_I$ is a $\sigma $ -stable standard Levi subgroup. Then we have $\epsilon _G=\epsilon _M$ , $M_0^F\cong \operatorname {\mathrm {GL}}_{n_0}(q)$ , $M_1^F\cong \prod _i(\operatorname {\mathrm {GL}}_{n_i}(q)\times \operatorname {\mathrm {GL}}_{n_i}(q))$ , $W_1\cong \prod _i(\mathfrak {S}_{n_i}\times \mathfrak {S}_{n_i})$ , and $W_1^{\sigma }\cong \prod _i\mathfrak {S}_{n_i}$ . The action of F on $W_1$ is trivial. A $\sigma $ -stable linear character of $M_1^F$ is of the form $(\theta _i,\theta _i{\!}^{-1})_i$ , where $\theta _i$ is a linear character of $\operatorname {\mathrm {GL}}_{n_i}(q)$ for each i. Then $\mathbf {1}\boxtimes \eta \boxtimes \theta $ is a regular linear character of $M_{00}^F\times M_1^F$ if and only if $\theta $ is regular in the following sense.

We will give a decomposition formula expressing the extension of $R^G_M(\chi _1\boxtimes \chi _0)$ to $G^F\sigma $ as a linear combination of generalised Deligne-Lusztig characters when $m\le 1$ . Therefore, we assume $m\le 1$ in what follows.

3.4.6

The set $\operatorname {\mathrm {Irr}}(W_1)^{\sigma }$ is in bijection with $\prod _i\mathcal {P}(n_i)$ , which is then in bijection with $\operatorname {\mathrm {Irr}}(W_1^{\sigma })$ . We define a bijection

(3.4.6.1) $$ \begin{align} \begin{aligned} \operatorname{\mathrm{Irr}}(W^{\sigma}_1)&\smash{\begin{array}{c}\sim\\[-1em] \longrightarrow\end{array}}\operatorname{\mathrm{Irr}}(W_1)^{\sigma}\\ \varphi&\longmapsto\{\varphi\} \end{aligned} \end{align} $$

as the composition of these two natural bijections.

Write $N_{\pm }=(n_{\pm }-m_{\pm }(m_{\pm }+1)/2)/2$ and so $n_0=m+2N_++2N_-$ . There exists a unique pair $(h_1,h_2)\in \mathbb {N}\times \mathbb {Z}$ such that

(3.4.6.2) $$ \begin{align} \begin{aligned} m_+&=\sup\{h_1+h_2,-h_1-h_2-1\},\\ m_-&=\sup\{h_1-h_2,h_2-h_1-1\}. \end{aligned} \end{align} $$

Note that exchanging $\mu _+$ and $\mu _-$ changes $(h_1,h_2)$ into $(h_1,-h_2)$ . More explicitly, if n is even, then $m=m_+=m_-=0$ , and so $h_1=h_2=0$ , and if n is odd, then $m=1$ , in which case $h_1$ is always equal to $0$ while $h_2=1$ if $m_+=1$ and $h_2=-1$ if $m_-=1$ . Therefore, $h_1$ is redundant, and we will write $\epsilon =h_2$ .