1. Introduction
Let $\mathbf{G}$ be a connected reductive algebraic group over an algebraic closure $k$ of the finite field $\mathbb{F}_{p}$ with $p$ elements, where $p$ is a prime. Let $\mathbf{W}$ be the Weyl group of $\mathbf{G}$ and ${\mathcal{N}}_\mathbf{G}$ be the set of all pairs $({\mathcal{O}},{\mathcal{E}})$ where ${\mathcal{O}}\subseteq \mathbf{G}$ is a unipotent conjugacy class and ${\mathcal{E}}$ is an irreducible local system on ${\mathcal{O}}$ (taken up to isomorphism) which is equivariant for the conjugation action of $\mathbf{G}$ . The Springer correspondence (originally defined in Reference 39 for $p$ not too small; for arbitrary $p$ see Reference 15 ) defines an injective map $\iota _\mathbf{G}\colon \operatorname {Irr}(\mathbf{W})\hookrightarrow {\mathcal{N}}_\mathbf{G}$ which plays a crucial role, for example, in the determination of the values of the Deligne–Lusztig Green functions of Reference 5 ; for recent surveys see Reference 7, Chap. 13 and Reference 10, §2.8 . However, the map $\iota _\mathbf{G}$ is not surjective in general. In order to understand the missing pairs in ${\mathcal{N}}_\mathbf{G}$ , Lusztig Reference 17 developed a generalisation of Springer’s correspondence. This in turn constitutes a substantial part of the general problem of computing the complete character tables of finite groups of Lie type.

With very few exceptions, the problem of determining explicitly the generalised Springer correspondence has been solved in Reference 17 , Reference 32 , Reference 38 (see also the further references there). The exceptions occur for $\mathbf{G}$ of type $E_6$ with $p\neq 3$ and $E_8$ with $p=3$ . Recently, Lusztig Reference 31 settled the case where $\mathbf{G}$ is of type $E_6$ and stated a conjecture concerning the last open cases in type $E_8$ . It is the purpose of this paper to prove that conjecture, thus completing the determination of the generalised Springer correspondence in all cases.

The paper is organised as follows. In Section 2 , we recall the definition of the generalised Springer correspondence, due to Reference 17 . In Section 3 , we explain two parametrisations of unipotent characters and unipotent character sheaves, one in terms of Lusztig’s Fourier transform matrices associated to families in the Weyl group, and the other one in terms of Harish-Chandra series, for simple groups of adjoint type with a split rational structure. In the last two sections, we focus on the specific case where $\mathbf{G}$ is simple of type $E_8$ in characteristic $p=3$ : Section 4 contains the description of the generalised Springer correspondence for this group, so that we can formulate Lusztig’s conjecture on the last open cases as indicated above. Finally, the proof of this conjecture is given in Section 5 . It is based on considering the Hecke algebra associated to the finite group $E_8(q)$ (where $q=3^n$ for some $n\geqslant 1$ ) and its natural $(B,N)\text{-pair}$ , exploiting a well-known formula relating characters of this Hecke algebra with the unipotent principal series characters of $E_8(q)$ . The fact that this formula carries subtle geometric information has been used before, for example in Reference 28 .

Notation.
As soon as a prime $p$ is fixed in a given setting, we denote by $k=\overline{\mathbb{F}}_{p}$ an algebraic closure of the finite field $\mathbb{F}_{p}$ with $p$ elements; furthermore, we tacitly assume to have fixed another prime $\ell \neq p$ , as well as an algebraic closure $\overline{\mathbb{Q}}_{\ell }$ of the field $\mathbb{Q}_\ell$ of $\ell$ -adic numbers. It will be convenient to assume the existence of an isomorphism $\overline{\mathbb{Q}}_{\ell }\simeq \mathbb{C}$ and to fix such an isomorphism once and for all, so that we can speak of “complex” conjugation or absolute values for elements of $\overline{\mathbb{Q}}_{\ell }$ using this isomorphism. In this way, we will also identify the rational numbers $\mathbb{Q}$ or the real numbers $\mathbb{R}$ as subsets of $\overline{\mathbb{Q}}_{\ell }$ and just write $\mathbb{Q}\subseteq \mathbb{R}\subseteq \overline{\mathbb{Q}}_{\ell }$ . In several places in this paper, we will need to fix a square root of $p$ (or of powers of $p$ ), so we do this right away:

$$\begin{equation} \begin{split} &\text{We fix, once and for all, a square root $\sqrt p$ of $p$ in $\overline{\mathbb{Q}}_{\ell }$.} \\ &\text{Whenever $q=p^e$ ($e\geqslant 1$), we set $\sqrt q\coloneq {(\sqrt p)}^e$.} \end{split} \cssId{sqrtp}{\tag{1.0.1}} \end{equation}$$
For any finite group $\Gamma$ , we denote by $\operatorname {CF}(\Gamma )$ the set of class functions $\Gamma \rightarrow \overline{\mathbb{Q}}_{\ell }$ and by $\operatorname {Irr}(\Gamma )\subseteq \operatorname {CF}(\Gamma )$ the subset consisting of irreducible characters of $\Gamma$ over $\overline{\mathbb{Q}}_{\ell }$ . Thus, $\operatorname {Irr}(\Gamma )$ is an orthonormal basis of $\operatorname {CF}(\Gamma )$ with respect to the scalar product

$$\begin{equation*} {\langle f, f'\rangle }_{\Gamma }\coloneq |\Gamma |^{-1}\sum _{g\in \Gamma }f(g)\overline{f'(g)}\quad (\text{for }f, f'\in \operatorname {CF}(\Gamma )). \end{equation*}$$
Now let $p$ be a prime and $\mathbf{G}$ be a connected reductive group over $k=\overline{\mathbb{F}}_{p}$ , defined over the finite subfield $\mathbb{F}_{q}\subseteq k$ where $q=p^n$ for some $n\geqslant 1$ , with corresponding Frobenius map $F\colon \mathbf{G}\rightarrow \mathbf{G}$ . With respect to this $F$ , we fix a maximally split torus $\mathbf{T}_0\subseteq \mathbf{G}$ and an $F$ -stable Borel subgroup $\mathbf{B}_0\subseteq \mathbf{G}$ such that $\mathbf{T}_0\subseteq \mathbf{B}_0$ . Let $\Phi$ be the set of roots of $\mathbf{G}$ with respect to $\mathbf{T}_0$ , and let $\Pi \subseteq \Phi$ be the subset of simple roots determined by $\mathbf{T}_0\subseteq \mathbf{B}_0$ . We denote by $\mathbf{W}=N_\mathbf{G}(\mathbf{T}_0)\big /\mathbf{T}_0$ the Weyl group of $\mathbf{G}$ (relative to $\mathbf{T}_0$ ). Given any closed subgroup $\mathbf{H}\subseteq \mathbf{G}$ (including the case where $\mathbf{H}=\mathbf{G}$ ), we denote by $\mathbf{H}^\circ \subseteq \mathbf{H}$ its identity component, by $\mathbf{Z}(\mathbf{H})\subseteq \mathbf{H}$ the centre of $\mathbf{H}$ and by $\mathbf{H}_{\mathrm{uni}}\subseteq \mathbf{H}$ the (closed) subvariety consisting of all unipotent elements of $\mathbf{H}$ . If $F(\mathbf{H})=\mathbf{H}$ , we set

$$\begin{equation*} \mathbf{H}^F\coloneq \{h\in \mathbf{H}\mid F(h)=h\}\subseteq \mathbf{H}, \end{equation*}$$
a finite subgroup of $\mathbf{H}$ . In particular, $\mathbf{G}^F$ is the finite group of Lie type associated to $(\mathbf{G},F)$ .

2. The generalised Springer correspondence
The notation is as in Section 1 . In this section, we give the definition of the generalised Springer correspondence for $\mathbf{G}$ , due to Lusztig Reference 17 .

2.1.
We start by briefly introducing some of the most important notions of Lusztig’s theory of character sheaves Reference 24 and the underlying theory of perverse sheaves Reference 1 . For the details we refer to Reference 17 , Reference 18 –Reference 22 . Recall that $\ell \neq p$ is a fixed prime. Let ${\mathscr{D}}\mathbf{G}$ be the bounded derived category of constructible $\overline{\mathbb{Q}}_{\ell }$ -sheaves on $\mathbf{G}$ in the sense of Beilinson–Bernstein–Deligne Reference 1 . To each $K\in {\mathscr{D}}\mathbf{G}$ and each $i\in \mathbb{Z}$ is associated the $i$ th cohomology sheaf ${\mathscr{H}}^i(K)$ , whose stalks ${\mathscr{H}}^i_g(K)$ (for $g\in \mathbf{G}$ ) are finite-dimensional $\overline{\mathbb{Q}}_{\ell }$ -vector spaces. The support of such a $K$ is defined as

$$\begin{equation*} \operatorname {supp}K\coloneq \overline{\{g\in \mathbf{G}\mid {\mathscr{H}}^i_g K\neq 0\text{ for some }i\in \mathbb{Z}\}}\subseteq \mathbf{G} \end{equation*}$$
(where the bar stands for the Zariski closure, here in $\mathbf{G}$ ). We set

$$\begin{equation} \hat{\varepsilon }_K\coloneq (-1)^{\dim \mathbf{G}-\dim \operatorname {supp}K}. \cssId{epsK}{\tag{2.1.1}} \end{equation}$$
Let ${\mathscr{M}}\mathbf{G}$ be the full subcategory of ${\mathscr{D}}\mathbf{G}$ consisting of the perverse sheaves on $\mathbf{G}$ ; the category ${\mathscr{M}}\mathbf{G}$ is abelian. Assume that $X$ is a locally closed subvariety of $\mathbf{G}$ and that ${\mathcal{L}}$ is an irreducible $\overline{\mathbb{Q}}_{\ell }$ -local system on $X$ . (We will just speak of a “local system” when we mean a $\overline{\mathbb{Q}}_{\ell }$ -local system from now on.) There is a unique extension of ${\mathcal{L}}$ to $\overline{X}$ , namely, the intersection cohomology complex $\operatorname {IC}(\overline{X},{\mathcal{L}})[\dim X]$ (where $[\,]$ denotes shift), due to Deligne–Goresky–MacPherson (see Reference 12 , Reference 1 ). The following notation will be convenient: For $X$ and ${\mathcal{L}}$ as above, and for any closed subvariety $Y\subseteq \mathbf{G}$ such that $X\subseteq Y$ , we denote by

$$\begin{equation*} {\operatorname {IC}(\overline{X},{\mathcal{L}})[\dim X]}^{\#Y}\in {\mathscr{D}}Y \end{equation*}$$
the extension of $\operatorname {IC}(\overline{X},{\mathcal{L}})[\dim X]$ to $Y$ , by $0$ on $Y\setminus \overline{X}$ . Now let us bring the Frobenius endomorphism $F\colon \mathbf{G}\rightarrow \mathbf{G}$ into the picture. Let $F^\ast \colon {\mathscr{D}}\mathbf{G}\rightarrow {\mathscr{D}}\mathbf{G}$ be its inverse image functor. Let $K\in {\mathscr{D}}\mathbf{G}$ be such that $F^\ast K\cong K$ in ${\mathscr{D}}\mathbf{G}$ , and let $\varphi \colon F^\ast K\xrightarrow {\sim }K$ be an isomorphism. For each $i\in \mathbb{Z}$ and $g\in \mathbf{G}^F$ , $\varphi$ induces linear maps

$$\begin{equation*} \varphi _{i,g}\colon {\mathscr{H}}^i_g(K)\rightarrow {\mathscr{H}}^i_g(K). \end{equation*}$$
Since ${\mathscr{H}}^i_g(K)$ is non-zero only for finitely many $i\in \mathbb{Z}$ , one can define a characteristic function Reference 19, 8.4

$$\begin{equation*} \chi _{K,\varphi }\colon \mathbf{G}^F\rightarrow \overline{\mathbb{Q}}_{\ell },\quad g\mapsto \sum _{i\in \mathbb{Z}}(-1)^i\operatorname {Trace}(\varphi _{i,g},{\mathscr{H}}^i_g(K)). \end{equation*}$$
In Reference 18, §2 , Lusztig defines the character sheaves on $\mathbf{G}$ as certain simple objects of ${\mathscr{M}}\mathbf{G}$ which are equivariant for the conjugation action of $\mathbf{G}$ on itself. We denote by $\hat{\mathbf{G}}$ a set of representatives for the isomorphism classes of character sheaves on $\mathbf{G}$ . An important subset of $\hat{\mathbf{G}}$ is the one consisting of cuspidal character sheaves, as defined in Reference 18, 3.10 . We denote by ${\hat{\mathbf{G}}}^\circ \subseteq \hat{\mathbf{G}}$ a set of representatives for the isomorphism classes of cuspidal character sheaves on $\mathbf{G}$ . The inverse image functor $F^\ast$ may also be regarded as a functor $F^\ast \colon {\mathscr{M}}\mathbf{G}\rightarrow {\mathscr{M}}\mathbf{G}$ , and we have $F^\ast (\hat{\mathbf{G}})=\hat{\mathbf{G}}$ . Thus, we can consider the subset

$$\begin{equation*} \hat{\mathbf{G}}^F\coloneq \{A\in \hat{\mathbf{G}}\mid F^\ast A\cong A\}\subseteq \hat{\mathbf{G}} \end{equation*}$$
consisting of the $F$ -stable character sheaves in $\hat{\mathbf{G}}$ . Of particular relevance for our purposes will be the set $\hat{\mathbf{G}}^{\mathrm{un}}\subseteq \hat{\mathbf{G}}$ of (representatives for the isomorphism classes of) unipotent character sheaves. These are, by definition, the simple constituents of a perverse cohomology sheaf ${\vphantom {H}}^pH^i(K_w^{{\mathcal{L}}_0})\in {\mathscr{M}}\mathbf{G}$ for some $w\in \mathbf{W}$ and some $i\in \mathbb{Z}$ , where $K_w^{{\mathcal{L}}_0}\in {\mathscr{D}}\mathbf{G}$ is as defined in Reference 18, 2.4 , with respect to the trivial local system ${\mathcal{L}}_0=\overline{\mathbb{Q}}_{\ell }$ on the torus $\mathbf{T}_0$ .

2.2.
Let ${\mathcal{N}}_\mathbf{G}$ be the set of all pairs $({\mathcal{O}},{\mathcal{E}})$ where ${\mathcal{O}}\subseteq \mathbf{G}$ is a unipotent conjugacy class and ${\mathcal{E}}$ is an irreducible local system on ${\mathcal{O}}$ (up to isomorphism) which is equivariant for the conjugation action of $\mathbf{G}$ on ${\mathcal{O}}$ . Let us consider a triple $(\mathbf{L},{\mathcal{O}}_0,{\mathcal{E}}_0)$ consisting of a Levi complement $\mathbf{L}$ of some parabolic subgroup of $\mathbf{G}$ , a unipotent class ${\mathcal{O}}_0$ of $\mathbf{L}$ and an irreducible local system ${\mathcal{E}}_0$ on ${\mathcal{O}}_0$ (up to isomorphism) which is equivariant for the conjugation action of $\mathbf{L}$ on ${\mathcal{O}}_0$ ; furthermore, assume that $({\mathbf{Z}(\mathbf{L})}^\circ .{\mathcal{O}}_0,1\boxtimes {\mathcal{E}}_0)$ is a cuspidal pair for $\mathbf{L}$ in the sense of Reference 17, 2.4 , where $1\boxtimes {\mathcal{E}}_0$ denotes the inverse image of ${\mathcal{E}}_0$ under the canonical map ${\mathbf{Z}(\mathbf{L})}^\circ .{\mathcal{O}}_0\rightarrow {\mathcal{O}}_0$ . The group $\mathbf{G}$ acts naturally on the set of all such triples by means of $\mathbf{G}$ -conjugacy:

$$\begin{equation*} {\vphantom {(}}^g(\mathbf{L},{\mathcal{O}}_0,{\mathcal{E}}_0)\coloneq (g\mathbf{L}g^{-1},g{\mathcal{O}}_0g^{-1},\operatorname {Int}(g^{-1})^\ast {\mathcal{E}}_0)\quad \text{for }g\in \mathbf{G}, \end{equation*}$$
where $\operatorname {Int}(g^{-1})\colon \mathbf{G}\rightarrow \mathbf{G}$ is the inner automorphism given by conjugation with $g^{-1}$ . Denote by ${\mathcal{M}}_\mathbf{G}$ the set of equivalence classes of triples $(\mathbf{L},{\mathcal{O}}_0,{\mathcal{E}}_0)$ as above under this action; however, by a slight abuse of notation, we will still just write $(\mathbf{L},{\mathcal{O}}_0,{\mathcal{E}}_0)\in {\mathcal{M}}_\mathbf{G}$ rather than $[(\mathbf{L},{\mathcal{O}}_0,{\mathcal{E}}_0)]\in {\mathcal{M}}_\mathbf{G}$ or the like. We typically write $\mathfrak{i},\mathfrak{i}',\dots$ for elements of ${\mathcal{N}}_\mathbf{G}$ and $\mathfrak{j},\mathfrak{j}',\dots$ for elements of ${\mathcal{M}}_\mathbf{G}$ . By Reference 17, §6 , any $\mathfrak{j}\in {\mathcal{M}}_\mathbf{G}$ gives rise to a certain perverse sheaf $K_\mathfrak{j}\in {\mathscr{M}}\mathbf{G}$ , whose endomorphism algebra ${\mathscr{A}}_\mathfrak{j}\coloneq \operatorname {End}_{{\mathscr{M}}\mathbf{G}}(K_\mathfrak{j})$ is isomorphic to the group algebra of the relative Weyl group ${\mathscr{W}}_\mathfrak{j}\coloneq W_\mathbf{G}(\mathbf{L})=N_\mathbf{G}(\mathbf{L})\big /\mathbf{L}$ , see Reference 17, Thm. 9.2 . Thus, the isomorphism classes of the simple direct summands of $K_\mathfrak{j}$ are naturally parametrised by $\operatorname {Irr}({\mathscr{W}}_\mathfrak{j})$ , so we have

$$\begin{equation} K_{\mathfrak{j}}\cong \bigoplus _{\phi \in \operatorname {Irr}({\mathscr{W}}_{\mathfrak{j}})}(A_\phi \otimes V_\phi ) \cssId{DecompKj}{\tag{2.2.1}} \end{equation}$$
where $A_\phi$ is the simple direct summand of $K_{\mathfrak{j}}$ corresponding to $\phi \in \operatorname {Irr}({\mathscr{W}}_{\mathfrak{j}})$ , and where $V_\phi =\operatorname {Hom}_{{\mathscr{M}}\mathbf{G}}(A_\phi , K_{\mathfrak{j}})$ . Then, for any $\phi \in \operatorname {Irr}({\mathscr{W}}_\mathfrak{j})$ , there exists a unique $({\mathcal{O}},{\mathcal{E}})\in {\mathcal{N}}_\mathbf{G}$ for which

$$\begin{equation} A_\phi |_{\mathbf{G}_{\mathrm{uni}}}\cong {\operatorname {IC}(\overline{{\mathcal{O}}}, {\mathcal{E}})[\dim {{\mathbf{Z}(\mathbf{L})}^\circ }+\dim {{\mathcal{O}}}]}^{\#\mathbf{G}_{\mathrm{uni}}}, \cssId{AphiGuni}{\tag{2.2.2}} \end{equation}$$
and the isomorphism class of $A_\phi$ is uniquely determined by this property among the simple perverse sheaves which are constituents of $K_{\mathfrak{j}}$ Reference 22, 24.1 . So for each $\mathfrak{j}\in {\mathcal{M}}_\mathbf{G}$ , the above procedure gives rise to an injective map

$$\begin{equation*} {\operatorname {Irr}({\mathscr{W}}_{\mathfrak{j}})}\hookrightarrow {{\mathcal{N}}_\mathbf{G}}. \end{equation*}$$
Conversely, given any $({\mathcal{O}},{\mathcal{E}})\in {\mathcal{N}}_\mathbf{G}$ , there exists a unique $\mathfrak{j}\in {\mathcal{M}}_\mathbf{G}$ such that $({\mathcal{O}},{\mathcal{E}})$ is in the image of the map ${\operatorname {Irr}({\mathscr{W}}_{\mathfrak{j}})}\hookrightarrow {{\mathcal{N}}_\mathbf{G}}$ just defined. So there is an associated surjective map

$$\begin{equation*} \tau \colon {\mathcal{N}}_\mathbf{G}\rightarrow {\mathcal{M}}_\mathbf{G}, \end{equation*}$$
whose fibres are called the blocks of ${\mathcal{N}}_\mathbf{G}$ . For any $\mathfrak{j}=(\mathbf{L},{\mathcal{O}}_0,{\mathcal{E}}_0)\in {\mathcal{M}}_\mathbf{G}$ , the elements in the block $\tau ^{-1}(\mathfrak{j})\subseteq {\mathcal{N}}_\mathbf{G}$ are thus parametrised by the irreducible characters of ${\mathscr{W}}_\mathfrak{j}=W_\mathbf{G}(\mathbf{L})$ . The collection of the bijections

$$\begin{equation} \operatorname {Irr}({\mathscr{W}}_\mathfrak{j})\xrightarrow {\sim }\tau ^{-1}(\mathfrak{j})\qquad (\text{for }\mathfrak{j}\in {\mathcal{M}}_\mathbf{G}) \cssId{GenSpring}{\tag{2.2.3}} \end{equation}$$
is called the generalised Springer correspondence. Given a pair $\mathfrak{i}=({\mathcal{O}},{\mathcal{E}})\in {\mathcal{N}}_\mathbf{G}$ and $\phi \in \operatorname {Irr}({\mathscr{W}}_{\tau (\mathfrak{i})})$ corresponding to $\mathfrak{i}$ under Equation 2.2.3 , we will write $A_{\mathfrak{i}}\coloneq A_\phi$ . Considering the element $\mathfrak{j}=(\mathbf{T}_0, \{1\}, \overline{\mathbb{Q}}_{\ell })\in {\mathcal{M}}_\mathbf{G}$ , map Equation 2.2.3 defines an injection

$$\begin{equation} \operatorname {Irr}(\mathbf{W})\hookrightarrow {\mathcal{N}}_\mathbf{G}, \cssId{OrdSpring}{\tag{2.2.4}} \end{equation}$$
which is called the (ordinary) Springer correspondence. The problem of determining the generalised Springer correspondence (that is, explicitly describing bijections Equation 2.2.3 for all $\mathfrak{j}\in {\mathcal{M}}_\mathbf{G}$ ) can be reduced to considering simple algebraic groups $\mathbf{G}$ of simply connected type, thus can be approached by means of a case-by-case analysis. This has been accomplished for almost all such $\mathbf{G}$ , thanks to the work of Lusztig Reference 17 , Lusztig–Spaltenstein Reference 32 , Spaltenstein Reference 38 (see also the references there for earlier results concerning the ordinary Springer correspondence), and again Lusztig Reference 31 , the only remaining open problems occur for $\mathbf{G}$ of type $E_8$ in characteristic $p=3$ (for which a conjecture is made in Reference 31, §6 ). In particular, the ordinary Springer correspondence Equation 2.2.4 is explicitly known in complete generality.

2.3.
We keep the setting of 2.2 and consider the Frobenius map $F\colon \mathbf{G}\rightarrow \mathbf{G}$ . It defines actions on ${\mathcal{N}}_\mathbf{G}$ and ${\mathcal{M}}_\mathbf{G}$ , given by

$$\begin{equation*} {\mathcal{N}}_\mathbf{G}\rightarrow {\mathcal{N}}_\mathbf{G},\quad ({\mathcal{O}},{\mathcal{E}})\mapsto (F^{-1}({\mathcal{O}}),F^\ast {\mathcal{E}}), \end{equation*}$$
and

$$\begin{equation*} {\mathcal{M}}_\mathbf{G}\rightarrow {\mathcal{M}}_\mathbf{G},\quad (\mathbf{L},{\mathcal{O}}_0,{\mathcal{E}}_0)\mapsto (F^{-1}(\mathbf{L}),F^{-1}({\mathcal{O}}_0),F^\ast {\mathcal{E}}_0). \end{equation*}$$
Let ${\mathcal{N}}_\mathbf{G}^F$ , ${\mathcal{M}}_\mathbf{G}^F$ be the respective sets of fixed points under these actions where, in terms of the local systems, this is only meant up to isomorphism, and for the triples in ${\mathcal{M}}_\mathbf{G}$ in addition only up to $\mathbf{G}$ -conjugacy. The map $\tau$ commutes with the action of $F$ on ${\mathcal{N}}_\mathbf{G}$ , ${\mathcal{M}}_\mathbf{G}$ , so it gives rise to a surjective map ${\mathcal{N}}_\mathbf{G}^F\rightarrow {\mathcal{M}}_\mathbf{G}^F$ . Furthermore, the generalised Springer correspondence Equation 2.2.3 induces bijections

$$\begin{equation*} {\operatorname {Irr}({\mathscr{W}}_\mathfrak{j})}^F\xrightarrow {\sim }\tau ^{-1}(\mathfrak{j})\cap {\mathcal{N}}_\mathbf{G}^F\qquad (\text{for }\mathfrak{j}\in {\mathcal{M}}_\mathbf{G}^F). \end{equation*}$$
(Here, we denote by ${\operatorname {Irr}({\mathscr{W}}_\mathfrak{j})}^F\subseteq \operatorname {Irr}({\mathscr{W}}_\mathfrak{j})$ the subset consisting of all irreducible characters of ${\mathscr{W}}_\mathfrak{j}$ which are invariant under the automorphism of ${\mathscr{W}}_\mathfrak{j}$ induced by $F$ .) Let $\mathfrak{j}=(\mathbf{L},{\mathcal{O}}_0,{\mathcal{E}}_0)\in {\mathcal{M}}_\mathbf{G}^F$ and $\phi \in {\operatorname {Irr}({\mathscr{W}}_{\mathfrak{j}})}^F$ , and assume that $\mathfrak{i}=({\mathcal{O}},{\mathcal{E}})$ is the corresponding element of $\tau ^{-1}(\mathfrak{j})\cap {\mathcal{N}}_\mathbf{G}^F$ . Choosing an isomorphism $\varphi _0\colon F^\ast {\mathcal{E}}_0\xrightarrow {\sim }{\mathcal{E}}_0$ which induces a map of finite order at the stalk of ${\mathcal{E}}_0$ at any element of ${\mathcal{O}}_0^F$ allows the definition of a unique isomorphism $\varphi _\mathfrak{i}\colon {F^\ast A_{\mathfrak{i}}}\xrightarrow {\sim }{A_{\mathfrak{i}}}$ (see Reference 22, 24.2 and also 3.11 ). We set

$$\begin{equation*} a_\mathfrak{i}\coloneq -\dim {\mathcal{O}}-\dim {\mathbf{Z}(\mathbf{L})}^\circ ,\quad b_\mathfrak{i}\coloneq \dim \operatorname {supp}A_{\mathfrak{i}},\quad d_{\mathfrak{i}}\coloneq \tfrac{1}{2}(a_\mathfrak{i}+b_\mathfrak{i}). \end{equation*}$$
We then have

$$\begin{equation*} {\mathscr{H}}^a(A_{\mathfrak{i}})|_{{\mathcal{O}}}\cong \begin{cases} {\mathcal{E}}&\text{if }a=a_\mathfrak{i},\\ 0&\text{if }a\neq a_\mathfrak{i}, \end{cases} \end{equation*}$$
so we can define an isomorphism $\psi _{\mathfrak{i}}\colon {F^\ast {\mathcal{E}}}\xrightarrow {\sim }{{\mathcal{E}}}$ by the requirement that $q^{d_\mathfrak{i}}\psi _{\mathfrak{i}}$ is equal to the isomorphism ${F^\ast {{\mathscr{H}}^{a_\mathfrak{i}}(A_{\mathfrak{i}})|_{{\mathcal{O}}}}}\xrightarrow {\sim }{{{\mathscr{H}}^{a_\mathfrak{i}}(A_{\mathfrak{i}})}|_{{\mathcal{O}}}}$ induced by $\varphi _\mathfrak{i}$ . It is shown in Reference 22, (24.2.4) that for any $u\in {\mathcal{O}}^F$ , the induced map $\psi _{{\mathfrak{i}},u}\colon {\mathcal{E}}_u\rightarrow {\mathcal{E}}_u$ on the stalk of ${\mathcal{E}}$ at $u$ is of finite order. Now consider the two functions

$$\begin{equation*} X_{\mathfrak{i}}\colon \mathbf{G}^F_{\mathrm{uni}}\rightarrow \overline{\mathbb{Q}}_{\ell }\qquad \text{and}\qquad Y_{\mathfrak{i}}\colon \mathbf{G}^F_{\mathrm{uni}}\rightarrow \overline{\mathbb{Q}}_{\ell }, \end{equation*}$$
defined by

$$\begin{equation*} X_{\mathfrak{i}}(u)\coloneq (-1)^{a_\mathfrak{i}}q^{-d_{\mathfrak{i}}}\chi _{A_{\mathfrak{i}},\varphi _\mathfrak{i}}(u) \end{equation*}$$
and

$$\begin{equation*} Y_{\mathfrak{i}}(u)\coloneq \begin{cases} \mathrm{Trace}(\psi _{{\mathfrak{i}},u},{\mathcal{E}}_u)&\text{if }u\in {\mathcal{O}}^F, \\ 0&\text{if }u\notin {\mathcal{O}}^F, \end{cases} \end{equation*}$$
for $u\in \mathbf{G}^F_{\mathrm{uni}}$ . Both $X_{\mathfrak{i}}$ and $Y_{\mathfrak{i}}$ are invariant under the conjugation action of $\mathbf{G}^F$ on $\mathbf{G}^F_{\mathrm{uni}}$ .

Theorem 2.4 ( Lusztig Reference 22, §24) .
In the setting of 2.3 , the following hold.

(a) The functions $Y_{\mathfrak{i}}$ , $\mathfrak{i}\in {\mathcal{N}}_\mathbf{G}^F$ , form a basis of the vector space consisting of all functions $\mathbf{G}^F_{\mathrm{uni}}\rightarrow \overline{\mathbb{Q}}_{\ell }$ which are invariant under $\mathbf{G}^F$ -conjugacy.

(b) There is a system of equations$$\begin{equation*} X_{\mathfrak{i}}=\sum _{\mathfrak{i}'\in {\mathcal{N}}_\mathbf{G}^F}p_{\mathfrak{i}',\mathfrak{i}}Y_{\mathfrak{i}'},\quad \mathfrak{i}\in {\mathcal{N}}_\mathbf{G}^F, \end{equation*}$$

for some uniquely determined $p_{\mathfrak{i}',\mathfrak{i}}\in \mathbb{Z}$ .

Proof.
See Reference 22, (24.2.7) and (24.2.9) . Note that the restrictions Reference 22, (23.0.1) on the characteristic $p$ of $k$ can be removed, thanks to the remarks in Reference 28, 3.10 .

■
2.5.
As an immediate consequence of Theorem 2.4 (and of 2.3 ), we see that:

(i) We have $p_{\mathfrak{i},\mathfrak{i}}=1$ for all $\mathfrak{i}\in {\mathcal{N}}_\mathbf{G}^F$ .

(ii) If $\mathfrak{i}'=({\mathcal{O}}',{\mathcal{E}}')\neq \mathfrak{i}=({\mathcal{O}},{\mathcal{E}})$ , then $p_{\mathfrak{i}',\mathfrak{i}}\neq 0$ implies ${\mathcal{O}}'\neq {\mathcal{O}}$ and ${\mathcal{O}}'\subseteq \overline{{\mathcal{O}}}$ .

(iii) If $\mathfrak{i}',\mathfrak{i}\in {\mathcal{N}}_\mathbf{G}^F$ belong to different blocks, we have $p_{\mathfrak{i}',\mathfrak{i}}=0$ .

Let us fix any total order $\leqslant$ on ${\mathcal{N}}_\mathbf{G}^F$ such that for $\mathfrak{i}=({\mathcal{O}},{\mathcal{E}}), \mathfrak{i}'=({\mathcal{O}}',{\mathcal{E}}')\in {\mathcal{N}}_\mathbf{G}^F$ , we have

$$\begin{equation*} \mathfrak{i}'\leqslant \mathfrak{i}\text{ whenever }{\mathcal{O}}'\subseteq \overline{{\mathcal{O}}}. \end{equation*}$$
(Note that the latter defines a partial order on the set of unipotent classes of $\mathbf{G}$ .) Then the matrix $(p_{\mathfrak{i}',\mathfrak{i}})_{\mathfrak{i}',\mathfrak{i}\in {\mathcal{N}}_\mathbf{G}^F}$ has upper unitriangular shape with respect to $\leqslant$ . In Reference 22, §24 , Lusztig provides an algorithm for computing this matrix $(p_{\mathfrak{i}',\mathfrak{i}})$ , which entirely relies on combinatorial data. This algorithm is implemented in Michel’s development version of CHEVIE Reference 33 and is accessible via the functions UnipotentClasses and ICCTable .