The character table of the finite Chevalley group $F_4(q)$ for q a power of 2

Abstract

Let q be a prime power and $F_4(q)$ be the Chevalley group of type $F_4$ over a finite field with q elements. Marcelo and Shinoda (Tokyo J Math 18:303–340, 1995) determined the values of the unipotent characters of $F_4(q)$ on all unipotent elements, extending earlier work by Kawanaka and Lusztig to small characteristics. Assuming that q is a power of 2, we explain how to construct the complete character table of $F_4(q)$.

1 Introduction

Let p be a prime and $k=\overline{{\mathbb {F}}}_p$ be an algebraic closure of the field with p elements. Let ${\textbf{G}}$ be a connected reductive algebraic group over k and assume that ${\textbf{G}}$ is defined over the finite subfield ${\mathbb {F}}_q\subseteq k$, where q is a power of p. Let $F:{\textbf{G}}\rightarrow {\textbf{G}}$ be the corresponding Frobenius map. The finite group of fixed points ${\textbf{G}}^F$ is called a “finite group of Lie type”. We are concerned with the problem of computing the character table of ${\textbf{G}}^F$. The work of Lusztig [11, 14] has led to a general program for solving this problem.

However, in concrete examples, there are still a certain number of technical—and sometimes quite intricate—issues to be resolved. In this paper, we show how this can be done for the groups ${\textbf{G}}^F=F_4(q)$, where q is a power of 2. The conjugacy classes have been classified by Shinoda [20]; the values of all unipotent characters on unipotent elements were already determined by Marcelo–Shinoda [17]. A further crucial ingredient is the fact that the characteristic functions of the F-invariant cuspidal character sheaves of ${\textbf{G}}$ (for the definition, see [14] and the references therein) are explicitly known as linear combinations of the irreducible characters of ${\textbf{G}}^F$. Building on earlier work of Shoji [21, 22], this has been achieved in [5, 17].

In Section 2, we introduce basic notation and collect some general results from Lusztig’s theory, where we use the books [2, 6] as our references. In Sections 3 and 4, we focus on ${\textbf{G}}^F= F_4(q)$. First we consider the unipotent characters of ${\textbf{G}}^F$. Then we address some issues concerning the two-variable Green functions involved in Lusztig’s cohomological induction functor which allows us, finally, to consider the non-unipotent characters.

The special feature of ${\textbf{G}}^F=F_4(q)$ as above is that the possible root systems of centralisers of semisimple elements are rather restricted. (See Remark 3.1 below.) There is a similar situation for ${\textbf{G}}$ of adjoint type $E_6$ and $p=2$. This, as well as the case of type $E_7$ and $p=2$, will be discussed in a sequel to this paper. The values of the unipotent characters on unipotent elements have been recently determined by Hetz [7] for these groups.

I understand that Frank Lübeck has already prepared an electronic “generic” character table of $F_4(q)$, based on some assumptions concerning the values of the characteristic functions of certain F-invariant character sheaves on ${\textbf{G}}$. With the results of this paper, it should now be possible to verify those assumptions (or adjust them appropriately).

1.1. Notation and conventions. The set of (complex) irreducible characters of a finite group $\Gamma$ is denoted by ${\text {Irr}}(\Gamma )$. We work over a fixed subfield ${\mathbb {K}}\subseteq {\mathbb {C}}$, which is algebraic over ${\mathbb {Q}}$, invariant under complex conjugation, and “large enough”, that is, ${\mathbb {K}}$ contains sufficiently many roots of unity and ${\mathbb {K}}$ is a splitting field for $\Gamma$ and all of its subgroups. In particular, $\chi (g)\in {\mathbb {K}}$ for all $\chi \in {\text {Irr}}(\Gamma )$ and $g\in \Gamma$. Let $\text{ CF }(\Gamma )$ be the space of ${\mathbb {K}}$-valued class functions on $\Gamma$. There is a standard inner product $\langle \; ,\; \rangle _\Gamma$ on $\text{ CF }(\Gamma )$ given by $\langle f,f' \rangle _\Gamma := |\Gamma |^{-1}\sum _{g \in \Gamma } f(g)\overline{f'(g)}$ for $f,f' \in \text{ CF }(\Gamma )$, where $x\mapsto \overline{x}$ denotes the automorphism of ${\mathbb {K}}$ given by complex conjugation. We denote by ${\mathbb {Z}}{\text {Irr}}(\Gamma )\subseteq \text{ CF }(\Gamma )$ the subset consisting of all integral linear combinations of ${\text {Irr}}(\Gamma )$. Finally, if $C\subseteq \Gamma$ is any (non-empty) subset that is a union of conjugacy classes of $\Gamma$, then we denote by $\varepsilon _C \in \text{ CF }(\Gamma )$ the (normalised) indicator function of C, that is, we have

$$\begin{aligned} \varepsilon _C(g)=\left\{ \begin{array}{cl} |\Gamma |/|C| &{} \qquad \text{ if }\, g\in C,\\ 0 &{} \qquad \text{ otherwise }.\end{array}\right. \end{aligned}$$

Note that, if C is a single conjugacy class of $\Gamma$ and $g\in C$, then $f(g)=\langle f,\varepsilon _C\rangle _{\Gamma }$ for any $f\in \text{ CF }(\Gamma )$. Thus, the problem of computing the values of $\rho \in {\text {Irr}}(\Gamma )$ is equivalent to working out the inner products of $\rho$ with the indicator functions of the various conjugacy classes of $\Gamma$.

2 Lusztig induction and uniform functions

Let ${\textbf{G}},F$ be as in the introduction. Given an F-stable maximal torus ${\textbf{T}}$ of ${\textbf{G}}$ and $\theta \in {\text {Irr}}({\textbf{T}}^F)$, we have a generalised character $R_{{\textbf{T}},\theta }^{\textbf{G}}\in {\mathbb {Z}}{\text {Irr}}({\textbf{G}}^F)$ as introduced by Deligne and Lusztig [1] (see also [6, §2.2]). We shall also need the following generalisation of $R_{{\textbf{T}},\theta }^{\textbf{G}}$.

2.1. An F-stable closed subgroup ${\textbf{L}}\subseteq {\textbf{G}}$ is called a “regular subgroup” if ${\textbf{L}}$ is a Levi complement in some (not necessarily F-stable) parabolic subgroup ${\textbf{P}}\subseteq {\textbf{G}}$. Given such a pair $({\textbf{L}},{\textbf{P}})$, we obtain an operator

$$\begin{aligned} R_{{\textbf{L}}\subseteq {\textbf{P}}}^{{\textbf{G}}} :{\mathbb {Z}}{\text {Irr}}({\textbf{L}}^F)\rightarrow {\mathbb {Z}}{\text {Irr}}({\textbf{G}}^F) \qquad \hbox {(``Lusztig induction''; see [2,} \S 9.1]). \end{aligned}$$

Denoting by ${\textbf{G}}_{\text {uni}}^F$ and ${\textbf{L}}_{\text {uni}}^F$ the sets of unipotent elements of ${\textbf{G}}^F$ and ${\textbf{L}}^F$, respectively, there is a corresponding two-variable Green function

$$\begin{aligned} Q_{{\textbf{L}}\subseteq {\textbf{P}}}^{{\textbf{G}}}:{\textbf{G}}_{\text {uni}}^F\times {\textbf{L}}_{\text {uni}}^F\rightarrow {\mathbb {Q}}\qquad \hbox {(see [2,} \S 10.1]). \end{aligned}$$

If ${\textbf{L}}={\textbf{T}}$ is an F-stable maximal torus of ${\textbf{G}}$ (and ${\textbf{B}}\subseteq {\textbf{G}}$ is a Borel subgroup containing ${\textbf{T}}$), then ${\textbf{T}}_{\text {uni}}^F=\{1\}$ and $Q_{{\textbf{T}}}^{\textbf{G}}:{\textbf{G}}_{\text {uni}}^F \rightarrow {\mathbb {Q}}$, $u\mapsto Q_{{\textbf{T}}\subseteq {\textbf{B}}}^{{\textbf{G}}}(u,1)$, is the “usual” Green function originally introduced in [1], that is, we have $Q_{\textbf{T}}^{\textbf{G}}(u)=R_{{\textbf{T}},1}^{{\textbf{G}}}(u)$ for all $u\in {\textbf{G}}_{\text {uni}}^F$.

2.2. Let ${\textbf{L}}\subseteq {\textbf{P}}$ be as above and $\psi \in {\text {Irr}}({\textbf{L}}^F)$. There is a character formula which expresses the values of $R_{{\textbf{L}}\subseteq {\textbf{P}}}^{\textbf{G}}(\psi )$ in terms of the values of $\psi$ and the two-variable Green functions for ${\textbf{G}}$ and for groups of the form $C_{{\textbf{G}}}^\circ (s)$ where $s\in {\textbf{G}}^F$ is semisimple; see [2, Prop. 10.1.2], [13, Prop. 6.2] for the precise formulation. For later reference, we only state here the following special case:

$$\begin{aligned} R_{{\textbf{L}}\subseteq {\textbf{P}}}^{{\textbf{G}}}(\psi )(u)=\sum _{v \in {\textbf{L}}_{\text {uni}}^F} Q_{{\textbf{L}}\subseteq {\textbf{P}}}^{{\textbf{G}}}(u,v^{-1})\psi (v) \qquad \text{ for } \text{ all }\, u\in {\textbf{G}}_{\text {uni}}^F. \end{aligned}$$

(a)

We also state the following useful formula. Let $g\in {\textbf{G}}^F$ and write $g=su=us$ where $s\in {\textbf{G}}^F$ is semisimple and $u\in {\textbf{G}}^F$ is unipotent (Jordan decomposition). By [2, Prop. 3.5.3], we have $g\in C_{\textbf{G}}^\circ (s)$. If $C_{{\textbf{G}}}^\circ (s)\subseteq {\textbf{L}}$, then

$$\begin{aligned} \rho (g)=\sum _{\psi \in {\text {Irr}}({\textbf{L}}^F)} \big \langle R_{{\textbf{L}}\subseteq {\textbf{P}}}^{{\textbf{G}}}(\psi ),\rho \big \rangle _{{\textbf{G}}^F}\psi (g)\qquad \text{ for } \text{ all }\, \rho \in {\text {Irr}}({\textbf{G}}^F). \end{aligned}$$

(b)

This appeared in K.D. Schewe’s dissertation [19]; see the remark following [6, Cor. 3.3.13] for a proof.

2.3. Let us denote by ${\mathfrak {X}}({\textbf{G}},F)$ the set of all pairs $({\textbf{T}},\theta )$ where ${\textbf{T}}\subseteq {\textbf{G}}$ is an F-stable maximal torus and $\theta \in {\text {Irr}}({\textbf{T}}^F)$. Following [10, p. 16], a class function $f\in \text{ CF }({\textbf{G}}^F)$ is called “uniform” if f can be written as a ${\mathbb {K}}$-linear combination of the generalised characters $R_{{\textbf{T}},\theta }^{{\textbf{G}}}$ for various pairs $({\textbf{T}}, \theta )\in {\mathfrak {X}}({\textbf{G}},F)$. If f is uniform, then we have (see [2, Prop. 10.2.4])

$$\begin{aligned} f=|{\textbf{G}}^F|^{-1}\sum _{({\textbf{T}},\theta )\in {\mathfrak {X}}({\textbf{G}},F)} |{\textbf{T}}^F| \langle f,R_{{\textbf{T}},\theta }^{\textbf{G}}\rangle _{{\textbf{G}}^F}R_{{\textbf{T}},\theta }^{\textbf{G}}. \end{aligned}$$

For example, if C is a conjugacy class of semisimple elements of ${\textbf{G}}^F$, then the indicator function $\varepsilon _C$ (as in 1.1) is uniform; see [2, Cor. 10.3.4].

Theorem 2.4

Let ${\textbf{C}}$ be an arbitrary F-stable conjugacy class of ${\textbf{G}}$. Then the indicator function $\varepsilon _{{\textbf{C}}^F}$ of the set ${\textbf{C}}^F$ is a uniform function.

(Note that, in general, ${\textbf{C}}^F$ is a union of conjugacy classes of ${\textbf{G}}^F$.)

Proof

See the appendix of [4]; this was conjectured by Lusztig [10, 2.16]. See also [2, Cor. 13.3.5] and [6, Theorem 2.7.11]. $\square$

Example 2.5

Let $g\in {\textbf{G}}^F$ and assume that $C_{\textbf{G}}(g)$ is connected. Let ${\textbf{C}}$ be the ${\textbf{G}}$-conjugacy class of g. Since $C_{\textbf{G}}(g)$ is connected, $C:={\textbf{C}}^F$ is a single conjugacy class of ${\textbf{G}}^F$; see [6, Example 1.4.10]. Now $\varepsilon _C$ is uniform by Theorem 2.4. Let $\rho \in {\text {Irr}}({\textbf{G}}^F)$. Recall from 1.1 that $\rho (g)= \langle \rho ,\varepsilon _C\rangle _{{\textbf{G}}^F}$ and $\langle \varepsilon _C,R_{{\textbf{T}},\theta }^{\textbf{G}}\rangle _{{\textbf{G}}^F}=R_{{\textbf{T}}, \theta ^{-1}}^{\textbf{G}}(g)$ for any $({\textbf{T}},\theta )\in {\mathfrak {X}}({\textbf{G}},F)$. Hence, using 2.3, we obtain the formula

$$\begin{aligned} \rho (g)=|{\textbf{G}}^F|^{-1}\sum _{({\textbf{T}},\theta )\in {\mathfrak {X}}({\textbf{G}},F)} |{\textbf{T}}^F|\, \langle R_{{\textbf{T}},\theta }^{\textbf{G}},\rho \rangle _{{\textbf{G}}^F}\,R_{{\textbf{T}},\theta ^{-1}}^{\textbf{G}}(g). \end{aligned}$$

This shows that the value $\rho (g)$ is determined by the multiplicities $\langle R_{{\textbf{T}},\theta }^{\textbf{G}},\rho \rangle _{{\textbf{G}}^F}$ and the values $R_{{\textbf{T}},\theta }^{\textbf{G}}(g)$, where $({\textbf{T}},\theta )$ runs over all pairs in ${\mathfrak {X}}({\textbf{G}},F)$.

2.6. We say that $\rho \in {\text {Irr}}({\textbf{G}}^F)$ is “unipotent” if $\langle R_{{\textbf{T}},1}^{{\textbf{G}}},\rho \rangle _{{\textbf{G}}^F} \ne 0$ for some F-stable maximal torus ${\textbf{T}}\subseteq {\textbf{G}}$. We denote by ${\text {Uch}}({\textbf{G}}^F)$ the set of unipotent characters of ${\textbf{G}}^F$. As shown in Lusztig’s book [11], these characters play a special role in the character theory of ${\textbf{G}}^F$; many questions about arbitrary characters of ${\textbf{G}}^F$ can be reduced to unipotent characters.

3 The unipotent characters for $F_4$ in characteristic 2

We assume from now on that $p=2$ and ${\textbf{G}}$ is simple of type $F_4$. Let $F:{\textbf{G}}\rightarrow {\textbf{G}}$ be a Frobenius map such that ${\textbf{G}}^F=F_4(q)$ where q is a power of 2. In order to compute the characters of ${\textbf{G}}^F$, we shall assume that the following information is known and available in the form of tables:

(A1):

Parametrisations of ${\mathfrak {X}}({\textbf{G}},F)$ and of all the conjugacy classes of ${\textbf{G}}^F$.

(A2):

The multiplicities $\langle R_{{\textbf{T}},\theta }^{{\textbf{G}}},\rho \rangle$ for all $\rho \in {\text {Irr}}({\textbf{G}}^F)$ and $({\textbf{T}},\theta )\in {\mathfrak {X}}({\textbf{G}},F)$.

(A3):

The values $R_{{\textbf{T}},\theta }^{\textbf{G}}(g)$ for all $g\in {\textbf{G}}^F$ and all $({\textbf{T}},\theta ) \in {\mathfrak {X}}({\textbf{G}},F)$.

(A4):

For every regular ${\textbf{L}}\subsetneqq {\textbf{G}}$, the values $\psi (u)$ for $\psi \in {\text {Irr}}({\textbf{L}}^F)$, $u\in {\textbf{L}}_{\text {uni}}^F$.

It will be convenient to also introduce the set ${\mathfrak {Y}}({\textbf{G}},s)$ of all pairs $({\textbf{T}},s)$ where ${\textbf{T}}\subseteq {\textbf{G}}$ is an F-stable maximal torus and $s\in {\textbf{T}}^F$. There are natural actions of ${\textbf{G}}^F$ on ${\mathfrak {X}}({\textbf{G}},F)$ and on ${\mathfrak {Y}}({\textbf{G}},F)$; see [6, 2.3.20 and 2.5.12]. Since ${\textbf{G}}\cong {\textbf{G}}^*$ is “self-dual” (in the sense of [6, Def. 1.5.17]), there is a bijective correspondence

$$\begin{aligned} {\mathfrak {X}}({\textbf{G}},F) \;\, \text{ mod }\, {\textbf{G}}^F \;\;\leftrightarrow \;\; {\mathfrak {Y}}({\textbf{G}},F)\;\, \text{ mod }\, {\textbf{G}}^F \qquad \text{(see } \text{[6, } \text{ Cor. } \text{2.5.14]) }. \end{aligned}$$

Remark 3.1

The conjugacy classes of ${\textbf{G}}^F$ are determined by Shinoda [20]. The tables in [20] provide the required classifications and parametrisations in (A1), where we use the above-mentioned bijection to pass from ${\mathfrak {Y}}({\textbf{G}},F)$ to ${\mathfrak {X}}({\textbf{G}},F)$. Since the center of ${\textbf{G}}$ is trivial, the information in (A2) is available via Lusztig’s “Main Theorem 4.23” in [11]; see also [6, §2.4, §4.2]. In order to obtain (A3), one uses the character formula in [1, §4] (see also [6, Theorem 2.2.16]) for the evaluation of $R_{{\textbf{T}},\theta }^{\textbf{G}}(g)$. This involves the Green functions for ${\textbf{G}}$ and for groups of the form ${\textbf{H}}_s=C_{\textbf{G}}(s)$ where $s\in {\textbf{G}}^F$ is semisimple; note that, for our ${\textbf{G}}$, the centraliser of any semisimple element is connected. By inspection of [20, Table III], we see that ${\textbf{H}}_s$ is either a maximal torus, or a regular subgroup (with a root system of type $F_4$, $B_3$, $C_3$, $A_1\times A_2$, $B_2$, $A_2$, $A_1 \times A_1$, or $A_1$) or ${\textbf{H}}_s$ has a root system of type $A_2\times A_2$. The Green functions for ${\textbf{G}}^F$ itself have been determined by Malle [15]; for the other cases, see Lübeck [9, Tabelle 16]. The further technical issues in the evaluation of $R_{{\textbf{T}},\theta }^{\textbf{G}}(su)$ are discussed in [5, §3] and [9, §2] (for example, one has to deal with a sum over all $x\in {\textbf{G}}^F$ such that $x^{-1}sx\in {\textbf{T}}$); in [9, §6], this is explained in detail for the groups ${\textbf{G}}^F= \text{ CSp}_6(q)$. Finally, the required values in (A4) can be extracted from Enomoto [3] (type $B_2$), Looker [8], Lübeck [9, Tabelle 27] (type $B_3,C_3$), and Steinberg [23] (type $A_1,A_2$).

Representatives for the ${\textbf{G}}^F$-conjugacy classes of semisimple elements are denoted by $h_0,h_1,\ldots ,h_{76}$ in [20, Table II], where $h_0=1$; note that some of the $h_i$ only occur according to whether $3\mid q-1$ or $3\mid q+1$, or when q is sufficiently large. We now go through the list of these elements and explain how to determine the values of any unipotent character $\rho \in {\text {Uch}}({\textbf{G}}^F)$ on elements of the form $h_iu$ where $u \in C_{\textbf{G}}(h_i)^F$ is unipotent.

In our group ${\textbf{G}}$, there are 37 unipotent characters, where we use the notation in Lusztig’s book [11, pp. 371/372]).

3.2. If $s=h_0=1$, then the values $\rho (u)$ for $\rho \in {\text {Uch}}({\textbf{G}}^F)$ and $u\in {\textbf{G}}_{\text {uni}}^F$ have been explicitly determined by Marcelo–Shinoda; see [17, Table 6.A]. This relies on the Green functions of ${\textbf{G}}^F$ (available from [15]) and also on the knowledge of the “generalised Green functions” arising from Lusztig’s theory of character sheaves. An algorithm for the computation of those functions is described in [12, §24]; it involves the delicate matter of normalising certain “$Y_\iota$-functions” (defined in [12, (24.2.3)]). Marcelo–Shinoda [17] do not explain in detail how they found those normalisations. But using the argument of Hetz [7, §4.1.4] (where the analogous problem is solved for groups of type $E_6$ in characteristic 2), one obtains an independent verification that the values in [17, Table 5] are correct.

3.3. Let $s=h_3$ (if $3\mid q-1$) or $s=h_{15}$ (if $3\mid q+1$). Then ${\textbf{H}}_s=C_{\textbf{G}}(s)$ has a root system of type $A_2 \times A_2$. Let $u\in {\textbf{H}}_s^F$ be unipotent and ${\textbf{C}}$ be the ${\textbf{G}}$-conjugacy class of su.

(a) Assume first that u is not regular unipotent. By inspection of [20, Table IV], we see that $C_{\textbf{G}}(su)$ is connected. So we can apply Example 2.5, together with (A2), (A3), to determine $\rho (su)$ even for all $\rho \in {\text {Irr}}({\textbf{G}}^F)$.

(b) Now assume that u is regular unipotent. We recall some facts from [5, §7.6]. (Note that, in [5, §7.6], it is assumed that $p\ne 2,3$ but the discussion works verbatim also for $p=2$.) The set ${\textbf{C}}^F$ splits into 3 classes in ${\textbf{G}}^F$, which we simply denote by $C_1,C_2,C_3$. We can choose the notation such that $C_1=C_1^{-1}$ and $C_2^{-1}=C_3$. Explicit representatives are described in [20, Table IV]; we have $|C_{\textbf{G}}(g_i)^F|=3q^4$ for $g_i\in C_i$ and $i=1,2,3$. Let $\chi _0:=\varepsilon _{{\textbf{C}}^F}$ be the indicator function on the set ${\textbf{C}}^F$ (as in 1.1). Let $1\ne \theta \in {\mathbb {K}}$ be a fixed third root of unity. Then we consider the following linear combinations of unipotent characters of ${\textbf{G}}^F$:

As discussed in [5, §7.6], the class functions $\chi _1,\chi _2$ are (scalar multiples of) characteristic functions of F-invariant cuspidal character sheaves on ${\textbf{G}}$; furthermore, the values of $\chi _0,\chi _1,\chi _2$ are given as follows:

$$\begin{aligned} \begin{array}{ccccc} \hline &{} C_1 &{}\quad C_2 &{}\quad C_3 &{}\quad g \in {\textbf{G}}^F{\setminus } {\textbf{C}}_s^F\\ \hline \chi _0 &{}\quad q^4 &{}\quad q^4 &{}\quad q^4 &{}\quad 0 \\ \chi _1 &{}\quad q^4 &{}\quad q^4\theta &{}\quad q^4\theta ^2 &{}\quad 0 \\ \chi _2 &{}\quad q^4 &{}\quad q^4\theta ^2 &{}\quad q^4 \theta &{}\quad 0 \\ \hline \end{array} \end{aligned}$$

Hence, $\;\;\varepsilon _{C_1}=\chi _0+\chi _1+\chi _2$, $\; \varepsilon _{C_2}=\chi _0+\theta ^2\chi _1+\theta \chi _2$, $\;\varepsilon _{C_3}=\chi _0+ \theta \chi _1+\theta ^2\chi _2$.

Now let $\rho \in {\text {Irr}}({\textbf{G}}^F)$ be arbitrary and $g_i\in C_i$ for $i=1,2,3$. Since $\chi _0$ is uniform by Theorem 2.4, we can determine $\langle \rho ,\chi _0\rangle _{{\textbf{G}}^F}$ using (A2), (A3), and the formula in 2.3. The inner products of $\rho$ with $\chi _1,\chi _2$ are known by the definition of $\chi _1,\chi _2$. Hence, we can explicitly work out $\rho (g_i)=\langle \rho ,\varepsilon _{C_i} \rangle _{{\textbf{G}}^F}$.

3.4. Let $s=h_i$ where $i\not \in \{0,3,15\}$. In these cases, ${\textbf{L}}=C_{{\textbf{G}}}(s)$ either is a maximal torus, or a proper regular subgroup with a root system of type $B_3$, $C_3$, $A_1\times A_2$, $B_2$, $A_2$, $A_1\times A_1$, or $A_1$. Let $u\in {\textbf{L}}^F$ be unipotent and ${\textbf{C}}$ be the ${\textbf{G}}$-conjugacy class of su. Let $\rho \in {\text {Uch}}({\textbf{G}}^F)$. In order to compute $\rho (su)$, we use Schewe’s formula in 2.2. First note that, if $\psi \in {\text {Irr}}({\textbf{L}}^F)$ is such that $\big \langle R_{{\textbf{L}}\subseteq {\textbf{P}}}^{{\textbf{G}}} (\psi ),\rho \big \rangle _{{\textbf{G}}^F}\ne 0$, then we must have $\psi \in {\text {Uch}}({\textbf{L}}^F)$; see [6, Prop. 3.3.21]. Furthermore, since s is in the centre of ${\textbf{L}}^F$, we have $\psi (su)= \psi (u)$. (This is a general property of unipotent characters; see [6, Prop. 2.2.20].) Hence, Schewe’s formula reads:

$$\begin{aligned} \rho (su)=\sum _{\psi \in {\text {Uch}}({\textbf{L}}^F)} \big \langle R_{{\textbf{L}}\subseteq {\textbf{P}}}^{{\textbf{G}}}(\psi ),\rho \big \rangle _{{\textbf{G}}^F}\psi (u). \end{aligned}$$

By (A4), the values $\psi (u)$ for $\psi \in {\text {Uch}}({\textbf{L}}^F)$ and $u\in {\textbf{L}}_{\text {uni}}^F$ are explicitly known. The multiplicities $\langle R_{{\textbf{L}}\subseteq {\textbf{P}}}^{{\textbf{G}}}(\psi ),\rho \rangle _{{\textbf{G}}^F}$ (for $\rho \in {\text {Uch}}({\textbf{G}}^F)$ and $\psi \in {\text {Uch}}({\textbf{L}}^F)$) can also be determined explicitly; see [6, §4.6], especially [6, Prop. 4.6.18]. In Michel’s version of CHEVIE [18], this is available through the function LusztigInductionTable. Let us illustrate this with an example.

Example 3.5

Let (a cuspidal unipotent character). Let $s=h_{53}$; then ${\textbf{L}}=C_{\textbf{G}}(s)$ is a regular subgroup of type $B_2$, where $|{\textbf{L}}^F|= q^4(q^2+1)(q^2-1)(q^4-1)$; see [20, Table III]. We would like to determine the values $\rho (h_{53}u)$ where $u\in {\textbf{L}}^F$ is unipotent. The values of the unipotent characters of ${\textbf{L}}^F$ on unipotent elements are given by Table 1. Using Michel’s LusztigInductionTable, we find that

$$\begin{aligned} \langle R_{{\textbf{L}}\subseteq {\textbf{P}}}^{{\textbf{G}}}(\psi _{10}),\rho \rangle _{{\textbf{G}}^F}=1 \qquad \text{ and }\qquad \langle R_{{\textbf{L}}\subseteq {\textbf{P}}}^{{\textbf{G}}}(\psi _i), \rho \rangle _{{\textbf{G}}^F}=0\quad \text{ for }\, i\ne 10. \end{aligned}$$

Hence, by Schewe’s formula, we have $\rho (h_{53}u)=\psi _{10}(u)$. — A completely analogous procedure works for any $s=h_i$ as in 3.4.

4 Non-unipotent characters for $F_4$ in characteristic 2

We keep the notation of the previous section, where ${\textbf{G}}$ is simple of type $F_4$ in characteristic 2. We now explain how to determine the values of the non-unipotent characters of ${\textbf{G}}^F$. First we recall some facts from Lusztig’s classification of ${\text {Irr}}({\textbf{G}}^F)$. Let $s\in {\textbf{G}}^F$ be semisimple. Then we define ${\mathscr {E}}({\textbf{G}}^F,s)$ to be the set of all $\rho \in {\text {Irr}}({\textbf{G}}^F)$ such that $\langle R_{{\textbf{T}},\theta }^{\textbf{G}}, \rho \rangle \ne 0$ for some pair $({\textbf{T}},\theta )\in {\mathfrak {X}}({\textbf{G}},F)$ in correspondence with $({\textbf{T}},s)\in {\mathfrak {Y}}({\textbf{G}},F)$. It is known that every $\rho \in {\text {Irr}}({\textbf{G}}^F)$ belongs to ${\mathscr {E}}({\textbf{G}}^F,s)$ for some s; furthermore, ${\mathscr {E}}({\textbf{G}}^F,s)$ only depends on the ${\textbf{G}}^F$-conjugacy class of s. If $s,s'\in {\textbf{G}}^F$ are such that ${\mathscr {E}}({\textbf{G}}^F,s)\cap {\mathscr {E}}({\textbf{G}}^F,s')\ne \varnothing$, then $s,s'$ are ${\textbf{G}}^F$-conjugate. (For all this, see, for example, [6, §2.6]; also recall that ${\textbf{G}}\cong {\textbf{G}}^*$.) Finally, by the “Main Theorem 4.23” of [11], there is a bijection ${\mathscr {E}}({\textbf{G}}^F,s)\leftrightarrow {\text {Uch}}({\textbf{H}}_s^F)$, where ${\textbf{H}}_s=C_{\textbf{G}}(s)$; this is called the “Jordan decomposition” of characters. We now proceed in 4 steps, where we determine the following information:

Table 1 Unipotent characters for type $B_2$ in characteristic 2

Step 1: The values of all the two-variable Green functions $Q_{{\textbf{L}}\subseteq {\textbf{P}}}^{\textbf{G}}$.

Step 2: The values $\rho (u)$ for all $\rho \in {\text {Irr}}({\textbf{G}}^F)$ and $u \in {\textbf{G}}_{\text {uni}}^F$.

Step 3: The decomposition of $R_{{\textbf{L}}\subseteq {\textbf{P}}}^{\textbf{G}}(\psi )$ for any $\psi \in {\text {Irr}}({\textbf{L}}^F)$.

Step 4: The values $\rho (g)$ for any $\rho \in {\text {Irr}}({\textbf{G}}^F)$ and any $g\in {\textbf{G}}^F$.

4.1. We show how Step 1 can be resolved. Assume that ${\textbf{L}}\subsetneqq {\textbf{G}}$ and let ${\text {Uch}}({\textbf{L}}^F)=\{\psi _1,\ldots ,\psi _n\}$. The information in (A4) (see Section 3) shows, in particular, that n is also the number of conjugacy classes of unipotent elements of ${\textbf{L}}^F$. Let $v_1,\ldots ,v_n$ be representatives of these classes. Then, again using (A4), we can also check that the matrix $(\psi _i(v_j))_{1\leqslant i,j\leqslant n}$ is invertible. (For an example, see Table 1.) Let $u_1,\ldots ,u_N$ be representatives of the conjugacy classes of unipotent elements of ${\textbf{G}}^F$; we have $N=35$ by [20, Theorem 2.1]. Then we write the character formula 2.2(a) as a system of equations:

where $c_j:=[{\textbf{L}}^F:C_{{\textbf{L}}}(v_j)^F]$ for all j. On the other hand, as explained in 3.4, we can determine the multiplicities $m(\psi _i,\rho ):=\langle R_{{\textbf{L}}\subseteq {\textbf{P}}}^{\textbf{G}}(\psi _i), \rho \rangle _{{\textbf{G}}^F}$ for any $\rho \in {\text {Uch}}({\textbf{G}}^F)$. Hence, we obtain equations

$$\begin{aligned} R_{{\textbf{L}}\subseteq {\textbf{P}}}^{\textbf{G}}(\psi _i)(u_k)=\sum _{\rho \in {\text {Uch}}({\textbf{G}}^F)} m(\psi _i,\rho )\rho (u_k) \qquad \text{ for }\, 1\leqslant i \leqslant n, 1\leqslant k\leqslant N. \end{aligned}$$

Consequently, since the values $\rho (u_k)$ for $\rho \in {\text {Uch}}({\textbf{G}}^F)$ are known by 3.2, the values $R_{{\textbf{L}}\subseteq {\textbf{P}}}^{{\textbf{G}}}(\psi _i) (u_k)$ can be computed explicitly. We can now invert ($\spadesuit$) and obtain all the values $Q_{{\textbf{L}}\subseteq {\textbf{P}}}^{\textbf{G}}(u_k,v_j^{-1})$ for $1\leqslant j \leqslant n$, $1\leqslant k\leqslant N$. (A similar argument appears in Malle–Rotilio [16, §2.2].)

4.2. We show how Step 2 can be resolved. As in the previous section, we consider the list of semisimple elements $h_0,h_1, \ldots ,h_{76}\in {\textbf{G}}^F$. Let $\rho \in {\text {Irr}}({\textbf{G}}^F)$. There is some $s\in \{h_0,h_1,\ldots ,h_{76}\}$ such that $\rho \in {\mathscr {E}}({\textbf{G}}^F,s)$. If $s=h_0$ (the identity element), then $\rho$ is unipotent and the required values are known by 3.2. Now assume that $s\in \{h_3,h_{15}\}$ where $C_{\textbf{G}}(s)$ has a root system of type $A_2\times A_2$. Then, by the discussion in [6, Lemma 2.4.18] (which is drawn from Lusztig’s book [11]), we know that $\rho$ is a uniform class function. (The group ${\textbf{W}}_{\lambda ,n}$ occurring in that discussion is isomorphic to the Weyl group of $C_{\textbf{G}}(s)$; see [6, (2.5.10)] and note again that ${\textbf{G}}\cong {\textbf{G}}^*$.) Hence, the values $\rho (u)$ for $u\in {\textbf{G}}_{\text {uni}}^F$ are known by (A2), (A3) in Section 3. Finally, let $s=h_i$ where $i\not \in \{0,3,15\}$. Then, as in 3.4, ${\textbf{L}}:=C_{\textbf{G}}(s)\subsetneqq {\textbf{G}}$ is a regular subgroup. In that case, Lusztig has shown that $\rho =\pm R_{{\textbf{L}}\subseteq {\textbf{P}}}^{\textbf{G}}(\psi )$ for some $\psi \in {\mathscr {E}}({\textbf{L}}^F,s)$; see [6, Theorem 3.3.22]. So, in order to determine $\rho (u)$ for $u\in {\textbf{G}}_{\text {uni}}^F$, we can use again the character formula 2.2(a), combined with the knowledge of $Q_{{\textbf{L}}\subseteq {\textbf{P}}}^{\textbf{G}}$ (see Step 1) and the values $\psi (v)$ for $v\in {\textbf{L}}_{\text {uni}}^F$ (see (A4)).

4.3. We show how Step 3 can be resolved. Assume that ${\textbf{L}}\subsetneqq {\textbf{G}}$ and let $\psi \in {\text {Irr}}({\textbf{L}}^F)$ be arbitrary. There is some semisimple $s\in {\textbf{L}}^F$ such that $\psi \in {\mathscr {E}}({\textbf{L}}^F,s)$. Let ${\mathscr {E}}({\textbf{G}}^F,s)=\{\rho _1,\ldots ,\rho _r\}$. Then, by [6, Prop. 3.3.20], we have

If $s=1$ and $\psi \in {\text {Uch}}({\textbf{L}}^F)$, we can use Michel’s LusztigInductionTable, as in 3.4. Now assume that $s\ne 1$. Then one could use the fact that $R_{{\textbf{L}}\subseteq {\textbf{P}}}^{\textbf{G}}$ commutes with the Jordan decomposition of characters; see [6, Theorem 4.7.2]. But having the results of Steps 1 and 2 at our disposal, we can also argue as follows. Let again $u_1,\ldots ,u_N$ be representatives of the conjugacy classes of unipotent elements of ${\textbf{G}}^F$. Using 2.2(a), (A4), and Step 1, we can compute the values:

$$\begin{aligned} R_{{\textbf{L}}\subseteq {\textbf{P}}}^{{\textbf{G}}}(\psi )(u_k)=\sum _{v\in {\textbf{L}}_{\text {uni}}^F} Q_{{\textbf{L}}\subseteq {\textbf{P}}}^{{\textbf{G}}}(u_k,v^{-1})\psi (v) \qquad \text{ for }\, 1\leqslant k \leqslant N. \end{aligned}$$

Comparing with ($*$), we obtain equations

$$\begin{aligned} \sum _{i=1}^r m(\psi ,\rho _i)\rho _i(u_k)=R_{{\textbf{L}}\subseteq {\textbf{P}}}^{{\textbf{G}}} (\psi )(u_k)=\text{ known } \text{ value } \qquad \text{ for }\, 1\leqslant k \leqslant N. \end{aligned}$$

Using Step 2, we can check that the matrix $(\rho _i(u_k))_{1\leqslant i \leqslant r,1\leqslant k\leqslant N}$ has rank r, where $r\leqslant N$. (This would not be true for $s=1$.) Hence, the above equations uniquely determine the numbers $m(\psi ,\rho _i)$ for $1\leqslant i\leqslant r$.

4.4. We show how Step 4 can be resolved. Let $\rho \in {\text {Irr}}({\textbf{G}}^F)$ and $g\in {\textbf{G}}^F$ be arbitrary. Let $i\in \{0,1,\ldots ,76\}$ be such that $\rho \in {\mathscr {E}}({\textbf{G}}^F,h_i)$. If $i=0$, then $h_0=1$, $\rho$ is unipotent, and we know the values of $\rho$ by Section 3. Next, let $i\in \{3,15\}$. Then, as already mentioned in 4.2, $\rho$ is uniform and so the values of $\rho$ are computable via (A2), (A3). Finally, let $i \not \in \{0,3,15\}$. Write $g=su=us$ where $s\in {\textbf{G}}^F$ is semisimple and $u\in {\textbf{G}}^F$ is unipotent. If $s=1$, then the values $\rho (u)$ for $u\in {\textbf{G}}_{\text {uni}}^F$ are known by Step 2. Now let $s\ne 1$. If $C_{\textbf{G}}(s)$ has type $A_2\times A_2$, then $\rho (su)$ is already known by 3.3. Otherwise, we are in the situation of 3.4 where ${\textbf{L}}:=C_{\textbf{G}}(s)\subsetneqq {\textbf{G}}$ is a regular subgroup. Let $\psi \in {\text {Irr}}({\textbf{L}}^F)$ and $({\textbf{T}},\theta )\in {\mathfrak {X}}({\textbf{L}},F)$ be such that $\langle R_{{\textbf{T}},\theta }^{\textbf{L}},\psi \rangle _{{\textbf{L}}^F} \ne 0$; then, by [6, Prop. 2.2.20], we have $\psi (su)=\theta (s) \psi (u)$. So Schewe’s formula, together with (A4) and the result of Step 3, yields the value $\rho (su)$.