The character table of the finite Chevalley group F4(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_4(q)$$\end{document} for q a power of 2

Let q be a prime power and F4(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_4(q)$$\end{document} be the Chevalley group of type F4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_4$$\end{document} 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 F4(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_4(q)$$\end{document} 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 F4(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_4(q)$$\end{document}.


Introduction
Let p be a prime and k = F p be an algebraic closure of the field with p elements.Let G be a connected reductive algebraic group over k and assume that G is defined over the finite subfield F q ⊆ k, where q is a power of p.Let F : G → G be the corresponding Frobenius map.The finite group of fixed points G F is called a "finite group of Lie type".We are concerned with the problem of computing the character table of 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 G F = F 4 (q), where q is a power of 2. The conjugacy classes have been classified by Shinoda [19]; 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 G (for the definition, see [14] and the references there) are explicitly known as linear combinations of the irreducible characters of G F .Building on earlier work of Shoji [20], [21], this has been achieved in [17], [5].
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 Section 3 and 4 we focus on G F = F 4 (q).First we consider the unipotent characters of 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 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 completely similar situation for G of type E 6 in characteristic 2, assuming that G has a connected centre and a simply connected derived subgroup.This, as well as the case of groups of type E 7 in characteristic 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 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 Γ is denoted by Irr(Γ).We work over a fixed subfield K ⊆ C, which is algebraic over Q, invariant under complex conjugation and "large enough", that is, K contains sufficiently many roots of unity and K is a splitting field for Γ and all of its subgroups.In particular, χ(g) ∈ K for all χ ∈ Irr(Γ) and g ∈ Γ.Let CF(Γ) be the space of K-valued class functions on Γ.There is a standard inner product , Γ on CF(Γ) given by f, f ′ , where x → x denotes the automorphism of K given by complex conjugation.We denote by Z Irr(Γ) ⊆ CF(Γ) the subset consisting of all integral linear combinations of Irr(Γ).Finally, if C ⊆ Γ is any (non-empty) subset that is a union of conjugacy classes of Γ, then we denote by ε C ∈ CF(Γ) the (normalised) indicator function of C, that is, we have Note that, if C is a single conjugacy class of Γ and g ∈ C, then f (g) = f, ε C Γ for any f ∈ CF(Γ).Thus, the problem of computing the values of ρ ∈ Irr(G F ) is equivalent to working out the inner products of ρ with the indicator functions of the various conjugacy classes of Γ.

Lusztig induction and uniform functions
Let G, F be as in the introduction.Given an F -stable maximal torus T of G and θ ∈ Irr(T F ), we have a generalised character R G T ,θ ∈ Z Irr(G F ) as introduced by Deligne and Lusztig [1] (see also [6, §2.2]).We shall also need the following generalisation of R G T ,θ .
2.1.An F -stable closed subgroup L ⊆ G is called a "regular subgroup" if L is a Levi complement in some (not necessarily F -stable) parabolic subgroup P ⊆ G. Given such a pair (L, P ) we obtain an operator Denoting by G F uni and L F uni the sets of unipotent elements of G F and L F , respectively, there is a corresponding two-variable Green function 2.2.Let L ⊆ P be as above and ψ ∈ Irr(L F ).There is a character formula which expresses the values of R G L⊆P (ψ) in terms of the values of ψ and the two-variable Green functions for G and for groups of the form [13,Prop. 6.2] for the precise formulation.For later reference we only state here the following special case: We also state the following useful formula.Let g ∈ G F and consider the Jordan decomposition of g, that is, we write This appeared in K. D. Schewe's dissertation (Bonner Mathematische Schriften, vol.165,1985); see the remark following [6,Cor. 3.3.13]for a proof.

2.3.
Let us denote by X(G, F ) the set of all pairs (T , θ) where T ⊆ G is an F -stable maximal torus and θ ∈ Irr(T F ).Following [10, p. 16], a class function f ∈ CF(G F ) is called "uniform" if f can be written as a K-linear combination of the generalised characters R G T ,θ for various pairs (T , θ) ∈ X(G, F ).If f is uniform, then we have (see [2,Prop. 10.2.4]): Hence, using (2.3), we obtain the formula: This shows that the value ρ(g) is determined by the multiplicities R G T ,θ , ρ G F and the values R G T ,θ (g), where (T , θ) runs over all pairs in X(G, F ).
, ρ G F = 0 for some F -stable maximal torus T ⊆ G.We denote by Uch(G F ) the set of unipotent characters of G F .As shown in Lusztig's book [11], these characters play a special role in the character theory of G F ; many questions about arbitrary characters of G F can be reduced to unipotent characters.

The unipotent characters for F 4 in characteristic 2
We assume from now on that p = 2 and G is simple of type F 4 .Let F : G → G be a Frobenius map such that G F = F 4 (q) where q is a power of 2. Let Y(G, s) be the set of all pairs (T , s) where T ⊆ G is an F -stable maximal torus and s ∈ T F .There are natural actions of G F on X(G, F ) and on Y(G, F ); see [6, 2.3.20 and 2.5.12].Since G ∼ = G * is "self-dual" (in the sense of [6, Def.1.5.17]),there is a bijective correspondence If (T , θ) ↔ (T , s) correspond in this way, we write R G T ,s := R G T ,θ (see [6, Def.2.5.17]).In order to compute the characters of G F , we shall assume that the following information is known and available in the form of tables: Remark 3.1.The conjugacy classes of G F are determined by Shinoda [19].The tables in [19] provide the required classifications and parametrisations in (A1).Since the center of 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 G T ,s (g).This involves the Green functions for G and for groups of the form H s = C G (s) where s ∈ G F is semisimple; note that, for our G, the centraliser of any semisimple element is connected.By inspection of [19,Table III], we see that H s is either a maximal torus, or a regular subgroup (with a root system of type or H s has a root system of type A 2 ×A 2 .The Green functions for 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 G T ,θ (su) are discussed in [5, §3] and [9, §2] (for example, one has to deal with a sum over all x ∈ G F such that x −1 sx ∈ T ); in [9, §6], this is explained in detail for the groups G F = 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 [22] (type A 1 , A 2 ).
Representatives for the G F -conjugacy classes of semisimple elements are denoted by h 0 , h 1 , . . ., h 76 in [19,Table II], where h 0 = 1; note that some of the h i only occur according to whether 3 | q − 1 or 3 | 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 ρ ∈ Uch(G F ) on elements of the form h i u where u ∈ C G (h i ) F is unipotent.
In our group G, there are 37 unipotent characters, where we use the notation in Lusztig's book [11, p. 371/372]).

If s
uni have been explicitly determined by Marcelo-Shinoda; see [17, Table 6.A].This relies on the Green functions of 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 ι -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.

Let
s be unipotent and C be the G-conjugacy class of su.(a) Assume first that u is not regular unipotent.By inspection of [19,Table IV], we see that C G (su) is connected.So we can apply Example 2.5, together with (A2), (A3), to determine ρ(su) even for all ρ ∈ Irr(G F ).
4. Non-unipotent characters for F 4 in characteristic 2 We keep the notation of the previous section, where G is simple of type F 4 in characteristic 2. We now explain how to determine the values of the non-unipotent characters of G F .First we recall some facts from Lusztig's classification of Irr(G F ). Let s ∈ G F be semisimple.Then we define E (G F , s) to be the set of all ρ ∈ Irr(G F ) such that R G T ,s , ρ = 0 for some F -stable maximal torus T ⊆ G with s ∈ T .It is known that every ρ ∈ Irr(G F ) belongs to E (G F , s) for some s; furthermore, E (G F , s) only depends on the (For all this see, for example, [6, §2.6]; also recall that G ∼ = G * .)Finally, by the "Main Theorem 4.23" of [11], there is a bijection E (G F , s) ↔ Uch(H F s ), where H s = C G (s); this is called the "Jordan decomposition" of characters.We now proceed in 4 steps, where we determine the following information: Step 1: The values of all the two-variable Green functions Q G L⊆P .
Step 2: The values ρ(u) for all ρ ∈ Irr(G F ) and u ∈ G F uni .
Step 3: The decomposition of R G L⊆P (ψ) for any ψ ∈ Irr(L F ).
Step 4: The values ρ(g) for any ρ ∈ Irr(G F ) and any g ∈ G F .

We show how
Step 1 can be resolved.Assume that L G and let Uch(L F ) = {ψ 1 , . . ., ψ n }.The information in (A4) (see Section 3) shows, in particular, that n is also the number of conjugacy classes of unipotent elements of L F .Let v 1 , . . ., v n be representatives of these classes.Then, again using (A4), we can also check that the matrix (ψ i (v j )) 1 i,j n is invertible.(For an example, see Table 1.)Let u 1 , . . ., u N be representatives of the conjugacy classes of unipotent elements of G F ; we have N = 35 by [19,Theorem 2.1].Then we write the character formula (2.2)(a) as a system of equations: where c j := [L F : C L (v j ) F ] for all j.On the other hand, as explained in (3.4), we can determine the multiplicities m(ψ i , ρ) := R G L⊆P (ψ i ), ρ G F for any ρ ∈ Uch(G F ). Hence,

Using
Step 2, we can check that the matrix (ρ i (u k )) 1 i r,1 k N has rank r, where r N.

( A1 )
Parametrisations of Y(G, F ) and of all the conjugacy classes of G F .(A2) The multiplicities R G T ,s , ρ for all ρ ∈ Irr(G F ) and (T , s) ∈ Y(G, F ). (A3) The values R G T ,s (g) for all g ∈ G F and all (T , s) ∈ Y(G, F ). (A4) For every regular L G, the values ψ(u) for ψ ∈ Irr(L F ), u ∈ L F uni .

4. 4 .
We show how Step 4 can be resolved.Let ρ ∈ Irr(G F ) and g ∈ G F be arbitrary.Let i ∈ {0, 1, . . ., 76} be such that ρ ∈ E (G F , h i ).If i = 0, then h 0 = 1, ρ is unipotent and we know the values of ρ by Section 3. Next, let i ∈ {3, 15}.Then, as already mentioned in (4.2), ρ is uniform and so the values of ρ are computable via (A2), (A3).Finally, let i ∈ {0, 3, 15}.Write g = su = us where s ∈ G F is semisimple and u ∈ G F is unipotent.If s = 1, then the values ρ(u) for u ∈ G F uni are known by Step 2. Now let s = 1.If C G (s) has type A 2 × A 2 , then ρ(su) is already known by(3.3).Otherwise, we are in the situation of(3.4)where L := C G (s) Then the indicator function ε C F of the set C F is a uniform function.(Note that, in general, C F is a union of conjugacy classes of G F .) then the indicator function ε C (as in (1.1)) is uniform; see [2, Cor.10.3.4].Theorem 2.4.Let C be an arbitrary F -stable conjugacy class of G. Example 2.5.Let g ∈ G F and assume that C G (g) is connected.Let C be the Gconjugacy class of g.Since C G (g) is connected, C := C F is a single conjugacy class of G F ; see [6, Example 1.4.10].Now ε C is uniform by Theorem 2.4.Let ρ ∈ Irr(G F ). Recall from (1.1) that ρ