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 \(F_4(q)\).
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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
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
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
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:
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
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])
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
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
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:
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:
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
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:
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
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:
Comparing with (\(*\)), we obtain equations
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 \(\rho (su)\).
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I thank Jonas Hetz and Gunter Malle for comments on an earlier version. This article is a contribution to SFB-TRR 195 by the DFG (Deutsche Forschungsgemeinschaft), Project-ID 286237555.
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Geck, M. The character table of the finite Chevalley group \(F_4(q)\) for q a power of 2. Arch. Math. 121, 669–679 (2023). https://doi.org/10.1007/s00013-023-01879-6
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DOI: https://doi.org/10.1007/s00013-023-01879-6