FOURIER TRANSFORMS AND FROBENIUS EIGENVALUES FOR FINITE COXETER GROUPS

MEINOLF GECK AND GUNTER MALLE

To Robert Steinberg on his 80th birthday

Abstract. Lusztig’s classification of the unipotent characters of a finite Chevalley or Steinberg group involves a certain non-abelian Fourier transformation. We construct analogous transformations for the Suzuki and Ree groups, based on a set of axioms derived from Lusztig’s theory of character sheaves. We also determine Fourier matrices for the “spetses” (in the sense of Broué, Michel and the second author) associated with twisted dihedral groups. This completes the determination of Fourier matrices for all “spetses” associated with finite Coxeter groups. We end by collecting common properties of these Fourier matrices and the eigenvalues of Frobenius of character sheaves and unipotent characters.

1. Introduction

Let G be a connected reductive algebraic group defined over the finite field with q elements, and let F: G → G be the corresponding Frobenius map. Let G^{F} be the finite group of fixed points and W be the Weyl group of G, with respect to some F -stable maximal torus of G which is contained in some F -stable Borel subgroup. Then for each w ∈ W, we have a corresponding Deligne–Lusztig generalized character R_{w}; see, for example, the exposition in [7, Chap. 7]. Following Deligne and Lusztig, we define

Uch(G^{F}) := {χ ∈ Irr(G^{F}) | ⟨R_{w}, χ⟩ ≠ 0 for some w ∈ W }

to be the set of unipotent characters of G^{F}. Here, ⟨ , ⟩ denotes the usual Hermitian product on the space of complex-valued class functions of G^{F}. Lusztig [19] classified the characters in Uch(G^{F}) and found explicit formulas for the multiplicities ⟨R_{w}, χ⟩. This involves a transformation of the unipotent characters by a certain non-abelian Fourier matrix. Subsequently, Lusztig gave a geometric meaning to that transformation in his theory of character sheaves [20].

Now, the definition of unipotent characters also makes sense for the Suzuki and Ree groups, which are defined using certain exceptional isogenies F: G → G. These groups do have features which definitely do not occur in the case where F is a Frobenius map. (For example, there are families of unipotent characters which contain exactly two elements; see Example 5.3.) In particular, it is not clear how to construct Fourier matrices for them. Using some axioms derived from the theory of character sheaves and the explicit knowledge of the character tables (see Suzuki [31], Ward [32] and Malle [24]), we construct here Fourier matrices for the Suzuki and Ree groups. (This is inspired from the constructions in the proof of [12, 7.5]. For type I_{2}(e) such matrices were already given combinatorially in [25, 6C].) We also determine the eigenvalues of Frobenius associated with those unipotent characters of G^{F} which are not covered by Lusztig’s results [16] and [19], in particular for the non-split even-dimensional orthogonal groups; see Theorem 4.11.

1991 Mathematics Subject Classification. According to the 2000 classification: Primary 20C15; Secondary 20G40.

More generally, in the theory of “spetses” developed by Broué, Michel and the second author in [5], one would like to associate a Fourier matrix to any pair (W, ϕ) consisting of a finite spetsial complex reflection group W and an automorphism ϕ stabilising the reflection representation of W. We remark that the idea that many properties of finite Coxeter groups should have extensions to the case of complex reflection groups has been already present in the early work of Steinberg; see [30]. In Section 6, we determine such Fourier matrices (and analogues of the Frobenius eigenvalues) for the twisted dihedral groups. In Section 7, we give an interpretation of these Fourier transforms in the spirit of Lusztig’s exotic Fourier transforms [23].

Combined with previous work by Lusztig, Broué–Malle and Malle, the results in this paper complete the determination of Fourier matrices for all pairs (W, ϕ) where W is a finite Coxeter group. (See (6.8) for precise references.) In Theorem 6.9, we also present a list of known properties of these Fourier matrices and the Frobenius eigenvalues. It is our hope that this might give a clue to a possible axiomatic setting and the construction of Fourier matrices for the general “spetses” associated with complex reflection groups.

2. Families of unipotent characters

Let G be a connected reductive algebraic group over \overline{𝔽}_{p} (where p is a prime number). Let F: G → G be an endomorphism such that some power of F is a Frobenius map corresponding to a rational structure of G over some finite subfield of \overline{𝔽}_{p}. We fix an F -stable Borel subgroup B ⊆ G and an F -stable maximal torus T ⊆ G contained in G. Let W = N_{G}(T)/T be the Weyl group of G with respect to T and S ⊆ W be the set of simple reflections determined by B.

The endomorphism F: G → G induces an automorphism of W which leaves S invariant and which we also denote by the symbol F. Throughout, we denote by δ ⩾ 1 the order of F: W → W and let q ∈ ℝ_{>0} be such that F^{δ} is the Frobenius map corresponding to an 𝔽_{q^{δ}}-rational structure on G.

We begin by recalling some results on zeta functions of Deligne–Lusztig varieties, following Lusztig [19, Chap. 3] and Digne–Michel [8].

2.1. Zeta functions and uniform almost characters. For any w ∈ W, the corresponding Deligne–Lusztig variety is defined by

X_{w} := {gB ∈ G/B | g^{−1}F (g) ∈ BwB} ⊆ G/B;

see [19, 2.1]. There is a natural action of G^{F} on X_{w} given by left multiplication. This action commutes with the natural action of F^{m} on X_{w}, where m ⩾ 1 is any multiple of δ. We set

N_{w}^{m}(g) := |{gB ∈ X_{w} | F^{m}(gB) = gB}| for all g ∈ G^{F} .

By Grothendieck’s trace formula, we can express N_{w}^{m}(g) in terms of the cohomology spaces H_{c}^{i}(X_{w}, \overline{ℚ}_{ℓ}) (where ℓ is a prime not equal to p). The natural actions of G^{F} and F^{m} on X_{w} induce a (G^{F} , F^{m})-bimodule structure on H_{c}^{i}(X_{w}, \overline{ℚ}_{ℓ}). Then we have

N_{w}^{m}(g) = ∑_{i⩾0}(−1)^{i}Trace(gF^{m}, H_{c}^{i}(X_{w}, \overline{ℚ}_{ℓ})) for any g ∈ G^{F};

see [8, I.4.1]. If we set m = 0 in the right hand side of the above formula, we obtain a class function which is denoted R_{w} and which is nothing but the Deligne–Lusztig generalized character R_{T_{w},1}, as defined in [7, Chap. 6], where T_{w} ⊆ G is an F -stable maximal torus obtained by twisting T with w. As mentioned in the introduction, the characters in

Uch(G^{F}) := {χ ∈ Irr(G^{F}) | ⟨R_{w}, χ⟩ ≠ 0 for some w ∈ W }

are called the unipotent characters of G^{F}. Now let Irr(W)^{F} be the set of irreducible characters of W which are fixed under the action of F. For each ρ ∈ Irr(W)^{F}, we can choose an extension ρ̃ to the semidirect product W ⋊ ⟨F ⟩ and define

Rρ̃ := \frac{1}{|W |} ∑_{w∈W} ρ̃(wF) R_{w}.

The characters Rρ̃ are called the uniform almost characters of G^{F}. Note that the choice of a different extension of ρ just leads to a non-zero scalar multiple of Rρ̃. We have

Uch(G^{F}) = {χ ∈ Irr(G^{F}) | ⟨Rρ̃, χ⟩ ≠ 0 for some ρ ∈ Irr(W)^{F} }.

This follows from [17, 3.19].

2.2. Uniform almost characters for the Suzuki and Ree groups. Assume that (G, F) is such that G^{F} is the Suzuki group ^{2}B_{2}(q^{2}), the Ree group ^{2}G_{2}(q^{2}) or the Ree group ^{2}F_{4}(q^{2}), of order

|^{2}B_{2}(q^{2})| = q^{4}(q^{2} − 1)(q^{4} + 1)  where q^{2} = 2^{2m+1},
|^{2}G_{2}(q^{2})| = q^{6}(q^{2} − 1)(q^{6} + 1)  where q^{2} = 3^{2m+1},
|^{2}F_{4}(q^{2})| = q^{24}(q^{2} − 1)(q^{6} + 1)(q^{8} − 1)(q^{12} + 1)  where q^{2} = 2^{2m+1}.

Table 1. Uniform almost characters for ^{2}B_{2}, ^{2}G_{2} and ^{2}F_{4}

\begin{aligned}^{2}B_{2} & (±1) & (ζ′′_{8}) & (ζ′_{8}) \\ & F & sF & stsF \\ ˜1_{1} & 1 & 1 & 1 \\ ˜1_{ε} & 1 & −1 & −1 \\ ˜2_{1} & 0 & \sqrt{2} & −\sqrt{2} \end{aligned}  \begin{matrix}R˜1_{1} = 1_{G}, \\ R˜1_{ε} = St_{G}, \\ R˜2_{1} = \frac{1}{\sqrt{2}}(W_{1} + W_{2}) \end{matrix}

\begin{aligned}^{2}G_{2} & (±1) & (ζ′_{12}) & (i) & (ζ′′_{12}) \\ & F & sF & stsF & ststsF \\ ˜1_{1} & 1 & 1 & 1 & 1 \\ ˜1_{ε} & 1 & −1 & −1 & −1 \\ ˜2_{1} & 0 & −\sqrt{3} & 0 & \sqrt{3} \\ ˜2_{2} & 0 & 1 & −2 & 1 \end{aligned}  \begin{matrix}R˜1_{1} = 1_{G},  R˜1_{ε} = St_{G}, \\ R˜2_{1} = \frac{1}{2\sqrt{3}}(ξ_{5} + ξ_{6} + ξ_{7}+ \\ +ξ_{8} + 2ξ_{9} + 2ξ_{10}), \\ R˜2_{2} = \frac{1}{2}(ξ_{5} − ξ_{6} + ξ_{7} − ξ_{8}). \end{matrix}

2F_{4} (±1) (ζ′′_{8}, ±1) (ζ′_{8}, ±1) (i, ±1) (ζ′′_{8}) (ζ′_{8}) (ζ′_{8}, ζ′′_{8}) (i) (ζ_{12}) (ζ′′_{24}) (ζ′_{24})
w_{1}F w_{6}F w_{2}F w_{3}F w_{11}F w_{10}F w_{7}F w_{9}F w_{8}F w_{5}F w_{4}F
˜1_{1}  1  1  1  1  1  1  1 1 1 1 1
˜9_{1}  1  1  1 −1  3  3  −1 −3 0 0 0
˜9_{4}  1  −1  −1  1  3  3  −1 −3 0 0 0
˜1_{4}  1  −1  −1 −1  1  1  1 1 1 1 1
˜4_{2}  0 −\sqrt{2}  \sqrt{2}  0 −2\sqrt{2} 2\sqrt{2}  0 0 0 −\sqrt{2} \sqrt{2}
˜4_{5}  0 −\sqrt{2}  \sqrt{2}  0 2\sqrt{2} −2\sqrt{2}  0 0 0 \sqrt{2} −\sqrt{2}
˜12_{1} 2  0  0  0 −2 −2  −2 2 −1 1 1
˜4_{1}  2  0  0  0  2  2  2 2 −1 −1 −1
˜6_{1}  0  0  0  0 −2 −2  2 −4 −1 1 1
˜6_{2} −2  0  0  0  4  4  0 2 −1 1 1
˜16_{1} 0  0  0  0 4\sqrt{2} −4\sqrt{2}  0 0 0 −\sqrt{2} \sqrt{2}
R˜1_{1} = χ_{1},  R˜9_{1} = χ_{2},  R˜9_{4} = χ_{3},  R˜1_{4} = χ_{4},
R˜4_{2} = \frac{1}{\sqrt{2}}(χ_{5} + χ_{6}),  R˜4_{5} = \frac{1}{\sqrt{2}}(χ_{7} + χ_{8}),
R_{˜12_{1}} = \frac{1}{12}(3χ_{9} + 3χ_{10} + 6χ_{11} + χ_{12} + χ_{13} + 2χ_{14} + 3χ_{15}

+3χ_{16} + 3χ_{17} + 3χ_{18} + 4χ_{19} + 4χ_{20} + 4χ_{21}),

R˜4_{1} = \frac{1}{4}(χ_{9} + χ_{10} + 2χ_{11} − χ_{12} − χ_{13} − 2χ_{14} − χ_{15} − χ_{16} − χ_{17} − χ_{18}),
R˜6_{1} = \frac{1}{3}(χ_{12} + χ_{13} − χ_{14} + χ_{19} + χ_{20} − 2χ_{21}),
R˜6_{2} = \frac{1}{6}(−3χ_{9} − 3χ_{10} − χ_{12} − χ_{13} − 2χ_{14} + 2χ_{19} + 2χ_{20} + 2χ_{21}),
R_{˜16_{1}} = \frac{1}{2\sqrt{2}}(χ_{9} − χ_{10} − χ_{12} + χ_{13} + χ_{15} + χ_{16} − χ_{17} − χ_{18}).

(Note that q > 0 is a real number, not an integer!) For the construction and basic properties of these groups, see Steinberg’s lecture notes [29, §11]. In each of these cases, F acts as an automorphism of order 2 on the Weyl group W and so there are precisely two extensions of an F -invariant irreducible character of W. (The two extensions differ only by a sign on the elements wF for w ∈ W.) In Table 1, we show the values of extensions of the F -invariant characters of W (B_{2}), W (G_{2}), W (F_{4}) and the corresponding uniform almost characters. The decomposition of the Deligne–Lusztig generalized characters R_{w} into irreducible characters is specified in the tables in [19, Appendix].

The classes of W (B_{2}) and W (G_{2}) are labelled as in [19, p. 373 and p. 376]; we also give representatives as products of the two simple reflections s, t of W (B_{2}) or W (G_{2}) (as in [14, Table IV]). Furthermore, 1_{1} denotes the trivial character, 1_{ε} denotes the sign character and 2_{1} is the character of the standard reflection representation. The unipotent characters of ^{2}B_{2}(q^{2}) and ^{2}G_{2}(q^{2}) are denoted as by Suzuki [31, p. 143] and Ward [32, p. 87], respectively. In both cases, St_{G} denotes the Steinberg character.

The classes and characters of W (F_{4}) are labelled as in [19, 14.2]; we also show the identification with the representatives w_{j} in [14, Table VI]. The unipotent characters of ^{2}F_{4}(q^{2}) are denoted as by Malle [24]; the labelling is such that χ_{i} is the i-th character in Carter’s list [7, p. 489] (but note that Φ_{6} should read Φ_{12} in that list).

2.3. Families. We have the following fundamental “Disjointness Theorem”: if the uniform almost characters Rρ̃ and R_{}_{ρ̃′} have an irreducible character in common, then ρ and ρ′ must lie in the same family of Irr(W), as defined in [19, Chap. 4]. In the case where F is a Frobenius map, this is proved in [19, 6.17]. The fact that this is also true for the Suzuki and Ree groups is implicitly contained in the remarks in [19, §14.2]. It can be seen explicitly in our Table 1.

Now let F ⊆ Irr(W) be an F -stable family. If F is a Frobenius map, then each element of F is invariant under F. (This is remarked in [19, 4.17].) If G^{F} is a Suzuki or a Ree group, this is no longer true in general but, at least, F contains some characters which are fixed by F. We define the corresponding family of Uch(G^{F}) by

Uch(G^{F} | F) := {χ ∈ Uch(G^{F}) | ⟨Rρ̃, χ⟩ ≠ 0 for some ρ ∈ F^{F} }.

Then we have a partition Uch(G^{F}) = ∐_{F} Uch(G^{F} | F), where F runs over the F -stable families of Irr(W). For the Suzuki and Ree groups, this is verified by inspection of Table 1.

Type Families
2B_{2} {1_{G}}, {St_{G}}, {W_{1}, W_{2}}
2G_{2} {1_{G}}, {St_{G}}, {ξ_{5}, ξ_{6}, ξ_{7}, ξ_{8}, ξ_{9}, ξ_{10}}
2F_{4} {χ_{1}}, {χ_{2}}, {χ_{3}}, {χ_{4}}, {χ_{5}, χ_{6}}, {χ_{7}, χ_{8}}, {χ_{i} | 9 ⩽ i ⩽ 21}

In the case where F is a Frobenius map, the families are described explicitly in the tables in the appendix of [19].

2.4. Twisting operators. Following Asai [2], we denote by t_{1}^{∗} the twisting operator on the space of class functions on G^{F}. Given a class function f on G^{F}, the value of t_{1}^{∗}(f) on x ∈ G^{F} is given by the formula

t_{1}^{∗}(f)(x) = f(αxα^{−1}) where α ∈ G is such that x = α^{−1}F (α).

Now it is known that any uniform almost character Rρ̃ as above is an eigenvector for t_{1}^{∗}; in fact, we have

t_{1}^{∗}(Rρ̃) = Rρ̃ for any ρ ∈ Irr(W)^{F} .

This follows from the fact that t_{1}^{∗} commutes with Deligne–Lusztig induction R_{T}^{G}; see [3, 2.4.1] (and also [8, IV.2.1] in good characteristic).

Now let A be a character sheaf on G, as defined by Lusztig [20]. Assume that A is F -stable and let χ_{A} be a corresponding characteristic function. (χ_{A} is a class function on G^{F}, well-defined up to non-zero scalar multiples.) Then Eftekhari [10] (see also Shoji [28, I.§3]) proved that χ_{A} is an eigenvector of t_{1}^{∗}, where the corresponding eigenvalue does not depend on the Frobenius endomorphism.

3. Cuspidal almost characters

In Section 2, we have recalled the construction of uniform almost characters. These will form a part of the desired Fourier transformation that we wish to construct. Now Lusztig’s theory of character sheaves provides another source for the explicit construction of almost characters. We will explain how this works for the so-called “cuspidal pairs” in G. Throughout this section, we assume that G is semisimple.

3.1. Characteristic functions. For any g ∈ G, we denote by A(g) the finite group of components of the centralizer of g in G. Now assume that g ∈ G^{F}. Then F induces a group automorphism of A(g) which we denote by the same symbol. For each y ∈ A(g), we denote by g_{y} an element in G^{F} obtained by twisting the given representative g with y. Then it is wellknown that the correspondence y ↦ g_{y} induces a bijection between the F -conjugacy classes of A(g) and the G^{F} -conjugacy classes contained in C, where C denotes the G-conjugacy class of g.

Now consider a pair A = (g, ψ) where g ∈ G^{F} and ψ ∈ Irr(A(g)) is F - invariant. Then we can extend ψ to a character ψ̃ of the semidirect product A(g) ⋊ ⟨F ⟩. We define a class function Ψ_{A} on G^{F} by the requirement that

Ψ_{A}(x) = \{ \begin{matrix}q^{(dim G−dim C)/2}ψ̃(yF) & if x = g_{y} for some y ∈ A(g), \\ 0 & otherwise. \end{matrix}

We call Ψ_{A} a characteristic function of A. Note that it depends on the choice of the extension ψ̃; thus, it is only well-defined up to non-zero scalar multiples. (The definition of Ψ_{A} is analogous to the construction in [20, 24.2.7].) We have a canonical choice for ψ̃ in the case where ψ is a linear character, that is, the map ψ: A(g) → ℂ is a group homomorphism. Indeed, the condition that ψ is F -invariant means that ψ(F^{−1}yF) = ψ(y) for any y ∈ A(g). Then the map ψ̃: A(g) ⋊ ⟨F ⟩ → ℂ, yF ↦ ψ(y), defines a group homomorphism extending ψ.

Furthermore, it is easily checked (see the proof of [28, I.3.8]) that Ψ_{A} is an eigenvector for the twisting operator t_{1}^{∗}, where the corresponding eigenvalue is determined as follows. Denote by ḡ the image of g in A(g). Then we have

t_{1}^{∗}(Ψ_{A}) = λ_{A} Ψ_{A} where λ_{A} = ψ(ḡ)/ψ(1).

Note that this characterisation of λ_{A} is independent of F!

Remark 3.2. Assume that A = (g, ψ) is F -stable and cuspidal, in the sense of [20, §7]. Consider a corresponding characteristic function Ψ_{A} and assume (for technical simplicity) that Ψ_{A} is a linear combination of unipotent characters of G^{F}. Furthermore, assume that F is a Frobenius map. Then, under some conditions on the characteristic, Lusztig has shown that there exists a root of unity ξ ∈ ℂ such that ξΨ_{A} coincides with some almost character as defined in [19, 4.24]; in particular, all constituents of Ψ_{A} lie in a fixed family of unipotent characters. Subsequently, the same problem has been solved by Shoji [28] under weaker assumptions on the characteristic.

We shall take these results as axioms, that is, we make the following definition.

Definition 3.3. Assume that A is an F -stable cuspidal pair. Let Ψ_{A} be a corresponding characteristic function and assume that Ψ_{A} is a linear combination of unipotent characters of G^{F}. Then we say that Ψ_{A} is a cuspidal almost character. The corresponding eigenvalue λ_{A} is defined as in (3.1).

We shall now determine the cuspidal almost characters for the Suzuki and Ree groups. This will be based on the known results concerning cuspidal pairs (see Lusztig [20], Shoji [28]) and the character tables of these groups (see Suzuki [31], Ward [32], Malle [24]). All actual computations were performed using the MAPLE part of the CHEVIE system [13]. We also specify the corresponding eigenvalues of the twisting operator.

In the tables below, we use the following notation. In the set-up of (3.1), let a_{1} = 1, a_{2}, …, a_{r} ∈ A(g) be representatives for the F -conjugacy classes of A(g) and denote g_{j} := g_{a_{j}} for 1 ⩽ j ⩽ r where g_{1} = g. Then we usually specify the values of Ψ_{A} by the following array.

\begin{matrix}& g = g_{1} & g_{2} & … & g_{r} \\ Ψ_{A} & q^{d/2}ψ̃(g) & q^{d/2}ψ̃(g_{2}) & … & q^{d/2}ψ̃(g_{r}) \end{matrix} (d = dim G − dim C).

(It is understood that Ψ_{A}(x) = 0 unless x is conjugate to some g_{j}.)

3.4. Cuspidal almost characters in ^{2}B_{2}. Let (G, F) be such that G^{F} is the Suzuki group ^{2}B_{2}(q^{2}) where q is an odd power of \sqrt{2}. According to Lusztig [20, 19.3(a”) and 22.2], we have precisely one cuspidal pair in G, and this is of the form A = (u, ψ) where u is a regular unipotent element and ψ is the unique non-trivial character of A(u) ≅ ℤ/2ℤ. This pair is automatically F -stable (F acts trivially on A(u)) and the values of the corresponding characteristic function are given as follows.

λ_{A} ρ ρ^{−1}
Ψ_{A} −1 q −q

Here, ρ denotes a certain regular unipotent element (of order 4), as in Suzuki’s character table [31]. We have that A(ρ) is generated by the image of ρ in A(ρ); this determines λ_{A}. Using Suzuki’s table, we see that

Ψ_{A} = \frac{i}{\sqrt{2}}(W_{2} − W_{1}) where i = \sqrt{−1} ∈ ℂ.

(Note that this identifies W_{1} and W_{2}.) In particular, we see that all constituents of Ψ_{A} lie in a fixed family of unipotent characters. Furthermore, Ψ_{A} together with the three uniform almost characters in Table 1 yield as many almost characters as we have unipotent characters; the transition matrix from unipotent characters to these almost characters has full rank.

3.5. Cuspidal almost characters in ^{2}G_{2}. Let (G, F) be such that G^{F} is the Ree group ^{2}G_{2}(q^{2}) where q is an odd power of \sqrt{3}. According to Shoji [28, I.7.2], we have 4 cuspidal pairs A_{1}, A_{2}, A_{3} and A_{4}. Here, A_{1} and A_{2} are supported on the class of regular unipotent elements of G (where A(u) ≅ ℤ/3ℤ), A_{3} is supported on the unique unipotent class of G of dimension 10 (where A(u) ≅ ℤ/2ℤ) and A_{4} is supported on the conjugacy class of su where s ∈ G is semisimple, C_{G}(s) is isogenous to SL_{2} × SL_{2} and u is regular unipotent in C_{G}(s) (where A(su) ≅ ℤ/2ℤ). Shoji’s description shows that all A_{k} are F -stable and that, in each case, F acts trivially on A(g). Using the notation of Ward’s character table [32], we see that the corresponding characteristic functions are given as follows (where we set Ψ_{k} = Ψ_{A_{k}} and λ_{k} = λ_{A_{k}} for all k and where θ = exp(2πi/3) ∈ ℂ denotes a primitive third root of unity).

\begin{aligned}& λ_{k} & Y & Y T & Y T^{−1} \\ Ψ_{1} & θ & q & qθ & qθ^{2} \\ Ψ_{2} & θ^{2} & q & qθ^{2} & qθ \end{aligned}  \begin{aligned}& λ_{k} & T & T^{−1} \\ Ψ_{3} & 1 & q^{2} & −q^{2} \end{aligned}  \begin{aligned}& λ_{k} & JT & JT^{−1} \\ Ψ_{4} & −1 & q & −q \end{aligned}

Here, Y, Y T, Y T^{−1} are regular unipotent elements, J is an involution with centralizer isogenous to SL_{2} × SL_{2} and T is a regular unipotent element in that centralizer. The eigenvalues λ_{k} are determined by Shoji [28, I.7.7]. Using Ward’s table, we find the following expressions of the above functions as linear combinations of the unipotent characters of G^{F}:

Ψ_{1} = \frac{1}{\sqrt{3}}(ξ_{7} + ξ_{8} − ξ_{10}),
Ψ_{2} = \frac{1}{\sqrt{3}}(ξ_{5} + ξ_{6} − ξ_{9}),
Ψ_{3} = \frac{i}{2\sqrt{3}}(−ξ_{5} − ξ_{6} + ξ_{7} + ξ_{8} − 2ξ_{9} + 2ξ_{10}),
Ψ_{4} = \frac{i}{2}(ξ_{5} − ξ_{6} − ξ_{7} + ξ_{8}).

In particular, we see that all constituents of these almost characters lie in a fixed family of unipotent characters. Furthermore, Ψ_{1}, Ψ_{2}, Ψ_{3}, Ψ_{4} together with the four uniform almost characters in Table 1 yield as many almost characters as we have unipotent characters; the transition matrix from unipotent characters to these almost characters has full rank.

3.6. Cuspidal almost characters in ^{2}F_{4}. Let (G, F) be such that G^{F} is the Ree group ^{2}F_{4}(q^{2}) where q is an odd power of \sqrt{2}. According to Shoji [28, I.7.2], we have 7 cuspidal pairs in G, denoted by A_{1}, …, A_{7}. Each of these pairs is of the form (g, ψ) where g ∈ G and ψ is a linear character of A(g). Shoji’s description shows that A_{1}, …, A_{7} are all F -stable. Let Ψ_{k} be the characteristic function of A_{k} with respect to the canonical extension of ψ described in (3.1). The following table shows the values of the corresponding characteristic functions. The eigenvalues λ_{k} are determined by Shoji [28, I.7.7].

\begin{aligned}& λ_{k} & u_{15} & u_{16} & u_{17} & u_{18} \\ Ψ_{1} & i & q^{2} & iq^{2} & −q^{2} & −iq^{2} \\ Ψ_{2} & −i & q^{2} & −iq^{2} & −q^{2} & iq^{2} \end{aligned}  \begin{aligned}& λ_{k} & t_{4}u_{7} & t_{4}u_{8} & t_{4}u_{9} \\ Ψ_{6} & θ & q^{2} & q^{2}θ & q^{2}θ^{2} \\ Ψ_{7} & θ^{2} & q^{2} & q^{2}θ^{2} & q^{2}θ \end{aligned}

\begin{aligned}& λ_{k} & u_{13} & u_{14} \\ Ψ_{3} & −1 & q^{3} & −q^{3} \end{aligned}  \begin{aligned}& λ_{k} & u_{10} & u_{11} & u_{12} \\ Ψ_{4} & 1 & q^{4} & −q^{4} & −q^{4} \end{aligned}  \begin{aligned}& λ_{k} & u_{7} & u_{8} & u_{9} \\ Ψ_{5} & 1 & q^{6} & −q^{6} & q^{6} \end{aligned}

Here, we use Shinoda’s notation [27] for unipotent elements in G^{F}. For example, by [28, I.7.2], A_{1} and A_{2} are cuspidal pairs supported on the class of regular unipotent elements. In Shinoda’s notation, these are the elements conjugate to u_{15}, u_{16}, u_{17} and u_{18}. We choose u_{15} as our standard representative. Then A(u_{15}) ≅ ℤ/4ℤ is generated by the image of u_{15} in A(u_{15}). Denoting that image by ū_{15}, we have a unique linear character ψ of A(u_{15}) such that ψ(ū_{15}) = ±i. Then A_{1} := (u_{15}, ψ) and A_{2} := (u_{15}, ψ̄) are F -stable cuspidal pairs. Now, using the arguments in the proofs of [20, 9.4.1 and 15.2], it follows that Ψ_{1} and Ψ_{2} must be orthogonal to the characters R_{L}^{G}(ψ) where L ⊆ G is any proper split Levi subgroup and ψ ∈ Irr(L^{F}). We apply this to the Levi subgroup of type ^{2}B_{2}(q^{2}) and a cuspidal unipotent character of L^{F}, that is, W_{1} or W_{2} in Suzuki’s table [31]. We have

R_{L}^{G}(W_{1}) = χ_{5} + χ_{7}  and  R_{L}^{G}(W_{2}) = χ_{6} + χ_{8}.

and the values of R_{L}^{G}(W_{1} − W_{2}) on u_{15}, u_{16}, u_{17}, u_{18} are given as follows.

u_{15}  u_{16} u_{17}  u_{18}
R_{L}^{G}(W_{1} − W_{2}) i\sqrt{2}q −i\sqrt{2}q i\sqrt{2}q −i\sqrt{2}q

Hence the condition that Ψ_{1} is orthogonal to R_{L}^{G}(W_{1} − W_{2}) implies that Ψ_{1}(u_{17}) = −q^{2}. Then we can choose the notation such that Ψ_{1}(u_{16}) = iq^{2} and Ψ_{1}(u_{18}) = −iq^{2}.

The remaining characteristic functions are obtained similarly.

Using Malle’s table, we can decompose the above functions as linear combinations of the unipotent characters of G^{F}:

Ψ_{1} = \frac{1}{4}(1 + i)(χ_{9} − χ_{10} + χ_{12} − χ_{13} − χ_{15} + χ_{16} + χ_{17} − χ_{18}),
Ψ_{2} = \frac{1}{4}(1 − i)(χ_{9} − χ_{10} + χ_{12} − χ_{13} + χ_{15} − χ_{16} − χ_{17} + χ_{18}),
Ψ_{3} = \frac{1}{4} \sqrt{2}(χ_{9} − χ_{10} − χ_{12} + χ_{13} − χ_{15} − χ_{16} + χ_{17} + χ_{18}),
Ψ_{4} = \frac{1}{4}(χ_{9} + χ_{10} − 2χ_{11} − χ_{12} − χ_{13} − 2χ_{14} + χ_{15} + χ_{16} + χ_{17} + χ_{18}),
Ψ_{5} = \frac{1}{12}(−3χ_{9} − 3χ_{10} + 6χ_{11} − χ_{12} − χ_{13} − 2χ_{14} + 3χ_{15}
+3χ_{16} + 3χ_{17} + 3χ_{18} − 4χ_{19} − 4χ_{20} − 4χ_{21}),
Ψ_{6} = \frac{1}{3}(χ_{12} + χ_{13} − χ_{14} + χ_{19} − 2χ_{20} + χ_{21}),
Ψ_{7} = \frac{1}{3}(χ_{12} + χ_{13} − χ_{14} − 2χ_{19} + χ_{20} + χ_{21}).

In particular, we see that all constituents of these almost characters lie in a fixed family of unipotent characters.

4. Eigenvalues of Frobenius

We keep the setting of Section 2. Lusztig [17, 3.9] has shown that one can attach to each unipotent character of G^{F} a corresponding “eigenvalue of Frobenius”. These eigenvalues are determined explicitly in [16] for those unipotent characters which occur in R_{T,1} where T ⊂ G is a “Coxeter torus”. Combined with the reduction arguments in [19, 11.3], this covers all unipotent characters of the Suzuki and Ree groups except for two cuspidal unipotent characters for ^{2}G_{2}(q^{2}) and five for ^{2}F_{4}(q^{2}). The purpose of this section is to determine the eigenvalues of Frobenius for these missing cases. We also consider the case of the twisted groups of type ^{2}D_{n} which does not seem to have been settled yet.

4.1. Zeta functions and eigenvalues. We recall Lusztig’s definition of the “eigenvalues of Frobenius” attached to a unipotent character χ ∈ Uch(G^{F}). First, since χ occurs with non-zero multiplicity in some Deligne– Lusztig generalized character, it is clear that there exist some w ∈ W, i ⩾ 0 and μ ∈ \overline{ℚ}_{ℓ}^{×} such that χ occurs in the character of the generalized μ-eigenspace of F^{δ} on H_{c}^{i}(X_{w}, \overline{ℚ}_{ℓ}). Now [17, 3.9] shows that μ is uniquely determined by χ up to a factor which is an integral power of q^{δ}. Furthermore, by [8, III.2.3], there is a well-defined root of unity ω_{χ} ∈ \overline{ℚ}_{ℓ} and a well-defined element λ_{χ} ∈ {1, q^{δ/2}} such that

(1)  μ = ω_{χ} λ_{χ} q^{sδ} for some integer s ⩾ 0.

The roots of unity ω_{χ} associated with the eigenvalues of Frobenius enter in the computation of the zeta function of X_{w}, as defined in (2.1). Indeed, by [8, III.2.3], we have the following fundamental formula.

(2)  N_{w}^{m} = ∑_{ρ∈Irr(W)^{F}} (ρ̃_{u^{1/2}}(T_{wF}))(q^{m/2}) ∑_{χ∈Uch(G^{F})}⟨Rρ̃, χ⟩ ω_{χ}^{m/δ}χ.

We have to explain the term ρ̃_{u^{1/2}}(T_{wF}) occurring in the above formula. Let A = \overline{ℚ}_{ℓ}[u^{1/2}, u^{−1/2}] be the ring of Laurent polynomials in an indeterminate u^{1/2} and H̃ be the generic Iwahori–Hecke algebra of W̃ = W ⋊ ⟨F ⟩. Thus, H̃ has an A-basis {T_{w} | w ∈ W̃ } where T_{wF} = T_{w}T_{F} for any w ∈ W and T_{F^{i}} = T_{F}^{i} for any i ⩾ 0; see [11, §1]. Furthermore, the multiplication is determined by T_{s}^{2} = uT_{1} +(u−1)T_{s} for s ∈ S and T_{w} = T_{s_{1}} ⋯ T_{s_{k}} whenever w = s_{1} ⋯ s_{k} (s_{i} ∈ S) is a reduced expression for w ∈ W. Let K be the field of fractions of A and denote H̃_{K} = K ⊗_{A} H. Then, by [11, 2.7 and 4.1], the algebra H̃_{K} is split semisimple and we have a canonical bijection between the irreducible characters of H̃ and those of W̃. Now let ρ ∈ Irr(W)^{F} and consider an extension ρ̃ of ρ to W̃. (Note that the result does not depend on this choice.) Let ρ̃_{u^{1/2}} be the corresponding irreducible character of H̃_{K}. Then we have

ρ̃_{u^{1/2}}(T_{wF}) ∈ A and (ρ̃_{u^{1/2}}(T_{wF}))(1) = ρ̃(wF) for w ∈ W;

see [11, 4.1]. (The proof is an extension of the arguments in [19, 3.3].) Thus, (ρ̃_{u^{1/2}}(T_{wF}))(q^{m/2}) is the result of specializing at q^{m/2} a certain character value of the extended Iwahori–Hecke algebra associated with W̃.

We have the following results which will be useful for the explicit computation of λ_{χ} and ω_{χ}. The first result provides a reduction to cuspidal unipotent characters.

Lemma 4.2 (Lusztig). Let χ ∈ Uch(G^{F}) and assume that χ occurs with non-zero multiplicity in the Harish–Chandra induction R_{L}^{G}(ψ), where L ⊆ G is an F -stable Levi complement in some F -stable parabolic subgroup of G and ψ ∈ Uch(L^{F}) is cuspidal. If L is a torus (that is, χ lies in the principal series), then we have ω_{χ}λ_{χ} = 1. Otherwise, we have

λ_{χ} = λ_{ψ}  and  ω_{χ} = ω_{ψ}.

Proof. This is proved in [17, 3.33]. □

Lemma 4.3 (Digne–Michel). Let χ ∈ Uch(G^{F}) and let τ be any field automorphism of \overline{ℚ}_{ℓ}. Then we have

τ(ω_{χ}λ_{χ}) = ω_{χ^{τ}} λ_{χ^{τ}} ,

where χ^{τ} is the unipotent character obtained by algebraic conjugation from χ; we have χ^{τ} (g) = τ(χ(g)) for all g ∈ G^{F}. Consequently, we have

ω_{χ}λ_{χ} ∈ ℚ(χ) := ℚ(χ(g) | g ∈ G^{F}).

Proof. The first statement is proved in [8, III.3.4]. It is based on the fact that, for any w ∈ W, there exists a polynomial P_{w,χ} ∈ ℤ[u] such that

⟨N_{w}^{m}, χ⟩ = P_{w,χ}(q^{m}) ω_{χ}^{m/δ} λ_{χ}^{m/δ} for any m ⩾ 0 such that δ | m;

see also the proof of [8, III.3.2]. Now consider the second statement. We know that ℚ(χ) is contained in a cyclotomic field and, hence, ℚ(χ) is a finite Galois extension of ℚ. Let τ be a field automorphism of \overline{ℚ}_{ℓ} which leaves ℚ(χ) invariant. Then we have χ^{τ} = χ and so τ(ω_{χ}λ_{χ}) = ω_{χ^{τ}} λ_{χ^{τ}} = ω_{χ}λ_{χ}. Since this is true for all such τ, we conclude that ω_{χ}λ_{χ} ∈ ℚ(χ), as desired. □

The following result provides a criterion for checking if λ_{χ} = 1 or q^{δ/2}.

Lemma 4.4. Let χ ∈ Uch(G^{F}) and w ∈ W be such that ⟨R_{w}, χ⟩ ≠ 0. Then

∑_{ρ∈Irr(W)^{F}} ρ̃_{u^{1/2}}(T_{wF}) ⟨Rρ̃, χ⟩ ≠ 0.

Assume that the coefficient of u^{i/2} in the above polynomial is non-zero. Then χ occurs in the generalized μ-eigenspace of F^{δ} on H_{c}^{j}(X_{w}, \overline{ℚ}_{ℓ}) (for some j) where |μ| = q^{i/2}. Thus, we have

λ_{χ} = \{ \begin{matrix}1 & if i is even, \\ q^{δ/2} & if i is odd. \end{matrix}

Proof. This is contained in [8] or [19, Chap. 3]. We explain the basic ingredients. Since R_{w} is obtained by formally setting m = 0 in N_{w}^{m}, the multiplicity of χ in R_{w} is obtained by setting u^{1/2} equal to 1 in the polynomial

Q_{w,χ} := ∑_{ρ∈Irr(W)^{F}} ρ̃_{u^{1/2}}(T_{wF}) ⟨Rρ̃, χ⟩ ∈ \overline{ℚ}_{ℓ}[u^{1/2}, u^{−1/2}].

Hence, if that multiplicity is non-zero, the above polynomial is non-zero. Consequently, we have ⟨N_{w}^{m}, χ⟩ ≠ 0 for all but finitely many m ⩾ 1 which are multiples of δ. Now, the formula in (4.1)(2) shows that

⟨N_{w}^{m}, χ⟩ = Q_{w,χ}(q^{m/2})ω_{χ}^{m/δ} for any m ⩾ 1 such that δ | m.

On the other hand, let P_{w,χ} ∈ ℤ[u] be the polynomial defined in the proof of [8, III.3.2]. We have

⟨N_{w}^{m}, χ⟩ = P_{w,χ}(q^{m}) ω_{χ}^{m/δ} λ_{χ}^{m/δ} for any m ⩾ 1 such that δ | m.

Furthermore, the construction shows that, if the coefficient of u^{i} in P_{w,χ} is non-zero, then χ occurs in the generalized μ-eigenspace of F^{δ} on H_{c}^{j}(X_{w}, \overline{ℚ}_{ℓ}) (for some j) where μ = q^{iδ}λ_{χ}ω_{χ}. Now, comparing the above two expressions for ⟨N_{w}^{m}, χ⟩, we find that Q_{w,χ}(q^{m/2}) = P_{w,χ}(q^{m})λ_{χ}^{m/δ} for any m ⩾ 1 which is a multiple of δ. Hence we see that, if λ_{χ} = 1, then Q_{w,χ} is a polynomial in u and, if λ_{χ} = q^{δ/2}, then Q_{w,χ} has non-zero coefficients at u^{i/2} with i odd. □

Let B^{+} be the braid monoid associated with (W, S); see [15, Chap. 4]. This monoid has the property that there exists a map f: W → B^{+} such that B^{+} is generated by f(W) and satisfies the condition that f(ww′) = f(w)f(w′) whenever w, w′ ∈ W are such that l(ww′) = l(w) + l(w′). Moreover, B^{+} is “universal” with this property. The automorphism F on W extends to an automorphism of B^{+} (which we denote by the same symbol) such that f(F (w)) = F (f(w)) for all w ∈ W. To simplify notation, we shall usually write w = f(w) for w ∈ W.

Lemma 4.5 (Lusztig, Broué–Michel). Let w_{0} ∈ W be the longest element. Assume that w ∈ W and d ⩾ 1 are such that

δ | d and wF (w)F^{2}(w) ⋯ F^{d−1}(w) = w_{0}^{2} where d l(w) = 2l(w_{0}).

Then we have N_{w}^{sδ}(1) = 0 for all 1 ⩽ s < d/δ.

Proof. If w is a (twisted or untwisted) Coxeter element, this is proved by Lusztig [16, 3.3]. A similar result is also obtained for a particular element in a Weyl group of type E_{8} in [19, p. 322]. (In this case, w^{12} = w_{0} and 12l(w) = l(w_{0}).) A general statement was given by Broué–Michel in [6, 5.2] (for trivial action of F) and in [6, 6.13] (for any F). Since the details in the latter case are not explicitly worked out in loc. cit., we add some comments for the convenience of the reader.

Let gB ∈ X_{w}. This means that g^{−1}F (g) ∈ BwB. Now, since δ divides d, we have l(wF (w)F^{2}(w) ⋯ F^{δ−1}(w)) = δl(w) and so

BwB.BF (w)B ⋯ BF^{δ−1}(w)B ⊆ B \underbrace{wF (w) ⋯ F^{δ−1}(w)}_{=:w_{1}∈W} B

by the Bruhat decomposition. Hence we have

g^{−1}F^{δ}(g) = (g^{−1}F (g))F (g^{−1}F (g)) ⋯ F^{δ−1}(g^{−1}F (g)) ∈ Bw_{1}B

and so gB ∈ X′_{w_{1}}, where X′_{w_{1}} ⊆ G/B is the Deligne–Lusztig variety defined with respect to the endomorphism F′ := F^{δ}: G → G. Now note that F′ acts trivially on W. Furthermore, we have

w_{1}^{d/δ} = w_{0}^{2} and (d/δ)l(w_{1}) = 2l(w_{0}).

Now we can argue as follows. Assume that N_{w}^{sδ}(1) = |X_{w}^{F^{sδ}}| ≠ 0 for some s ⩾ 1. We must show that s ⩾ d/δ. To see this, let g ∈ G be such that gB ∈ X_{w} is fixed by F^{sδ}. We have just seen that then we also have gB ∈ X′_{w_{1}}; clearly, gB is fixed by (F′)^{s}. Thus, the fixed point set of X′_{w_{1}} under (F′)^{s} is non-empty. We are now reduced to the untwisted case, where [6, 5.2] yields that we must have s ⩾ d/δ, as required. (Note that the proof in loc. cit. does not require the general definition of Deligne–Lusztig varieties associated with elements in the braid group.) □

Example 4.6. Consider the Suzuki group G^{F} = ^{2}B_{2}(q^{2}). By Table 1, we have four unipotent characters 1_{G}, St_{G}, W_{1} and W_{2}. The trivial character and the Steinberg character lie in the principal series and so we have ω_{1_{G}}λ_{1_{G}} = ω_{St_{G}}λ_{St_{G}} = 1; see Lemma 4.2. Furthermore, by [16, 7.3 and 7.4], the two cuspidal unipotent characters W_{1} and W_{2} occur in R_{w_{cox}} where w_{cox} is a Coxeter element as in [16, 1.7]. Furthermore, we have

\{ω_{W_{1}}λ_{W_{1}}, ω_{W_{2}}λ_{W_{2}}\} = \{\frac{i−1}{\sqrt{2}}q, \frac{−i−1}{\sqrt{2}}q\}.

In order to determine the sign, it would be necessary to compute explicitly N_{w}^{m}(g) for some element g ∈ G^{F} such that W_{1} and W_{2} have different values on g.

Example 4.7. Consider the Ree group G^{F} = ^{2}G_{2}(q^{2}). By Table 1, we have eight unipotent characters 1_{G}, St_{G}, ξ_{j} (5 ⩽ j ⩽ 10). The trivial character and the Steinberg character lie in the principal series and so we have ω_{1_{G}}λ_{1_{G}} = ω_{St_{G}}λ_{St_{G}} = 1; see Lemma 4.2. Furthermore, by [16, 7.3 and 7.4], four cuspidal unipotent characters occur in R_{w_{cox}} where w_{cox} is a Coxeter element as in [16, 1.7]. Using the formula in [16, 6.1], we can compute their degrees. Comparing the information in the table [19, p. 376] with Ward’s table [32], we see that

R_{w_{cox}} = 1_{G} − St_{G} + ξ_{5} + ξ_{7} + ξ_{9} + ξ_{10}.

Furthermore, we have

\{ω_{ξ_{5}}λ_{ξ_{5}}, ω_{ξ_{7}}λ_{ξ_{7}}\} = \{±iq\}, \{ω_{ξ_{9}}λ_{ξ_{9}}, ω_{ξ_{10}}λ_{ξ_{10}}\} = \{\frac{i−\sqrt{3}}{2}q, \frac{−i−\sqrt{3}}{2}q\}.

It remains to determine ω_{ξ_{6}}λ_{ξ_{6}} and ω_{ξ_{8}}λ_{ξ_{8}}. We claim that

{ω_{ξ_{6}}λ_{ξ_{6}}, ω_{ξ_{8}}λ_{ξ_{8}}} = {±iq}.

To prove this, we consider the element w := sts ∈ W (G_{2}) (the element with eigenvalues (i) in Table 1). By [19, p. 376], we have

R_{w} = 1_{G} − St_{G} − ξ_{5} + ξ_{6} − ξ_{7} + ξ_{8}.

Furthermore, wF (w) = w_{0} and so we are in the setting of Lemma 4.5, where d = 4. We consider the identity (4.1)(2) for m = 2. The left hand side is zero. To evaluate the right hand side, we need the values ρ̃_{u^{1/2}}(T_{wF}) (which can be found in [14, 7.6]) and the multiplicities ⟨Rρ̃, χ⟩ (which are printed in Table 1). Hence we obtain:

0 = ((˜1_{1})_{u^{1/2}}(T_{wF}))(q) 1_{G}(1) + ((˜1_{ε})_{u^{1/2}}(T_{wF}))(q) St_{G}(1)

+ \frac{1}{2}((˜2_{2})_{u^{1/2}}(T_{wF}))(q) (ω_{ξ_{5}}ξ_{5}(1) − ω_{ξ_{6}}ξ_{6}(1) + ω_{ξ_{7}}ξ_{7}(1) − ω_{ξ_{8}}ξ_{8}(1))

= q^{6} − q^{6} − q^{3}(−ω_{ξ_{6}} − ω_{ξ_{8}})ξ_{6}(1) = q^{3}(ω_{ξ_{6}} + ω_{ξ_{8}})ξ_{6}(1).

This yields ω_{ξ_{6}} = −ω_{ξ_{8}}. Now, by inspection of Ward’s table, we find that ℚ(ξ_{6}) = ℚ(ξ_{8}) = ℚ(\sqrt{−3}). Using Lemma 4.3, we conclude that ω_{ξ_{6}}, λ_{ξ_{6}} ∈ K := ℚ(\sqrt[4]{3}, i). Let τ be the field automorphism of K which sends i to −i and fixes \sqrt[4]{3}. Then Ward’s table shows that ξ_{6}^{τ} = ξ_{8} and so τ(ω_{ξ_{6}})λ_{ξ_{6}} = τ(ω_{ξ_{6}}λ_{ξ_{6}}) = ω_{ξ_{8}}λ_{ξ_{6}}. This yields τ(ω_{ξ_{6}}) = ω_{ξ_{8}}. Since ω_{ξ_{6}} = −ω_{ξ_{8}}, we conclude that ω_{ξ_{6}} = −ω_{ξ_{8}} = ±i. Now the fact that ω_{ξ_{6}}λ_{ξ_{6}} lies in ℚ(\sqrt{−3}) implies that λ_{ξ_{6}} cannot be rational. Hence we must have λ_{ξ_{6}} = q, as claimed. The same argument also applies to λ_{ξ_{8}}.

Example 4.8. Consider the Ree group G^{F} = ^{2}F_{4}(q^{2}). By Table 1, we have 21 unipotent characters χ_{i} (1 ⩽ i ⩽ 21). To simplify notation, let’s write ω_{i}, λ_{i} for ω_{χ_{i}}, λ_{χ_{i}}. The seven characters χ_{1}, χ_{2}, χ_{3}, χ_{4}, χ_{9}, χ_{10}, χ_{11} lie in the principal series and so, by Lemma 4.2, we have

ω_{i}λ_{i} = 1 for i = 1, 2, 3, 4, 9, 10, 11.

The four characters χ_{5}, χ_{6}, χ_{7}, χ_{8} occur in R_{L}^{G}(W_{j}) where L ⊂ G is a split Levi subgroup of type ^{2}B_{2} and j = 1, 2. We have

R_{L}^{G}(W_{1}) = χ_{5} + χ_{7}  and  R_{L}^{G}(W_{2}) = χ_{6} + χ_{8}.

So, by Lemma 4.2, we conclude that

\begin{matrix}ω_{5}λ_{5} = ω_{7}λ_{7} \\ ω_{6}λ_{6} = ω_{8}λ_{8} \end{matrix}  and  \{ω_{5}λ_{5}, ω_{6}λ_{6}\} = \{\frac{i−1}{\sqrt{2}}q, \frac{−i−1}{\sqrt{2}}q\}.

Thus, it remains to consider the ten cuspidal unipotent characters χ_{j} (12 ⩽ j ⩽ 21). By [16, 7.3 and 7.4], five of them occur in R_{w_{cox}} where w_{cox} is a Coxeter element as in [16, 1.7]. Comparing the information in the table [19, p. 374] with Malle’s table [24], we see that

R_{w_{cox}} = χ_{1}+χ_{4}−χ_{5}−χ_{6}+χ_{7}+χ_{8}−χ_{9}+χ_{12}+χ_{17}+χ_{18}+χ_{19}+χ_{20}.

Furthermore, we have

ω_{12}λ_{12} = −1, {ω_{17}λ_{17}, ω_{18}λ_{18}} = {±i},
{ω_{19}λ_{19}, ω_{20}λ_{20}} = {−θ, −θ^{2}}.

Here, θ denotes a primitive third root of unity. It remains to consider the five cuspidal unipotent characters χ_{13}, χ_{14}, χ_{15}, χ_{16}, χ_{21}. By inspection of Malle’s table, we find the character fields:

ℚ(χ_{15}) = ℚ(χ_{16}) = ℚ(i), ℚ(χ_{13}) = ℚ(χ_{14}) = ℚ(χ_{21}) = ℚ.

Using Lemma 4.3, this already implies that ω_{j}λ_{j} = ±1 for j = 13, 14, 21. We claim that

{ω_{15}λ_{15}, ω_{16}λ_{16}} = {±i} and ω_{j}λ_{j} = −1 for j = 13, 14, 21.

By [14, Table II], the assumptions of Lemma 4.5 are satisfied for the elements w_{5}, w_{8}, w_{9}, w_{11} in W (F_{4}). (The elements have F -order 24, 12, 4, 8, respectively). The element w_{5} is a Coxeter element, so it will not yield any new information. Using [14, Table VI], we obtain the following relations. First, we have

0 = N_{w_{8}}^{2}(1) = \frac{1}{6}(q^{24} − 3q^{20} + 4q^{16} − 3q^{12} + q^{8})(ω_{14} + 1),

which yields ω_{14} = −1. Next, we compute

0 = N_{w_{11}}^{4}(1) = f(q)(ω_{15}^{2} + ω_{16}^{2} + 2),

where f(q) is a non-zero polynomial expression in q. Since ω_{15} and ω_{16} are roots of unity, this implies that ω_{15} = ±i and ω_{16} = ±i. Next we compute that

0 = N_{w_{11}}^{2}(1) = f(q)(ω_{15} + ω_{16}) + h(q)(ω_{21} + 1),

where f, h are polynomials in q such that f(q) ≠ ah(q) for any non-zero constant a. It easily follows that ω_{21} = −1 and ω_{15} = −ω_{16}. Then we also see that {ω_{15}, ω_{16}} = {±i}. Finally, we compute

0 = N_{w_{9}}^{2}(1) = k(q)(ω_{13} + 1) where k(q) ≠ 0.

It follows that ω_{13} = −1, as claimed. These results and Lemma 4.3 also show that λ_{j} = 1 for j = 13, 14, 15, 16, 21.

Remark 4.9. The eigenvalues of all unipotent characters of a group G^{F} have been determined by Lusztig [19, Chap. 11] in the case where F acts trivially on W. The results in [16] cover some cases where F does not act trivially on W, but not all. For example, if G^{F} = ^{3}D_{4}(q) or G^{F} = ^{2}E_{6}(q), then there are cuspidal unipotent characters denoted by ^{3}D_{4}[1] and ^{2}E_{6}[1], respectively, which are not covered by these results; see [17, 3.29]. Using the above methods, one can show that

ω_{χ}λ_{χ} = 1 for χ = ^{3}D_{4}[1] or χ = ^{2}E_{6}[1].

In these cases, let us take the element w in [14, Table I or III] such that wF (w) ⋯ F^{5}(w) = w_{0}^{2}. Then we have ⟨R_{w}, χ⟩ ≠ 0. We first show that λ_{χ} = 1. If we are in type ^{3}D_{4}(q), then the character values of the corresponding Iwahori–Hecke algebre lie in ℤ[u, u^{−1}] by [8, II.3.3]. In type ^{2}E_{6}(q), the condition that ⟨Rρ̃, χ⟩ ≠ 0 implies that ρ cannot be any of the characters of degree 64 (see [19, 4.11]) and so the values of ρ̃_{u^{1/2}}(T_{wF}) lie again in ℤ[u, u^{−1}] by [8, II.3.3]. Hence Lemma 4.4 shows that λ_{χ} = 1. To see that ω_{χ} = 1, we apply Lemma 4.5 with s = 1. The character values of the corresponding Iwahori–Hecke algebra are determined by [8, II.3.3]; the decomposition of the uniform almost characters into unipotent characters is printed in [18, §1]. Performing a computation as in the examples above, we obtain ω_{χ} = 1, as required.

4.10. The non-split even-dimensional orthogonal groups. It seems that the Frobenius eigenvalues for groups of type ^{2}D_{n} have not been determined in the literature. Note that, by Lemma 4.3 and since all unipotent characters of ^{2}D_{n}(q) are rational, we necessarily have ω_{χ}λ_{χ} ∈ {±1} for all unipotent characters χ.

Theorem 4.11. All unipotent characters χ of ^{2}D_{n}(q) have Frobenius eigenvalue ω_{χ}λ_{χ} = 1.

Proof. By Lemma 4.2, it is enough to prove this for a cuspidal unipotent character. Now G^{F} = ^{2}D_{n}(q) has a cuspidal unipotent character if and only if n is an odd square; see [17, 3.22]. Thus, we can assume that n = (2k + 1)^{2} is an odd square. In this case, the Frobenius map acts by conjugation with the longest element w_{0} on the Weyl group W = W (D_{n}). Furthermore, the unique cuspidal unipotent character is parametrized by the symbol

\binom{0 … 4k + 1}{−} .

Clearly, removing 1-cohooks from this symbol we arrive at the 1-cocore

\binom{0 1}{−} ,

parametrizing the trivial character of the torus ^{2}D_{1}(q). Hence the cuspidal unipotent character lies in the principal Φ_{2}-Harish-Chandra series of G^{F}, so it occurs with non-zero multiplicity in R_{w_{0}}. Thus, we will obtain a relation involving ω_{χ} by evaluating N_{w_{0}}^{2}(1) using the formula in (4.1). So let us consider the terms involved in that formula. Let ρ̃ be the extension to W ⋊ ⟨F ⟩ of an F -stable irreducible character ρ ∈ Irr(W). By [19, 5.12.1], we have

ρ̃_{u^{1/2}}(T_{w_{0}F}) = u^{N−(a_{ρ}+A_{ρ})/2}ρ̃(w_{0}F),

where N = l(w_{0}) is the number of positive roots of W and where A_{ρ} and a_{ρ} denote the degree and the order of zero at u = 0, respectively, of the generic degree of ρ. Inserting this into (4.1)(2) we find

N_{w_{0}}^{2} = ∑_{ρ∈Irr(W)^{F}} q^{2N−a_{ρ}−A_{ρ}} ρ̃(w_{0}F) ∑_{χ∈Uch(G^{F})}⟨Rρ̃, χ⟩ω_{χ}χ

= ∑_{χ∈Uch(G^{F})} ω_{χ} ( ∑_{ρ∈Irr(W)^{F}} q^{2N−a_{ρ}−A_{ρ}}⟨Rρ̃, χ⟩ρ̃(w_{0}F)) χ.

By [19, 5.27], the invariants a_{ρ} and A_{ρ} are constant on families. So we can write a_{χ} = a_{ρ} and A_{χ} = A_{ρ} if χ ∈ Uch(G^{F} | F) and ρ ∈ F. Thus,

N_{w_{0}}^{2} = ∑_{χ∈Uch(G^{F})} ω_{χ}q^{2N−a_{χ}−A_{χ}} ( ∑_{ρ∈Irr(W)^{F}} ⟨Rρ̃, χ⟩ρ̃(w_{0}F)) χ

= ∑_{χ∈Uch(G^{F})} ω_{χ}q^{2N−a_{χ}−A_{χ}}⟨R_{w_{0}}, χ⟩ χ by the definition of Rρ̃.

Now let D_{G} denote the Alvis–Curtis–Kawanaka duality operation on the character ring of G^{F}. Combining [1, 3.6 and 3.7] with [19, 4.26.3 and 5.11.5], we see that

D_{G}(χ)(1) = ε_{χ}q^{N−a_{χ}−A_{χ}}χ(1) where ε_{χ} = ±1.

Now we know that a split Levi subgroup of ^{2}D_{n}(q) having a cuspidal unipotent character must have even 𝔽_{q}-rank. Hence [9, 8.15] shows that ε_{χ} = 1 for all χ ∈ Uch(G^{F}). Inserting the above degree formula into the formula for N_{w_{0}}^{2}, we obtain

N_{w_{0}}^{2}(1) = q^{N} ∑_{χ∈Uch(G^{F})} ω_{χ}⟨R_{w_{0}}, χ⟩ D_{G}(χ)(1).

By [9, 8.11], duality preserves Harish-Chandra series, so by Lemma 4.2 we have ω_{χ} = ω_{D_{G}(χ)}. Moreover

⟨R_{w_{0}}, D_{G}(χ)⟩ = ⟨D_{G}(R_{w_{0}}), χ⟩ = (−1)^{l(w_{0})}⟨R_{w_{0}}, χ⟩

by [9, 12.8 and 12.10]. But l(w_{0}) = N = n(n−1) is even, hence we conclude

(∗)  N_{w_{0}}^{2}(1) = q^{N} ∑_{χ∈Uch(G^{F})} ω_{χ}⟨R_{w_{0}}, χ⟩ χ(1).

On the other hand, an argument entirely analogous to that in the proof of [16, 3.3(ii)] shows that

N_{w_{0}}^{2}(1) = \frac{|G^{F} |}{|T_{w_{0}}^{F}|} = q^{N}R_{w_{0}}(1) = q^{N} ∑_{χ∈Uch(G^{F})}⟨R_{w_{0}}, χ⟩ χ(1).

Comparing this with (∗) and using induction shows that ω_{χ} = 1 for all χ. □

This result is in accordance with the general formula of Broué and Michel [6, 6.16], which was shown under certain (as yet unproven) hypotheses.

5. Fourier matrices

We have seen two sources of almost characters: the uniform ones (see (2.1)) and the cuspidal ones (see Definition 3.3). By pursuing the analogy between almost characters and character sheaves, one could in fact set up a kind of Harish–Chandra philosophy, based on the results in [20, §10.4]. (This would also yield almost characters which are intermediate between the uniform and the cuspidal ones.) However, we shall not need this here in our treatment of the Suzuki and Ree groups.

5.1. Axioms for Fourier matrices. More precisely, let F ⊆ Irr(W) be an F -stable family and Uch(G^{F} | F) the corresponding family of unipotent characters of G^{F}; see (2.3). Let us write Uch(G^{F} | F) = {χ_{λ} | λ ∈ Λ} for some finite indexing set Λ = Λ_{F}. We wish to describe a certain square matrix

M(F) = (a_{λμ})_{λ,μ∈Λ} where a_{λμ} ∈ ℝ.

For each λ ∈ Λ, we set

Ψ_{λ} := ∑_{μ∈Λ} a_{λμ} χ_{μ}.

The matrix M(F) should satisfy the following conditions.

(U) There exists a subset Λ_{U} ⊆ Λ and a bijection Λ_{U} ↔ F^{F}, λ ↔ ρ_{λ}, such that the class functions Ψ_{λ} for λ ∈ Λ_{U} are precisely the uniform almost characters of G^{F} which have their constituents in Uch(G^{F} | F).

(C) There exists a subset Λ_{C} ⊆ Λ such that the class functions Ψ_{λ} for λ ∈ Λ_{C} are precisely the characteristic functions of the F -stable cuspidal pairs in G which have their constituents in Uch(G^{F} | F).

(O) The rows of M(F) are orthogonal in the sense that, for any λ, μ ∈ Λ,

we have ⟨Ψ_{λ}, Ψ_{μ}⟩ = δ_{λμ}, where δ_{λμ} denotes the Kronecker delta. (T) For each λ ∈ Λ, the class function Ψ_{λ} is an eigenvector for the

twisting operator t_{1}^{∗}; see (2.4). A matrix M(F) satisfying these conditions will be called a Fourier matrix associated with F, F. Note that each row of M(F) may be multiplied by −1 without affecting the above properties. However, in general, the above conditions will not be sufficient to determine M(F) up to multiplication of the rows by ±1.

Remark 5.2. Assume that F is a Frobenius map corresponding to some 𝔽_{q}- rational structure on G. Then Fourier matrices for the various F -stable families of Irr(W) are constructed by Lusztig [19, Chap. 4]. Under some mild conditions on the characteristic, these matrices are also known to be Fourier matrices in the sense defined above. Indeed, conditions (U) and (O) are satisfied by the construction in [19, 4.21.14] and [19, 4.25], respectively. (Here, no assumption on q is required.) As already pointed out in Remark 3.2, condition (C) is satisfied by the results of Lusztig [21] and Shoji [28]. Finally, condition (T) holds by [28, I.3.3].

Example 5.3. Let F ⊆ Irr(W) be an F -stable family.

(a) Assume that Uch(G^{F} | F) is a singleton set. Then, up to sign, condition (O) forces that

M(F) = ( 1 ) .

Moreover, in this case, we have F = {ρ} and the unique unipotent character in Uch(G^{F} | F) is χ = ±Rρ̃. Thus, condition (T) is also satisfied by (2.4).

(b) Assume that Uch(G^{F} | F) contains exactly two characters, χ_{1} and χ_{2} say. By the description of families in [19, Chap. 4], this can only happen if G^{F} is the Suzuki group ^{2}B_{2}(q^{2}) or the Ree group ^{2}F_{4}(q^{2}). (If F is a Frobenius map, then the number of unipotent characters in a family is 2^{2e} (e ⩾ 0), 8, 21 or 39.) Now, Table 1 shows that there exists some F -invariant ρ ∈ F such that

Rρ̃ = \frac{1}{\sqrt{2}}(χ_{1} + χ_{2}).

By condition (U), this will give one row of M(F). The corresponding eigenvalue of t_{1}^{∗} is 1 by (2.4). Furthermore, the condition (O) implies that, up to multiplication of the rows by −1, we must have

M(F) = \frac{1}{\sqrt{2}} ( \begin{matrix}1 & 1 \\ 1 & −1 \end{matrix} ) .

Moreover, the effect of t_{1}^{∗} can be computed explicitly, using [8, IV.1.1] and the knowledge of the character tables from [31] and [24]. This yields that χ_{1} − χ_{2} is an eigenvector of t_{1}^{∗} with eigenvalue −1.

As far as the remaining families in the Suzuki and Ree groups are concerned, we have the following result.

Theorem 5.4. Let G^{F} be the Suzuki group ^{2}B_{2}(q^{2}), the Ree group ^{2}G_{2}(q^{2}) or the Ree group ^{2}F_{4}(q^{2}). Let F ⊆ Irr(W) be an F -stable family. Then, by (2.3), the number of unipotent characters in Uch(G^{F} | F) is 1, 2, 6 or 13.

(a) If Uch(G^{F} | F) contains 1 or 2 characters, the corresponding Fourier matrix is given by Example 5.3.

(b) In type ^{2}G_{2}(q^{2}) there is a six-element family of unipotent characters. Up to multiplication of the rows by ±1, the Fourier matrix M(F) associated with that family is given by

\frac{1}{2\sqrt{3}} ( \begin{matrix}1 & 1 & 1 & 1 & 2 & 2 \\ \sqrt{3} & −\sqrt{3} & \sqrt{3} & −\sqrt{3} & 0 & 0 \\ 0 & 0 & 2 & 2 & 0 & −2 \\ 2 & 2 & 0 & 0 & −2 & 0 \\ 1 & 1 & −1 & −1 & 2 & −2 \\ \sqrt{3} & −\sqrt{3} & −\sqrt{3} & \sqrt{3} & 0 & 0 \end{matrix} )

Here, the rows are labelled by R˜2_{1}, R˜2_{2}, Ψ_{1}, Ψ_{2}, −Ψ_{3}, Ψ_{4} and the columns are labelled by ξ_{5}, ξ_{6}, ξ_{7}, ξ_{8}, ξ_{9}, ξ_{10} . (Notation of Table 1 and (3.5).) The corresponding eigenvalues of t_{1}^{∗} are 1, 1, θ, θ^{2}, 1, −1, respectively.

(c) In type ^{2}F_{4}(q^{2}) there is a 13-element family of unipotent characters. Up to multiplication of the rows by ±1, the Fourier matrix M(F) associated with that family is given by the matrix in Table 2.

Proof. It only remains to consider the family with 6 unipotent characters in type ^{2}G_{2} and the family with 13 unipotent characters in type ^{2}F_{4}. Now, in type ^{2}G_{2}, we have two uniform almost characters and four cuspidal almost characters; see (3.5). It is readily checked that condition (O) is satisfied.

Table 2. The Fourier matrix in type ^{2}F_{4}

\frac{1}{12} ( \begin{matrix}3 & 3 & 6 & 1 & 1 & 2 & 3 & 3 & 3 & 3 & 4 & 4 & 4 \\ 3 & 3 & 6 & −3 & −3 & −6 & −3 & −3 & −3 & −3 & 0 & 0 & 0 \\ 0 & 0 & 0 & 4 & 4 & −4 & 0 & 0 & 0 & 0 & 4 & 4 & −8 \\ −6 & −6 & 0 & −2 & −2 & −4 & 0 & 0 & 0 & 0 & 4 & 4 & 4 \\ 3\sqrt{2} & −3\sqrt{2} & 0 & −3\sqrt{2} & 3\sqrt{2} & 0 & 3\sqrt{2} & 3\sqrt{2} & −3\sqrt{2} & −3\sqrt{2} & 0 & 0 & 0 \\ 3\sqrt{2} & −3\sqrt{2} & 0 & 3\sqrt{2} & −3\sqrt{2} & 0 & −3\sqrt{2} & 3\sqrt{2} & 3\sqrt{2} & −3\sqrt{2} & 0 & 0 & 0 \\ 3\sqrt{2} & −3\sqrt{2} & 0 & 3\sqrt{2} & −3\sqrt{2} & 0 & 3\sqrt{2} & −3\sqrt{2} & −3\sqrt{2} & 3\sqrt{2} & 0 & 0 & 0 \\ 3\sqrt{2} & −3\sqrt{2} & 0 & −3\sqrt{2} & 3\sqrt{2} & 0 & −3\sqrt{2} & −3\sqrt{2} & 3\sqrt{2} & 3\sqrt{2} & 0 & 0 & 0 \\ 3 & 3 & −6 & −3 & −3 & −6 & 3 & 3 & 3 & 3 & 0 & 0 & 0 \\ −3 & −3 & 6 & −1 & −1 & −2 & 3 & 3 & 3 & 3 & −4 & −4 & −4 \\ 0 & 0 & 0 & 4 & 4 & −4 & 0 & 0 & 0 & 0 & 4 & −8 & 4 \\ 0 & 0 & 0 & 4 & 4 & −4 & 0 & 0 & 0 & 0 & −8 & 4 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 6 & −6 & 6 & −6 & 0 & 0 & 0 \end{matrix} )

The rows are labelled by R_{˜12_{1}}, R˜4_{1}, R˜6_{1}, R˜6_{2}, R_{˜16_{1}}, Ψ_{1}, Ψ_{2}, Ψ_{3}, Ψ_{4}, Ψ_{5}, Ψ_{6}, Ψ_{7}, Ψ_{0}. The corresponding eigenvalues of t_{1}^{∗} are 1, 1, 1, 1, 1, i, −i, −1, 1, 1, θ, θ^{2}, −1, respectively. The columns are labelled by χ_{9}, …, χ_{21}. Notation of Table 1 and (3.6); Ψ_{0} is the almost character determined by condition (O).

Furthermore, after multiplication of the rows by certain roots of unity, we obtain precisely the Fourier matrix printed above.

Finally, assume that we are in type ^{2}F_{4} where we have to consider the family with 13 unipotent characters. We have 5 uniform almost characters and 7 cuspidal almost characters. Up to multiplication with certain roots of unity, these yield 12 rows of M(F). The remaining row is uniquely determined (up to multiplication with a root of unity) by condition (O). In Table 2, this is the row labelled by Ψ_{0}. Using Malle’s table [24], we can check explicitly that Ψ_{0} is an eigenvector of t_{1}^{∗} with eigenvalue −1. □

Remark 5.5. As mentioned in the introduction to this section, it is in fact possible to set up a kind of Harish–Chandra philosophy for almost characters. This would also yield almost characters which are intermediate between the uniform ones and the cuspidal ones. In general, given an almost character Ψ of G^{F}, there should exist a closed F -stable subgroup L ⊆ G, which is a Levi subgroup in some parabolic subgroup of G, and a cuspidal almost character Ψ_{L} of L^{F} such that Ψ is not orthogonal to R_{L}^{G}(Ψ_{L}) where R_{L}^{G} denotes the Lusztig induction. Furthermore, the pair (L, Ψ_{L}) should be unique up to conjugacy. Then [28, I.3.5] shows that both Ψ_{L} and Ψ should be eigenvalues of the twisting operators on L^{F} and G^{F}, respectively, with the same eigenvalue.

This would give a more conceptual explanation of the fact that the eigenvalue of the last almost character Ψ_{0} for ^{2}F_{4}(q^{2}) equals −1: it arises from the unique cuspidal almost character of the Levi subgroup of type ^{2}B_{2}(q^{2}).

6. Dihedral groups with automorphism

The groups of type ^{2}B_{2} and ^{2}G_{2} have a dihedral Weyl group with nontrivial action of the Frobenius of order 2. Now for any complex reflection group W with automorphism ϕ stabilizing the reflection representation the concept of fake degrees may be defined via the action of W, ϕ on its coinvariant algebra (see for example [5, 4B]). Lusztig has shown how to formally define unipotent degrees associated to an arbitrary dihedral group W = W (I_{2}(e)) (with arbitrary ϕ) [22, 4.1, 4.2]. This construction of unipotent degrees was subsequently generalized by the second author [25] to arbitrary imprimitive complex reflection groups G(e, 1, n) and G(e, e, n) (the dihedral groups being the special case G(e, e, 2)). Moreover, [25, Sect. 6] combinatorially defines a Fourier transform sending unipotent degrees to fake degrees. But due to the very general setup, the definition of the Fourier matrix in loc. cit. is quite difficult to evaluate in practice. Here we give closed formulas for the relevant data which for e = 4, 6 agree with the results of Theorem 5.4.

6.1. The Fourier matrix. Let W = W (I_{2}(e)) be a dihedral group of order 2e with e > 2. Then Irr(W) has three families, two of them singletons and one containing the remaining irreducible characters of W. Let us denote that family by F. Let ϕ be the non-trivial outer automorphism of W of order 2 stabilizing a chosen reflection representation. All but the linear characters in F are ϕ-stable (there exist two linear characters in F for e even, none for e odd). The corresponding set of almost characters Alm(F) has ⌊(e−1)^{2}/4⌋ elements. They can be indexed by pairs (i, j) of integers satisfying

0 < i < j < i + j < e  or  0 = i < j < e/2 .

Let us denote this set of pairs by I. For (i, j) ∈ I the corresponding fake degree is then given by

R_{i,j} = \{ \begin{matrix}x^{e−j} − x^{j} & if i = 0, \\ 0 & else. \end{matrix}

The unipotent degrees of the reflection datum (W, ϕ) lying in the family F can be indexed by pairs (k, l) of odd integers satisfying

0 < k < l < k + l < 2e .

The corresponding unipotent degrees are then given by

γ_{k,l} := \frac{ζ^{k} + ζ^{−k} − ζ^{l} − ζ^{−l}}{e} \frac{x(x^{2} − 1)(x^{e} + 1)}{(x − ζ^{k})(x − ζ^{−k})(x − ζ^{l})(x − ζ^{−l})}

(see [22, 4.2] and [25, (6.4)]), where we have noted ζ := exp(πi/e), a primitive 2eth root of unity. The Fourier matrix takes the form M(F) = (⟨(i, j), (k, l)⟩) with

⟨(i, j), (k, l)⟩ := \frac{ζ^{il+jk} + ζ^{−il−jk} − ζ^{ik+jl} − ζ^{−ik−jl}}{e}

(as follows by evaluating the construction in [25, 6C]). The following is an analogue of condition (U) in (5.1):

Theorem 6.2. Let e ⩾ 3 and (W, ϕ) the reflection datum with ϕ the nontrivial outer automorphism of W = W (I_{2}(e)) of order 2. Then the matrix M(F) = (⟨(i, j), (k, l)⟩) defined above takes the vector of unipotent degrees of the only non-trivial family F of Irr(W) to the corresponding vector of fake degrees.

Proof. Let’s compare the given parametrizations with the standard parametrizations of almost characters and unipotent degrees by e-symbols in [25]. To a parameter (i, j) ∈ I of an almost character we associate the e-symbol which has entries 0 in all lines except the ith and jth, and which has entries 1 in lines 0 and i + j. It is then easy to check that this map gives a bijection between parameters (i, j) for almost characters as above and e-symbols corresponding to ϕ-stable almost characters for the big family of W (I_{2}(e)). Similarly, for the index (k, l) of a unipotent degree γ_{k,l} let (k′, l′) be defined by k = 2k′ − 1, l = 2l′ − 1. Then we associate to (k, l) the e-symbol which has entries 0 in all lines except the k′th and l′th, and which has entries 1 in lines 0 and k′ + l′ − 1. Again, this gives a bijection between parameters for unipotent degrees and e-symbols of rank 2 and defect 1 up to cyclic shift (see [25, 6A]).

Moreover the formulas given above are precisely those for fake degrees and unipotent degrees as in loc. cit. (5.7) respectively (6.4), and for the entries of the Fourier matrix. Thus the theorem is just a rephrasing of [25, Satz 6.26]. □

Let’s check condition (O) from (5.1) (this is also implicit in [25, Lemma 6.19]):

Lemma 6.3. The rows of M(F) are orthogonal.

Proof. From the definition of M(F) in 6.1 we have

e^{2} ∑_{(i,j)∈I}⟨(i, j), (k, l)⟩⟨(i, j), (m, n)⟩

= T (\frac{k + m}{2}, \frac{l + n}{2}) + T (\frac{k − m}{2}, \frac{l − n}{2})

− T (\frac{k − n}{2}, \frac{l − m}{2}) − T (\frac{k + n}{2}, \frac{l + m}{2}),

where

T (a, b) := ∑_{(i,j)∈I} (ζ_{e}^{ia+jb} + ζ_{e}^{−ia−jb} + ζ_{e}^{ib+ja} + ζ_{e}^{−ib−ja})

with ζ_{e} := exp(2πi/e). Making appropriate changes of summation variables as in [23, 3.3] we find that

T (a, b) = −(−1)^{a} − (−1)^{b} + (−1)^{a+b} + ∑_{0<i,j<e} ζ_{e}^{ia+jb}

+ ∑_{i=1}^{e−1} (ζ_{e}^{ia} + ζ_{e}^{ib} − ζ_{e}^{i(a+b)} − ζ_{e}^{i(a−b)})

= δ_{a}δ_{b}e^{2} − δ_{a+b}e − δ_{a−b}e − (−1)^{a} − (−1)^{b} + (−1)^{a+b} + 1,

where δ_{u} := 1 if u ≡ 0 (mod e) and δ_{u} := 0 otherwise. The result follows from this. □

6.4. Eigenvalues. We may also formally associate eigenvalues of the ‘twisting operator t_{1}^{∗}’ to almost characters and eigenvalues of Frobenius to unipotent degrees in this case as follows: the eigenvalue of t_{1}^{∗} for the almost character indexed by (i, j) ∈ I is given by ζ^{−2ij} (this is just the eigenvalue associated to the almost character in the case ϕ = 1, see [23, 3.1] or [25, 6.27]). The eigenvalue of Frobenius ω_{k,l} of the unipotent character γ_{k,l} is given by ζ^{kl} (where still ζ = exp(πi/e)). We define an action of complex conjugation on Alm(F) by decreeing that it should fix all (0, j) ∈ I and interchange (i, j) ∈ I and (i, e − j) ∈ I for i > 0. Note that the induced action on the associated eigenvalues of Frobenius is indeed just complex conjugation.

Example 6.5. Let e = 3. Then the unique almost character in F is indexed by (0, 1), and has eigenvalue 1. The unipotent degree is indexed by (1, 3) with corresponding Frobenius eigenvalue −1. This describes the one-element family of the 3-dimensional unitary group containing the cuspidal unipotent character.

Example 6.6. Let e = 4. Then the parameters for almost characters in F are (0, 1), (1, 2), with corresponding 4-symbols

( \begin{matrix}1 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ − \\ − \\ 0 & 1 \end{matrix} ) ,

and corresponding eigenvalues 1, −1. The parameters for unipotent degrees are (1, 3), (1, 5) with corresponding 4-symbols

( \begin{matrix}0 & 1 \\ − \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ − \\ 0 \\ 1 \end{matrix} ) ,

and corresponding eigenvalues ζ_{8}^{3}, ζ_{8}^{5}. The Fourier matrix is now precisely the matrix given in Theorem 5.4(a), and the eigenvalues are those determined in Example 4.6.

Example 6.7. As a second example, let e = 6. Here the almost characters are indexed by (0, 1), (0, 2), (1, 2), (1, 3), (1, 4), (2, 3), with 6-symbols

( \begin{matrix}1 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}1 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ − \\ − \\ 0 & 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ − \\ 0 \\ − \\ 0 & 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ − \\ 0 \\ 0 \\ − \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ 0 \\ − \\ − \\ 0 \\ 0 & 1 \end{matrix} ) ,

and eigenvalues 1, 1, ζ_{3}, −1, ζ_{3}^{2}, 1. The parameters for the unipotent degrees are (1, 3), (1, 5), (1, 7), (1, 9), (3, 5), (3, 7) with corresponding 6-symbols

( \begin{matrix}0 & 1 \\ − \\ 1 \\ 0 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ − \\ 0 \\ 1 \\ 0 \\ 0 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ − \\ 0 \\ 0 \\ 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ − \\ 0 \\ 0 \\ 0 \\ 1 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ 0 \\ − \\ − \\ 0 & 1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ 0 \\ − \\ 0 \\ − \\ 0 & 1 \end{matrix} ) ,

and eigenvalues of Frobenius i, ζ_{12}^{5}, ζ_{12}^{7}, −i, i, −i. A short calculation shows that the Fourier matrix as defined above coincides with the matrix in Theorem 5.4(b) up to reordering rows and columns and adjusting signs of lines, and the eigenvalues are those determined in Example 4.7.

We may now summarize the known results on Fourier matrices as follows.

6.8. Finite Coxeter groups. Let W be a finite Coxeter group and assume that we are given an automorphism ϕ: W → W which leaves a set of simple reflections invariant. Let F ⊆ Irr(W) be a ϕ-invariant family and M(F) be the Fourier matrix for the corresponding family of unipotent degrees Uch(F). Then the columns of M(F) are indexed by Uch(F), the rows are indexed by the almost characters Alm(F) corresponding to this family. If W is a Weyl group and ϕ is ordinary in the sense of [19, 3.1], this matrix is constructed by Lusztig [19, Chap. 4]; for type I_{2}(e) and trivial ϕ, see Lusztig [23, §3]; for the types H_{3} and H_{4}, see Broué–Malle [4, (7.3)] and Malle [26]. The remaining cases (i.e., type F_{4} or I_{2}(e) with a non-trivial ϕ) are settled by the results in this paper; see Theorems 5.4 and 6.2. Let us write M = M(F, ϕ) = (a_{λμ}) where (λ, μ) ∈ Alm(F) × Uch(F) for the Fourier matrix.

Let F_{1} = F_{1}(F, ϕ) be the diagonal matrix of eigenvalues of the twisting operator t_{1}^{∗} on the almost characters Alm(F), and F_{2} := F_{2}(F, ϕ) the diagonal matrix with entries the eigenvalues of Frobenius of unipotent characters μ ∈ Uch(F). By the results of Lusztig [19, 11.2] and Shoji [28, I.§3] we have F_{1} = F_{2} when ϕ = 1, in which case we just write F for this matrix. These eigenvalues are known explicitly in all cases, see Section 4 and the references there. Finally, let Δ be the permutation matrix describing complex conjugation on Alm(F) (hence also on the eigenvalues of t_{1}^{∗} on Alm(F)). Then we may observe the following general properties satisfied by all these matrices:

Theorem 6.9. Let W be a finite Coxeter group, ϕ an automorphism of W leaving a set of simple reflections invariant, F ⊂ Irr(W) a ϕ-stable family and M, F_{1}, F_{2}, Δ the matrices attached to F, ϕ as above. Then we have:

(F1) M transforms the vector of unipotent degrees in Uch(F) to the vector

of fake degrees (extended by zeros) associated to F. (F2) All entries a_{λμ} of M are real. (F3) We have M · M^{tr} = M^{tr} · M = 1. (F4) Let λ_{0} be the row index of the uniform almost character correspond-

ing to the special character (in the sense of [19, 4.1.4]) in F. Then all entries in that row are non-zero.

(F5) The structure constants

a_{λμ}^{ν} := ∑_{κ∈Uch(F)} \frac{a_{λκ} a_{μκ} a_{νκ}}{a_{λ_{0}κ}}

are rational integers for all λ, μ, ν ∈ Alm(F). (F6) If ϕ = 1 then M = M^{tr}, Δ · M = M · Δ and (F · Δ · M)^{3} = 1. (F6’) If ϕ ≠ 1, ϕ^{2} = 1 then (F_{2} · M^{tr} · F_{1}^{−1} · M)^{2} = 1. (F6”) If ϕ ≠ 1, ϕ^{3} = 1 then (F_{2} · M^{tr} · F_{1}^{−1} · M)^{3} = 1.

Note that the products in (F6), (F6’) and (F6”) make sense, since in the first case there is a natural bijection between Alm(F) and Uch(F) (see [19, 4.21] for Weyl groups, respectively the explicit constructions for the other finite Coxeter groups) while in the other two cases the columns of F_{2} are indexed by the same set as the rows of M^{tr}, and so on. Moreover note that in the second case the rows and columns of M are determined only up to signs, but this does not affect the validity of (F6’) or (F6”). Also note that in both cases we have by definition F_{1}^{−1} = F_{1}^{Δ}.

Proof. In the case where W is a Weyl group and ϕ is trivial, the statements follow from the construction of the Fourier matrices and Frobenius eigenvalues starting from a finite group Γ(F) associated to the family F and its interpretation in terms of a certain tensor category, see [23, Prop. 1.6]. Previous proofs of assertions (F1)–(F4) can be found in [19, Chap. 4]. (Indeed, (F1) and (F4) are contained in [19, 4.26], (F2) follows from the explicit description of the matrices in [19, Chap. 4] and (F3) is the statement in [19, 4.25].) Property (F6) was shown by Digne and Michel [8, VII.3]. Note that this also holds for the exceptional families in E_{7} and E_{8} containing non-rational characters of the Hecke algebra.

If W is still a Weyl group and ϕ is non-trivial but ordinary, then all elements in a ϕ-stable family F are individually ϕ-stable, and the Fourier matrix is the same as for trivial ϕ. Moreover, the Frobenius eigenvalues for Alm(F) are the same as in the case ϕ = 1. This proves (F1)–(F5). Now assume that ϕ^{2} = 1. Statements (F6’) and (F6”) clearly reduce to the case where ⟨W, ϕ⟩ is irreducible. Let first W = W_{1} × W_{2} be a direct product of two factors interchanged by ϕ. The ϕ-stable families of W are in bijection with the families of W_{1}, in such a way that the Fourier matrices and Frobenius eigenvalues of almost characters agree. Furthermore, F_{2} is the diagonal matrix of eigenvalues of the square of the Frobenius, so we have to verify that (F^{2}M^{tr}F^{Δ}M)^{2} = 1. But this is a formal consequence of (F6) for the corresponding family of the untwisted group W_{1}. The same argument applies when W = W_{1}×W_{2}×W_{3} and ϕ cyclically permutes the three factors. If W itself is irreducible, (F6’) is checked by explicit computation in the case of ^{2}E_{6}. For the unitary groups, all families are singletons, and the Frobenius eigenvalues are ±1, so (F6’) is trivial. For ^{2}D_{n} the matrix F_{2} is the identity matrix by Theorem 4.11 and F_{1} has entries ±1, so there is nothing to prove. For^{3}D_{4} with ϕ of order 3, (F6”) can again be checked by direct computation.

Now let W be a Weyl group but ϕ ≠ 1 not ordinary. For^{2}F_{4} the properties can be checked from the matrix in Table 2 and the Frobenius eigenvalues in Example 4.8 by explicit calculation. The cases ^{2}B_{2} and ^{2}G_{2} will be dealt with in the context of dihedral groups.

If W is not a Weyl group, but ϕ = 1, (F1)–(F6) were verified by Lusztig [23, Sec. 3] for the dihedral groups (see also [25, Folg. 6.25]), and by the second author [26] for W (H_{4}). The non-trivial families of W (H_{3}) have the same data as the non-trivial family in W (H_{2}) = W (I_{2}(5)).

It only remains to consider the case that W = W (I_{2}(e)), e > 2, is a dihedral group and ϕ ≠ 1 has order 2. Then (F1) is Theorem 6.2, (F2) follows immediately from the defining formula in 6.1 for M(F), (F3) is Lemma 6.3. The special character of W (I_{2}(e)) is ϕ-stable and indexed by (0, 1). Thus (F4) is immediate. For (F5) we can argue as in [23, 3.7]. A more conceptual proof is given in the next section.

For (F6’) we verify that

F_{2} · M^{tr} · F_{1}^{−1} · M = M^{tr} · F_{1} · M · F_{2}^{−1},

that is,

Fr(k, l) ∑_{(i,j)∈I}⟨(i, j), (k, l)⟩ Fr(i, j)^{−1} ⟨(i, j), (m, n)⟩

= Fr(m, n)^{−1} ∑_{(i,j)∈I}⟨(i, j), (k, l)⟩ Fr(i, j) ⟨(i, j), (m, n)⟩

for all parameters (k, l), (m, n) of unipotent degrees. As in the proof of (F3) the sums on both sides can be split up into four parts T^{±}(a, b) with

T^{±}(a, b) = ∑_{(i,j)∈I} ζ_{e}^{±ij} (ζ_{e}^{ia+jb} + ζ_{e}^{−ia−jb} + ζ_{e}^{ib+ja} + ζ_{e}^{−ib−ja})

which can be evaluated as before to give

T^{±}(a, b) = eζ_{e}^{∓ab} +ζ_{e}^{e(2a+2b±e)/4} −ζ_{e}^{ea/2} −ζ_{e}^{eb/2} −1−∑_{i=1}^{e−1}(ζ_{e}^{i(a+b±i)} +ζ_{e}^{i(a−b∓i)}).

All terms except the first cancel when adding the four summands, and thus the result follows. □

7. Based rings and tensor categories

As Lusztig has pointed out (see [23, §1]), property (F5) allows to define an associative and commutative ring structure on the free ℤ-module R(M) generated by the rows of the Fourier matrix, the based ring associated to M(F), by taking the a_{λμ}^{ν} as the structure constants for multiplication of basis elements. (In fact Lusztig only considers the case where all structure constants are non-negative, in which case he can show the based ring to be semisimple. It would be interesting to know if the algebra is still semisimple in our more general setting.) The Fourier matrix is then, up to a scaling, the table of characters of the ring R(M), evaluated on the basis.

Note that the twisted dihedral groups W (I_{2}(e)) with e ⩾ 6 even and ^{2}F_{4} are the only cases where negative structure constants do occur; moreover, they are the only ones where the Fourier matrix fails to be symmetric (with respect to any ordering of rows or columns).

7.1. Lusztig’s tensor category. Lusztig has shown that the based ring constructed from the Fourier matrix of a family F of a Weyl group with trivial automorphism arises from the tensor category of Γ-equivariant vector bundles, where Γ is the finite group attached to F. Furthermore, for dihedral groups with trivial automorphism, he has given the following interpretation of the Fourier matrix via a tensor category (see [23, 3.8]): Let C_{e} be the Wess-Zumino-Witten tensor category (with simple objects the simple integrable modules with central charge e − 2 of the affine Lie algebra of type A_{1}). Let P_{1}, …, P_{e−1} be the simple objects in C_{e}. Then the based ring on P_{1}, …, P_{e−1} is the one obtained from a Fourier matrix M = (a_{ij}) with rows and columns indexed by {1, …, e − 1} and entries

a_{ij} := \frac{ζ^{ij} − ζ^{−ij}}{\sqrt{−2e}} ,

where ζ := exp(πi/e). Consider the tensor category consisting of external tensor products of objects in C_{e} with objects in C_{e} and let C′_{e} be the full subcategory consisting of objects isomorphic to direct sums of objects of the form P_{i} ⊙ P_{j} with i ≡ j (mod 2). Now Lusztig considers the category obtained from C′_{e} by adjoining an isomorphism between P_{1} ⊙ P_{1} and P_{e−1} ⊙ P_{e−1} and remarks that the based ring on the simple objects of this category, that is, the Grothendieck ring, gives rise to the Fourier matrix for the big family of the dihedral group W (I_{2}(e)) with trivial automorphism ϕ.

Our Fourier matrix for the dihedral group W (I_{2}(e)) with non-trivial automorphism ϕ can now be obtained as follows: We would like to declare P_{1} ⊙ P_{1} isomorphic to −(P_{e−1} ⊙ P_{e−1}). For this, let R̃ be the based ring on the simple objects of C′_{e}. Define R as the quotient R := R̃/I of R̃ by the ideal I generated by P_{1} ⊙ P_{1} + P_{e−1} ⊙ P_{e−1}. Now an easy calculation shows that

(P_{i} ⊙ P_{j}) ⊗ (P_{1} ⊙ P_{1} + P_{e−1} ⊙ P_{e−1}) = P_{i} ⊙ P_{j} + P_{e−i} ⊙ P_{e−j}

in C′_{e}, so the images of P_{i} ⊙ P_{j} and −(P_{e−i} ⊙ P_{e−j}) coincide in R for all 1 ≤ i, j ≤ e − 1. A ℤ-basis of R is given by the images in R of P_{i} ⊙ P_{j} with (i, j) in

J := {(i, j) | 0 < i < j < e, i ≡ j (mod 2), or 0 < i = j < e/2}.

Obviously the map ψ : J → I, (i, j) ↦ ((j − i)/2, (j + i)/2) is a bijection. Under this bijection, the Fourier matrix for the based ring R is just our matrix M(F) for the big family of W (I_{2}(e)) with non-trivial ϕ defined in 6.1, as can be seen easily from the defining formulas. In particular this shows axiom (F5): all structure constants are rational integers.

Thus, the only Fourier matrices for families of Coxeter groups for which an interpretation in terms of tensor categories is still missing is the 74 × 74matrix from [26] for the big family of W (H_{4}) and the matrix for ^{2}F_{4} from Theorem 5.4(c).

Example 7.2. We give the multiplication table for the basis elements of the based rings defined by the Fourier matrices for the 8-element family of G_{2} and the 6-element family of ^{2}G_{2}. For G_{2} the basis elements are ordered by (1, 1), (g_{2}, 1), (g_{3}, θ), (g_{3}, θ^{2}), (1, ϵ), (g_{2}, ϵ), (1, r), (g_{3}, 1). The non-trivial automorphism of W interchanges the two linear characters of W with fake degrees indexed by (1, r), (g_{3}, 1).

( \begin{matrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 2 & 1+3+4+7+8 & 2+6 & 2+6 & 6 & 3+4+5+7+8 & 2+6 & 2+6 \\ 3 & 2+6 & 1+3+5 & 7+8 & 3 & 2+6 & 4+8 & 4+7 \\ 4 & 2+6 & 7+8 & 1+4+5 & 4 & 2+6 & 3+8 & 3+7 \\ 5 & 6 & 3 & 4 & 1 & 2 & 7 & 8 \\ 6 & 3+4+5+7+8 & 2+6 & 2+6 & 2 & 1+3+4+7+8 & 2+6 & 2+6 \\ 7 & 2+6 & 4+8 & 3+8 & 7 & 2+6 & 1+5+7 & 3+4 \\ 8 & 2+6 & 4+7 & 3+7 & 8 & 2+6 & 3+4 & 1+5+8 \end{matrix} )

For ^{2}G_{2} the basis elements are ordered as in the Fourier matrix in Theorem 5.4.

( \begin{matrix}1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 1 + 3 + 4 & 2 − 6 & 2 + 6 & 6 & −3 + 4 + 5 \\ 3 & 2 − 6 & 1 + 3 − 5 & 0 & −3 & −2 + 6 \\ 4 & 2 + 6 & 0 & 1 + 4 + 5 & 4 & 2 + 6 \\ 5 & 6 & −3 & 4 & 1 & 2 \\ 6 & −3 + 4 + 5 & −2 + 6 & 2 + 6 & 2 & 1 + 3 + 4 \end{matrix} )

The similarity between these two tables is explained by the construction in 7.1 of the based rings starting from the category C′_{e}.

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Institut Girard Desargues, bat. Jean Braconnier, Université Lyon 1, 21 av Claude Bernard, F–69622 Villeurbanne cedex, France

  E-mail address: geck@desargues.univ-lyon1.fr

Fachbereich Mathematik/Informatik, Universität Kassel, Heinrich-Plett-Str. 40, D–34132 Kassel, Germany.

  E-mail address: malle@mathematik.uni-kassel.de