CUSPIDAL UNIPOTENT CLASSES AND CUSPIDAL BRAUER CHARACTERS

MEINOLF GECK AND GUNTER MALLE

1. Introduction

1.1. Let G be a connected reductive group defined over the finite field with q elements. We assume that the centre of G is connected and that q is a power of a good prime for G. In his theory of generalized Gelfand-Graev representations (GGGR’s for short), N. Kawanaka [Ka] has established a relation between the unipotent classes and the ordinary characters of the finite group G(𝔽_{∥}). This was used in [GHM] to derive results about the Harish-Chandra series of ℓ–modular representations of groups of type A_{n}, for primes ℓ not dividing q. The main tools were the Harish-Chandra induction of GGGR’s and the investigation of intersections of unipotent classes with Levi-complements of F -stable parabolic subgroups of G. In this paper we generalize these ideas to groups G of arbitrary type. As an application, we prove that the 2-modular reduction of a cuspidal unipotent character of a finite classical group (if it exists) is an irreducible Brauer character. (In [Ge] it is already shown that the 2-modular decomposition matrix of such a group has a lower unitriangular shape.)

1.2. The first main results of this paper, Propositions 2.5 and 2.6, assert that the method of Harish-Chandra induction of GGGR’s works in complete generality. Roughly speaking this means that, under suitable hypothesis, a GGGR may be replaced by the Harish-Chandra induction of a GGGR of a Levi complement of some proper F -stable parabolic subgroup of G. The proof of this result is based on Lusztig’s paper [L3] where the character of a GGGR is expressed in terms of intersection cohomology complexes of closures of unipotent classes with coefficients in various local systems. We also need a geometric property of unipotent classes due to R. Richardson and F. Digne, G. Lehrer, J. Michel [DLM].

This paper is a contribution to the DFG-research project on “Algorithmic Number Theory and Algebra”. The second author gratefully acknowledges financial support by the DFG..

In Section 3 we develop the concept of “cuspidal unipotent classes” in complete generality (see the definition in (3.1)). For type A_{n}, this was introduced in [GHM]. The main problem for arbitrary types arises from the fact that one also has to take into account the group of components of the centralizer of an element in a unipotent class. We give a classification of the cuspidal unipotent classes, individually for all types. Using this case–by–case analysis we state, in Theorems 3.2 and 3.3, some basic geometric properties of cuspidal unipotent classes for an arbitrary group G as in (1.1). This may also be of independent interest.

1.3. In Section 4 these results are applied to the 2-modular representations of groups G of classical type with connected centre and with odd q. The set-up is roughly similar to the one employed in [GHM] for groups of type A_{n}. Since we are in the case of non-defining characteristic, the GGGR’s of G(𝔽_{q}) are projective representations. In [Ge], it is shown that they give rise to a “basic set” of projective representations for the principal 2-block B_{1} of G(𝔽_{q}). Moreover, the matrix of scalar products with a suitably chosen “basic set” of ordinary characters is lower unitriangular. This implies that the decomposition matrix of B_{1} also is lower unitriangular.

Now assume that G(𝔽_{q}) has a (unique) cuspidal unipotent character ρ. We consider the scalar products of ρ with the characters of the various GGGR’s. If we can replace a GGGR by the Harish-Chandra induction of a suitably chosen GGGR of a proper Levi subgroup of G then the scalar product with ρ will be zero. This replacement can indeed be done for all GGGR’s which correspond to non-cuspidal unipotent classes of G. Our results in Sections 2,3 ensure that there will be precisely one non-zero scalar product left. This shows, as in [GHM], that the 2-modular reduction of ρ is an irreducible Brauer character.

1.4. An application of this method to the exceptional groups fails at present since first we do not have that the decomposition matrix of G(𝔽_{q}) is lower triangular (which would require further results about the GGGR’s for exceptional groups), and secondly since there are always at least two cuspidal unipotent characters. Also, for the classical groups with respect to primes ℓ ≠ 2, we again do not know that the decomposition matrix is triangular.

We also point out that all of the above results which involve the theory of GGGR’s are valid only under additional assumptions on G and q. More precisely, we have to assume that G is split and that q is a power of a sufficiently large prime. This is because such assumptions were already needed in [L3]. It is very likely, however, that the results on GGGR’s are true in general.

1.5. Notation. Let H be any linear algebraic group and H^{∘} the connected component of the identity. Given an element h ∈ H we denote by (h)_{H} its conjugacy class and by C_{H}(h) its centralizer in H. Moreover, A_{H}(h) will denote the group of components of C_{H}(h). (A_{H}(h) always is a finite group.) Given two elements h, h′ ∈ H we write h ∼_{H} h′ if h, h′ are conjugate in H. There is a partial order relation on the conjugacy classes of H: If C, C′ are two classes in H we write C ≤ C′ if C is contained in the Zariski closure of C′ in H.

If H is defined over the finite field 𝔽_{q} we always denote by F : H → H the corresponding Frobenius map. The ordinary representations of the finite group of fixed points H^{F} will always be taken over ℚ̄_{ℓ}, an algebraic closure of the field of ℓ-adic numbers for a prime ℓ not dividing q. We denote by x ↦ x̄ an automorphism of that field which maps roots of unity to their inverses. The scalar product between class functions f, f′ on H^{F} is given by (f, f′)_{H} = (1/|H^{F} |) ∑_{h∈H^{F}} f(h)\overline{f′(h)}.

2. Harish-Chandra induction of GGGR’s

2.1. In this section we generalize the results of [GHM, Section 5]. Let G be a connected reductive group defined over 𝔽_{q}, with corresponding Frobenius map F : G → G. Throughout this section, we fix an F - stable parabolic subgroup P in G with a decomposition P = UL where U is the unipotent radical of P and L is an F -stable Levi complement of P. We also assume that q is a power of a good prime for G.

If u ∈ L^{F} is a unipotent element then there are corresponding generalized Gelfand-Graev representations (GGGR’s) for L^{F} and G^{F} whose characters we denote by Γ_{u}^{L} and Γ_{u}^{G}, respectively. In [GHM], we studied the relation between the characters

D_{G}(Γ_{u}^{G}) and R_{L}^{G}(D_{L}(Γ_{u}^{L}))

where R_{L}^{G} denotes Harish-Chandra induction from L^{F} to G^{F} and D_{L}, D_{G} denote the standard duality operations on the character rings of L^{F} and G^{F}. However, in [loc. cit.] we were not able to treat this in general but only in the case where G has a connected centre and a root system of type A_{n}. Now, we consider the general case (arbitrary type and not necessarily connected centre) which is more complicated due to the fact that centralizers of unipotent elements need not be connected.

At some point we will have to use Lusztig’s results [L3] on GGGR’s and unipotent support of irreducible characters. These are proved under the additional assumption that G is split and that q is a sufficiently large power of a sufficiently large prime. It is very likely, however, that they remain true in general.

2.2. We assume that each F -stable unipotent class C contains a split element u ∈ C^{F}. (This is the case in classical groups, cf. [Sh, (5.1)].) This implies that the induced action of F on A_{G}(u) is trivial. We consider the set I^{G} of all pairs (C, ϕ) where C is an F -stable unipotent class in G and ϕ is an irreducible character of A_{G}(u). Let i = (C, ϕ) ∈ I^{G}. The class C will be called the support of i. We define a class function Y_{i}^{G} on G^{F} as follows. We fix a split element u ∈ C^{F}. Let 1 = a_{1}, …, a_{d} ∈ A_{G}(u) be representatives for the conjugacy classes of A_{G}(u), and u = u_{1}, …, u_{d} ∈ C^{F} be corresponding representatives of the G^{F} -orbits on C^{F}. Then

Y_{i}^{G}(g) := \{ \begin{matrix}ϕ(a_{r}) & if g ∼_{G^{F}} u_{r} for some r ∈ \{1, …, d\}, \\ 0 & otherwise. \end{matrix}

The functions Y_{i}^{G} (i ∈ I^{G}) form a basis of the space of class functions on G^{F} which are supported by unipotent elements only. Finally, we define the twisted GGGR Γ_{i}^{G} corresponding to i ∈ I^{G} by

Γ_{i}^{G} = ∑_{r=1}^{d} \frac{|A_{G}(u)|}{|A_{G}(u_{r})^{F} |}Y_{i}(u_{r})Γ_{u_{r}}^{G} (see [L3, (7.5a)]).

There are two basic results about GGGR’s that we need.

(a) For any i = (C, ϕ) ∈ I^{G} and any unipotent element v ∈ G^{F}, we

have

D_{G}(Γ_{i}^{G})(v) = 0 unless C ≤ (v)_{G}.

(b) For each i ∈ I^{G} there exists a non-zero constant c_{i}^{G} ∈ ℚ̄_{ℓ} such that

(D_{G}(Γ_{i}^{G}), Y_{j}) = c_{i}^{G}δ_{ij} where i, j ∈ I both have support C.

Throughout this section, we will assume that these two properties hold for the GGGR’s of G and L. This is known in the case where q is a sufficiently large power of a sufficiently large prime and where G is split over 𝔽_{q} (see [Ge, Proposition 3.5 and Corollary 3.6], and [L3]). Our aim will be to show that analogous properties hold for the Harish-Chandra induction of GGGR’s from L^{F} to G^{F}.

We start with the following result.

Proposition 2.3. Let u ∈ L^{F} be a unipotent element. Then, for all unipotent v ∈ G^{F}, we have

R_{L}^{G}(D_{L}(Γ_{u}^{L}))(v) = 0 unless (u)_{G} ≤ (v)_{G}.

(A similar result also holds for the twisted GGGR’s.)

Proof. The argument is almost the same as in the proof of [GHM, Lemma 5.4], using the induction formula for characters. Let v = p_{1}, …, p_{m} ∈ P^{F} be representatives for the P^{F} -conjugacy classes contained in (v)_{G^{F}} ∩ P^{F}, and write p_{r} = x_{r}l_{r} with x_{r} ∈ U^{F} and l_{r} ∈ L^{F}. (Note that all l_{r} are unipotent elements.) Using the definition of Harish-Chandra induction and the character formula for induction, we then have that

R_{L}^{G}(D_{L}(Γ_{u}^{L}))(v) = ∑_{r=1}^{m} \frac{|C_{G}(v)^{F} |}{|C_{P} (p_{r})^{F} |}D_{L}(Γ_{u}^{L})(l_{r}).

Let us fix an index r such that the corresponding term in this sum is non-zero. Property (2.2a) then implies that (u)_{L} ≤ (l_{r})_{L}. Hence we conclude that (u)_{G} ≤ (l_{r})_{G} ≤ (x_{r}l_{r})_{G} = (v)_{G}, using [GHM, Lemma 5.2] (which was proved for arbitrary G and not only for GL_{n}). This implies the assertion. Indeed, if the left hand of the above formula is non-zero then there exists some r giving a non-zero contribution in the sum on the right hand.

2.4. Now let u ∈ L^{F} be unipotent. Let C be the G-conjugacy class of u, and C′ the L-conjugacy class of u. Assume that u is split in C′.

Consider the canonical map π : C_{G}(u) → A_{G}(u). Its restriction to L has kernel C_{G}(u)^{∘} ∩ L which clearly contains C_{L}(u)^{∘}. As is shown in [Sp, (1.4)], we have in fact equality. Hence we obtain a canonical injective map

π′ : A_{L}(u) → A_{G}(u),

induced by restricting π to L. Moreover, the induced actions of F on A_{L}(u) and on A_{G}(u) are compatible with this map. To proceed we impose the following two conditions on the pair (L, u).

(a) We assume that the above map π′ : A_{L}(u) → A_{G}(u) is surjective (hence, an isomorphism).

(b) We assume that, if v ∈ C ∩L^{F} such that (u)_{L} ≤ (v)_{L} then v ∈ C′. Property (b) is a technical condition which was already used in [GHM]. Property (a) implies that, if 1 = a_{1}, …, a_{d} ∈ A_{L}(u) are representatives of the conjugacy classes of A_{L}(u) and 1 = u_{1}, …, u_{d} ∈ C′^{F} are corresponding representatives of the L^{F} -orbits on C′^{F} then u_{1}, …, u_{d} are also representatives of the G^{F} -orbits on C^{F}. Furthermore, the map

i = (C, ϕ) ↦ i′ = (C′, ϕ ∘ π′)

is a bijection between pairs in I^{G} with support C and pairs in I^{L} with support C′, and we have that (*) Y_{i}^{G}(v) = Y_{i′}^{L}(v) for all v ∈ C′^{F}.

Now we can state and prove the analogue of the second basic property (2.2b) for the Harish-Chandra induction of GGGR’s.

Proposition 2.5. With the assumptions in (2.4), let i, j ∈ I^{G} have the same support C. Then

(R_{L}^{G}(D_{L}(Γ_{i′}^{L})), Y_{j}^{G})_{G} = \frac{c_{i′}^{L}}{|C_{U}(u)^{F} |}δ_{ij}.

Proof. Let ^{∗}R_{L}^{G} be the operator adjoint to R_{L}^{G} with respect to the standard scalar product for class functions on G^{F} and L^{F}. Then, for any class function f on G^{F}, we have

∗R_{L}^{G}(f)(v) = \frac{1}{|U^{F} |} ∑_{x∈U^{F}} f(xv) for v ∈ L^{F}.

Writing out the scalar product between two class functions and using this formula, we obtain

(R_{L}^{G}(D_{L}(Γ_{i′}^{L})), Y_{j}^{G})_{G} = (D_{L}(Γ_{i′}^{L}),^{∗}R_{L}^{G}(Y_{j}^{G}))_{L} = \frac{1}{|L^{F} |} ∑_{v∈L^{F}} D_{L}(Γ_{i′}^{L})(v)^{∗}R_{L}^{G}(Ȳ_{j}^{G})(v)
= \frac{1}{|L^{F} |} ∑_{v∈L^{F}} \frac{1}{|U^{F} |} ∑_{x∈U^{F}} D_{L}(Γ_{i′}^{L})(v)Ȳ_{j}^{G}(xv).

Assume that the term corresponding to x ∈ U^{F} and v ∈ L^{F} gives a non-zero contribution to this sum. The condition that Ȳ_{j}^{G}(xv) is non-zero implies that xv ∼_{G} u (in particular, v is unipotent) and the condition that D_{L}(Γ_{i′}^{L})(v) is non-zero implies that (u)_{L} ≤ (v)_{L} (by property (2.2a)). Since also (v)_{G} ≤ (xv)_{G} (by [GHM, Lemma 5.2]) we deduce that

(u)_{G} ≤ (v)_{G} ≤ (xv)_{G} = (u)_{G}.

It follows that u, v and xv are conjugate in G.

On one hand this implies, by [DLM, Theorem 5.11.1], that v and xv are already conjugate in U^{F}. In particular, Ȳ_{j}^{G} has the same value on v and xv. On the other hand, (2.4b) implies that v ∈ C′, since v ∈ C ∩ L and (u)_{L} ≤ (v)_{L}. In particular, the above sum need only be extended over all v ∈ C′^{F} and all x ∈ S(v), where

S(v) := {x ∈ U^{F} | xv ∼_{U^{F}} v} for v ∈ C′^{F} .

Using (2.4)(*), the above sum therefore simplifies to

\frac{1}{|L^{F} |} ∑_{v∈C′^{F}} D_{L}(Γ_{i′}^{L})(v)Ȳ_{j′}^{L}(v)\frac{|S(v)|}{|U^{F} |}.

We now claim that |S(v)| = |U^{F} |/|C_{U}(u)^{F} |. Indeed, the map x ↦ xv defines a bijection between S(v) and (v)_{U^{F}}. Hence |S(v)| = |U^{F} |/|C_{U}(v)^{F} |.

Now there exists some l ∈ L such that v = lul^{−1}. Then conjugation by l defines an isomorphism of algebraic groups from C_{U}(v) onto C_{U}(u). Since these groups are connected (see [Sp, Lemma 1.4]), unipotent and defined over 𝔽_{q}, their fixed point sets under F have the same cardinality, by a theorem of Rosenlicht [Ro]. The claim is proved.

Our expression now has the form

\frac{1}{|C_{U}(u)^{F} |} \frac{1}{|L^{F} |} ∑_{v∈C′^{F}} D_{L}(Γ_{i′}^{L})(v)Ȳ_{j′}^{L}(v).

Since Ȳ_{j′}^{L} is zero on all elements of L^{F} outside C′^{F} we can extend the sum over all v ∈ L^{F}. Thus, we see that this expression equals

\frac{1}{|C_{U}(u)^{F} |}(D_{L}(Γ_{i′}^{L}), Y_{j′}^{L})_{L}.

By property (2.2b), the right hand side equals a constant times δ_{i′j′}. Using the bijection i ↦ i′ in (2.4), the result follows.

We remark that the sets S(v) also appeared in the work of Digne, Lehrer, Michel, see [DLM, Proposition (7.10)]. Next we study the question of how the constants c_{i}^{G} and c_{i′}^{L} are related.

Proposition 2.6. With the assumptions in (2.4), let i ∈ I^{G} with support C. Then

c_{i′}^{L} = q^{dim C_{U} (u)}c_{i}^{G}.

Furthermore, the restrictions of D_{G}(Γ_{i}^{G}) and R_{L}^{G}(D_{L}(Γ_{i′}^{L})) to C^{F} are equal.

Proof. The second statement follows from the first, since then the two functions have the same scalar product with all Y_{j} where j ∈ I has support C. In order to prove the first statement we have to recall the definition of c_{i}^{G}. By [L3], [Ge], there exists a fourth root of unity ζ_{i}^{G} such that

c_{i}^{G} = |A_{G}(u)|ζ_{i}^{G}q^{(− dim G−s_{i}^{G})/2}

where s_{i}^{G} = − dim C − dim Z(L_{i})^{∘} and Z(L_{i}) is the centre of a certain Levi subgroup in some parabolic subgroup of G associated with i. (The precise definition of ζ_{i}^{G} is given in [L3, Proposition 7.2], and [Ge]; we shall say more about the group L_{i} below.) Similarly and corresponding to i′ ∈ I^{L}, we have a Levi subgroup L_{i′} in some parabolic subgroup of L. Then we are done if we can show that

ζ_{i′}^{L} = ζ_{i}^{G} and − dim G − s_{i}^{G} = − dim L − s_{i′}^{L} − 2 dim C_{U}(u).

Now, by [L2, Lemma 2.9(a)], we have that dim C −dim C′ = 2(dim U − dim C_{U}(u)). Inserting this and the defining equation for s_{i}^{G} and s_{i′}^{L} into the above equation, we see that it is sufficient to prove that

ζ_{i′}^{L} = ζ_{i}^{G} and dim Z(L_{i})^{∘} = dim Z(L_{i′})^{∘}.

This really is a question about the generalized Springer correspondence. We use [Sp, (1.2) and (1.4)], as a convenient reference where a similar set-up as in (2.4) above is considered. At first we look at the situation inside L. The group L_{i′} is a Levi subgroup of some parabolic subgroup in L, and the pair (C′, i′) corresponds to a unique (up to conjugation in L) quadruple (L_{i′}, v, ψ, ρ′) where v ∈ L_{i′} is unipotent, ψ is an irreducible character of A_{L_{i′}}(v) such that (v, ψ) is a cuspidal pair in L_{i′} (in the sense of [L2]), and ρ′ is an irreducible character of N_{L}(L_{i′})/L_{i′}. We say that the pair i′ ∈ I^{L} lies above the pair (v, ψ). Looking at the situation in G we have, similarly, that the pair i ∈ I^{G} lies above some cuspidal pair of L_{i}.

Now we can also regard (v, ψ) as a cuspidal pair in G. Correspondingly, we have pairs in I^{G} lying above it. A relation between these and the pairs for L lying above (v, ψ) is established as follows.

Following [Sp, (1.4)], we consider the action of A_{G}(u) × A_{L}(u) on A_{G}(u) by (a, b).x = axb^{−1}, where a, x ∈ A_{G}(u), b ∈ A_{L}(u), and where we have identified A_{L}(u) ⊆ A_{G}(u) using the map π′. Let ε be the character of the corresponding permutation representation. Since A_{L}(u) = A_{G}(u), we have

ε(a, b) = \{ \begin{matrix}|C_{A_{G}(u)}(a)| & if a is conjugate to b, \\ 0 & otherwise. \end{matrix}

Now consider the character ϕ ⊗ ϕ̄ of A_{G}(u) × A_{L}(u), where i = (C, ϕ). Then the multiplicity of ϕ ⊗ ϕ̄ in ε is given by

(ϕ ⊗ ϕ̄, ε)_{A_{G}(u)×A_{L}(u)} = (ϕ, ϕ)_{A_{G}(u)} = 1.

The fact that this is non-zero implies, by [Sp, (1.2)(II)], that i = (C, ϕ) lies above the pair (v, ψ). (The corresponding quadruple is of the form (L_{i′}, v, ψ, ρ) where ρ is an irreducible character of N_{G}(L_{i′})/L_{i′}.) Since i lies above a cuspidal pair of L_{i} we conclude, using [L2, Proposition 6.3], that the groups L_{i} and L_{i′} are conjugate in G. In particular, dim Z(L_{i})^{∘} = dim Z(L_{i′})^{∘}.

The proof of [L3, Proposition 7.2], shows that the fourth roots of unity ζ_{i}^{G}, ζ_{i′}^{L} only depend on the cuspidal pair (v, ψ), which is the same for i and i′. Hence the roots of unity are equal. The proposition is proved.

The following result will be useful, in combination with [Ge], for our applications to 2-modular Brauer characters of finite classical groups. For this purpose, let us recall the notion of unipotent support (see [L3] and note that, here again, we need the stronger conditions on q and G mentioned above). Let χ be an irreducible character of G^{F}. Then there exists a unique F -stable unipotent class in G of maximal possible dimension such that ∑_{x∈C^{F}} χ(x) ≠ 0. This class has the property that, if C_{1} is any F -stable unipotent class in G and χ(y) ≠ 0 for some y ∈ C_{1}^{F} then dim C_{1} ≤ dim C with equality only if C = C_{1}.

Corollary 2.7. In the set-up of (2.4), assume that χ is an irreducible character of G^{F} with unipotent support C. Then

(R_{L}^{G}(D_{L}(Γ_{u}^{L})), χ)_{G} = (D_{G}(Γ_{u}^{G}), χ)_{G}

for all u ∈ C′^{F}.

Proof. Let f be a class function on G^{F} with support contained in the set of unipotent elements of G^{F} such that, if C_{1} is an F -stable unipotent class in G and f(x) ≠ 0 for some x ∈ C_{1}^{F} then C ≤ C_{1}. The scalar product of f with χ is computed by summing up the terms f(x)χ̄(x) over all x ∈ G^{F}. Assume that such a term is non-zero and that the element x lies in the F -stable unipotent class C_{1}. The assumption on f implies that C ≤ C_{1}, and the fact that C is the unipotent support of χ implies that dim C_{1} ≤ dim C. Hence C = C_{1}. We conclude that

(f, χ)_{G} = \frac{1}{|G^{F} |} ∑_{x∈C^{F}} f(x)χ̄(x).

This shows, in particular, that we only need to consider the restrictions of f and of χ to C^{F}. Now there exist constants a_{i} (for i ∈ I with support C) such that χ(x) = ∑_{i} a_{i}Y_{i}(x) for all x ∈ C^{F}. Hence we conclude that

(f, χ)_{G} = ∑_{i} ā_{i}(f, Y_{i})_{G}.

Thus, the scalar product of f and χ only depends on the scalar products of f and the various Y_{i} where i has support C.

Now the previous propositions show that the functions R_{L}^{G}(D_{L}(Γ_{i′}^{L})) and D_{G}(Γ_{i}^{G}) satisfy the assumptions on f and have the same scalar product with all Y_{i} where i has support C. Hence

(R_{L}^{G}(D_{L}(Γ_{i′}^{L})), χ)_{G} = (D_{G}(Γ_{i}^{G}), χ)_{G}.

Now the GGGR’s can be expressed as linear combinations of the twisted GGGR’s. Using the defining equation in (2.3) and the bijection i ↦ i′ in (2.4), we see that the coefficients in these linear combinations are the same with respect to L and with respect to G. The corollary follows.

3. Cuspidal unipotent classes

3.1. Let G be a connected reductive algebraic group defined over the finite field 𝔽_{∥}, q a power of the prime p, and F : G → G the corresponding Frobenius morphism. In this section we generalize the concept of cuspidal unipotent classes, which was introduced in [GHM] in the case where G has type A_{n}. We also give a classification of the cuspidal unipotent classes for all types in good characteristic.

Definition. An F -stable unipotent conjugacy class C of G is called non-cuspidal, if there exists an F -stable Levi complement L of a proper F -stable parabolic subgroup of G such that C has non-empty intersection with L^{F} and the canonical injective map π′ : A_{L}(u) → A_{G}(u) (see (2.4)) is an isomorphism for some u ∈ C ∩ L^{F}. Otherwise C is called a cuspidal class of (G, F).

(Note that for G of type A_{n} this definition coincides with the one given in [GHM, 6.1].) The remarks in (2.4) show that, if C is noncuspidal and L is a corresponding Levi subgroup then C ∩ L^{F} contains representatives for all G^{F} -classes contained in C^{F}.

We now state two basic properties of cuspidal unipotent classes. The proof of these results will occupy the remainder of this section.

Theorem 3.2. Assume that q is a power of a good prime for G and that the centre of G is connected. Let C be an F -stable unipotent conjugacy class of G. Let L be an F -stable Levi complement in some F -stable parabolic subgroup of G of minimal possible dimension with respect to the properties

L^{F} ∩ C ≠ ∅ , and A_{L}(u) ≅ A_{G}(u) for some u ∈ L^{F} ∩ C.

Let C′ := (u)_{L}. Then C′ is a cuspidal unipotent class in L. Moreover, the following property holds: If v ∈ C ∩ L^{F} such that (u)_{L} ≤ (v)_{L} then v ∈ C′ (cf. (2.4b)).

Theorem 3.3. Assume that q is a power of a good prime for G and that the centre of G is connected. Let W be the Weyl group of G (with respect to a maximally split torus) and γ the automorphism of W induced by F. Assume that Irr(W) contains a γ-stable cuspidal family F (in the sense of [L1, (8.1)]). Let C be the F -stable unipotent conjugacy class of G attached to F via the Springer correspondence. Then we have:

(a) C lies in the closure of all cuspidal unipotent classes of (G, F),

(b) C is itself cuspidal, (c) the group of connected components A_{G}(u) is isomorphic to the

finite group G_{F} attached to the cuspidal family F of Irr(W) by the method of [L1, Chapter 4].

3.4. At first we note that the proof of these results can be reduced to the case that G is simple of adjoint type. (A similar reduction process was used, for example, in [L1, (8.8)]). The reduction in our case is based on the following facts. Assume that G is of arbitrary type, with connected centre. Let ϕ : G → G_{ad} be the adjoint quotient of G. Then ϕ induces an isomorphism from the unipotent variety of G onto that of G_{ad}. This map is compatible with the Frobenius morphism, it preserves unipotent classes, the groups of components of centralizers, and hence the notion of cuspidality. As moreover G_{ad} has the same Weyl group as G, the Springer correspondence commutes with ϕ. It follows that we may assume that G is semisimple adjoint. Since the Springer correspondence and cuspidality of classes behave well with respect to direct products of F -stable factors, we may further reduce to the case where G is a direct product G_{1} × ⋯ × G_{s} of isomorphic simple adjoint groups, permuted cyclically by F. The F -stable unipotent classes of G are in natural bijection with the F^{s}-stable unipotent classes of G_{1}. Also, the F -stable families of the Weyl group of W are in natural bijection with the F^{s}-stable families of the Weyl group of G_{1}. Again since the Springer correspondence commutes with taking direct products, we can suppose that G is simple of adjoint type. We omit further details.

3.5. From now on, we assume that G is simple of adjoint type. Our first goal is a classification of the cuspidal unipotent classes of (G, F). For type A_{n} this was already done in [GHM, 6.2]. Let us first remind the reader of the parametrization of unipotent classes of classical groups by appropriate combinatorial objects (see [Ca, 13.1]):

(a) For A_{n}, the unipotent classes are parametrized by partitions α of n.

(b) For B_{n}, p ≠ 2, the unipotent classes are parametrized by pairs of partitions (α, β) with 2|α| + |β| = 2n + 1 such that all parts of β are odd and distinct.

(c) For C_{n}, p ≠ 2, the unipotent classes are parametrized by pairs of partitions (α, β) with |α| + |β| = n such that all parts of β are distinct.

(d) For D_{n}, p ≠ 2, the unipotent classes are parametrized by pairs of partitions (α, β) with 2|α| + |β| = 2n such that all parts of β are odd and distinct. If β is empty and all parts of α are even, then two different classes are parametrized by the same pair.

We will also write C(α), C(α, β), respectively, for the unipotent class parametrized by the indicated combinatorial object.

For exceptional groups in good characteristic, the unipotent classes C are best described by pairs (L, P_{L′}) consisting of a rational Levi subgroup L and a distinguished parabolic subgroup P_{L′} of the semisimple part L′ of L, such that C ∩ L contains a dense orbit of the action of P_{L′} on its unipotent radical (see for example [Ca, Chapter 5]).

Proposition 3.6. The cuspidal unipotent conjugacy classes of simple adjoint groups in good characteristic are as follows:

(a) For A_{n}, only C(α) with α = (n) is cuspidal.

(b) For ^{2}A_{n}, C(α) is cuspidal if and only if all parts of α are distinct.

(c) For B_{n}, p ≠ 2, C(α, β) is cuspidal if and only if the parts of α are odd, distinct and different from the parts of β.

(d) For C_{n}, p ≠ 2, C(α, β) is cuspidal if and only if the parts of α are distinct, even, and different from 2β_{i} for all parts β_{i} of β.

(e) For D_{n}, p ≠ 2, C(α, β) is cuspidal if and only if the parts of α are odd, distinct and different from the parts of β.

(f) For ^{2}D_{n}, p ≠ 2, C(α, β) is cuspidal if and only if it is cuspidal for D_{n}.

(g) For ^{3}D_{4}, p ≠ 2, C(α, β) is cuspidal if and only if

(α, β) ∈ {(31, −), (−, 53), (−, 71)} .

(h) For G_{2}, p ≥ 5, the classes G_{2}(a_{1}) and G_{2} are cuspidal.

(i) For F_{4}, p ≥ 5, there are 4 cuspidal classes:

F_{4}(a_{3}), F_{4}(a_{2}), F_{4}(a_{1}), F_{4}.

(j) For E_{6}, p ≥ 5, there are 4 cuspidal classes:

D_{4}(a_{1}), E_{6}(a_{3}), E_{6}(a_{1}), E_{6}.

(k) For ^{2}E_{6}, p ≥ 5, there are 7 cuspidal classes:

D_{4}(a_{1}), A_{4} + A_{1}, D_{5}(a_{1}), E_{6}(a_{3}), D_{5}, E_{6}(a_{1}), E_{6}.

(l) For E_{7}, p ≥ 5, there are 8 cuspidal classes:

A_{4} + A_{1}, E_{7}(a_{5}), E_{7}(a_{4}), E_{6}(a_{1}), E_{7}(a_{3}), E_{7}(a_{2}), E_{7}(a_{1}), E_{7}.

(m) For E_{8}, p ≥ 7, there are 15 cuspidal classes:

E_{8}(a_{7}), D_{5} + A_{2}, D_{7}(a_{2}), E_{6}(a_{1}) + A_{1}, E_{8}(b_{6}), D_{7}(a_{1}), E_{8}(a_{6}),

E_{8}(b_{5}), E_{8}(a_{5}), E_{8}(b_{4}), E_{8}(a_{4}), E_{8}(a_{3}), E_{8}(a_{2}), E_{8}(a_{1}), E_{8}.

In the course of the proof of this Proposition we will also verify the assertions of Theorems 3.2 and 3.3 for simple groups of adjoint type.

For type A_{n} this was done in [GHM, Proposition 6.2]. Before we treat the other cases, we first remind the reader of some useful facts. If G is connected reductive with connected centre, then all Levi subgroups L of G also have connected centre (see for example [Ca, Proposition 8.1.4]). This ensures that the classification of unipotent classes for proper Levi subgroups of the given group G is again as described above. Now let G = G_{n} be of classical type, hence G_{n}^{F} = G_{n}(q) is one of SO_{2n+1}(q), P CSp_{2n}(q), P CO_{2n}^{+}(q)^{∘} or P CO_{2n}^{−}(q)^{∘}.

Let first G_{n}(q) = SO_{2n+1}(q) with q odd. The class C(α, β), with α = (α_{1}, …, α_{h}), β = (β_{1}, …, β_{k}), contains elements with Jordan blocks of sizes

α_{1}, α_{1}, …, α_{h}, α_{h}, β_{1}, …, β_{k}

in the natural 2n + 1-dimensional matrix representation of G_{n}. It is then clear that the minimal Levi subgroup of G having non-trivial intersection with C(α, β) is of type

L ≅ ∏_{i=1}^{h} GL_{α_{i}} × G_{n−2m}, where m = ∑_{i=1}^{h} α_{i} .

In particular, u ∈ C(α, β) lies inside a proper split Levi subgroup unless α is empty. Denote the number of Jordan blocks of u of size i by r_{i}. The group of connected components of the centralizer of u ∈ C(α, β) is then isomorphic to

A_{G}(u) ≅ 2^{a(u)}, a(u) := \begin{cases}n(u) − 1, & if n(u) ≥ 1 \\ 0, & if n(u) = 0, \end{cases}

where n(u) counts the number of odd i such that r_{i} > 0 [Ca, 13.1]. For any subset I ⊆ {1, …, h}, G_{n} contains a Levi subgroup

L_{I} ≅ GL_{m} × G_{n−2m}, where m = ∑_{i∈I} α_{i} .

If α contains an even part α_{i}, then for L = L_{{i}} we have a_{L}(u) = a_{G}(u). The same observation applies if the part α_{i} of α coincides with a part α_{j}, i ≠ j, or with some β_{j}. On the other hand it is straightforward that for pairs (α, β) satisfying the conditions in the Proposition, a_{L}(u) is strictly smaller than a_{G}(u) for all proper L_{I}. This proves Proposition 3.6.

We now turn to the proof of Theorems 3.2 and 3.3 for G of type B_{n}. The closure relation on the set of unipotent classes of G_{n} can be described as follows [Ca, 13.4]: Let C, C′ denote two unipotent classes and λ, λ′ the corresponding partitions of 2n + 1 defined by the elementary divisors of elements in C, C′ respectively. Then C ≤ C′ if and only if

∑_{i=1}^{k} λ_{i} ≤ ∑_{i=1}^{k} λ′_{i} for all k ≥ 1.

From the first part of the proof it is easy to see that the minimal Levi subgroup described in Theorem 3.2 has the structure

L_{C} := L_{I(α,β)} = ∏_{i∈I(α,β)} GL_{α_{i}} × G_{n−2m}, m = ∑_{i∈I(α,β)} α_{i},

where

I(α, β) := {i | α_{i} ≡ 0 (mod 2), or α_{i} = α_{j} for some j > i, or α_{i} = β_{j} for some j}.

Note that C ∩ L_{C} contains a cuspidal class of L by Proposition 3.6, namely the class parametrized by

∏_{i∈I(α,β)}(α_{i}) × ((α_{i} | i ∉ I(α, β)), β) ,

which we denote C_{L_{C}}. This class satisfies a_{L_{C}}(C_{L_{C}}) = a_{G}(C). The other classes of L fusing into C are precisely those giving rise to the same elementary divisors as C_{L_{C}}. It is now immediate that no such class different from C_{L_{C}} contains C_{L_{C}} in its closure.

The proof for the other classical groups is rather similar and will be omitted. Just note that in the case of ^{2}D_{n}, the only unipotent classes for which the Levi subgroup constructed for the untwisted group is not F -stable are themselves not F -stable.

For type B_{n} it remains to verify Theorem 3.3 . The cuspidal families of Weyl groups W with automorphism γ are classified in [L1, (8.1)]. The closure relation on the set of unipotent classes of G_{n} was already described above. In particular, if n = d^{2}+d for some d ≥ 1, the cuspidal class C(α, β) with α = (1, 3, 5, …, 2d − 1), β = (2d + 1), lies in the closure of all other cuspidal unipotent classes by the explicit description of these classes in Proposition 3.6. Under the Springer correspondence [Ca, 13.3] this has as image the special character of W (B_{n}) indexed by the pair of partitions ((1, 2, …, d), (1, 2, …, d)). Finally, if W is of type B_{n}, n ≥ 2, there exists a cuspidal family F ⊂ Irr(W) if and only if n = d^{2} +d for some d ≥ 1. In this case the only cuspidal family is the one containing the special representation parametrized by the symbol

(\begin{matrix}0 & 2 & … & 2d \\ 1 & 3 & … & 2d − 1 \end{matrix}) .

But this symbol is just the one belonging to the special character obtained above. The family F contains 2^{2d} irreducible characters, and the group G_{F} attached to this family is elementary abelian of order 2^{d}. But we already saw above that the group of connected components A_{G}(u) of the centralizer of elements from C(α, β) is elementary abelian of order 2^{d}. This completes the proof of Theorem 3.3 for groups of type B_{n}.

For the other classical groups the arguments are similar and hence omitted.

For the exceptional groups we use the list in [Ca, 13.1] giving the classification of unipotent classes via dense orbits. We first remark that classes for which the centralizer does not contain a torus are certainly cuspidal, since every proper Levi subgroup contains a non-trivial central torus.

Next assume that a class C is parametrized by the F -stable proper Levi subgroup L, with corresponding parabolic subgroup P_{L} equal to L. Then C ∩ L meets the regular unipotent class of L. It follows that C ∩ L′ is empty for any proper Levi subgroup L′ of L. It is moreover well-known that A_{L}(u) = 1 for u regular unipotent in an adjoint group G in good characteristic. Thus if A_{G}(u) = 1 for u ∈ C, the class C is not cuspidal, and the class of regular unipotent elements of L meets the requirements of Theorem 3.2. Hence from the lists in loc. cit. we can immediately discard all cases with F -stable L = P_{L} and A_{G}(u) = 1.

The remaining cases are treated by ad hoc arguments. We give the details in the case of E_{6} and ^{2}E_{6}, the other ones being entirely similar. The following table lists those unipotent classes with A_{G}(u) ≠ 1 or L ≠ P_{L}, and for which the centralizer contains a non-trivial torus.

E_{6}  ^{2}E_{6}
class dim C A_{G}(u) L_{C}  cusp.  L_{C}  cusp.
A_{2}  36  S_{2}  D_{4}  D_{4}
A_{2} + A_{1} 32  1 A_{2} + A_{1}  A_{2} + 2A_{1}
D_{4}(a_{1}) 20  S_{3}  E_{6}  ∗  E_{6}  ∗
A_{4}  18  1  A_{4}  A_{5}
A_{4} + A_{1} 16  1 A_{4} + A_{1}  E_{6}  ∗
D_{5}(a_{1}) 14  1  D_{5}  E_{6}  ∗
D_{5}  10  1  D_{5}  E_{6}  ∗

It is easily verified that precisely the classes marked with a ∗ in the respective cases are cuspidal. This already gives Proposition 3.6. For the cuspidal classes, as well as for classes with A_{G}(u) = 1, Theorem 3.2 is vacuously satisfied. Thus there only remains the class of type A_{2} to be considered. From the description of the unipotent classes in classical groups it is immediate that the smallest Levi subgroup satisfying the condition in Theorem 3.2 is of type D_{4}. Listing the unipotent classes of D_{4} one sees that all those having larger dimension than the one of type A_{2} fuse into classes of E_{6} different from A_{2}. This gives the desired classification of cuspidal classes for groups of type E_{6}.

We turn to the proof of Theorem 3.3 for groups of exceptional type. The cuspidal families are listed in [L1, (8.1)]. The Springer correspondence and the closure relation on the set of unipotent classes can be found in [Ca, 13.3 and 13.4]. From these two references it is immediate to check that the first cuspidal class listed in Proposition 3.6 lies in the closure of all other cuspidal classes and under the Springer correspondence belongs to the unique cuspidal family of W. The assertion A_{G}(u) ≅ G_{F} can also be checked from the tables in [loc. cit.]. This completes the proof of the proposition and of the theorems.

Remark 3.7. Note that the statement in Theorem 3.2 can not be strengthened to: C_{L} has larger dimension than any other class of L fusing into C. Namely, for G = SO_{55}, C = ((1, 3, 5, 9, 9), (7)), we obtain L_{C} = GL_{9} × SO_{37}. The class in C ∩ L_{C} with A_{L_{C}} = A_{G} is C_{L_{C}} = (9) × ((1, 3, 5, 9), (7)), whose elements have centralizer dimension 128. But the class (1, 3, 5)×((9, 9), (7)) also fuses into C and has centralizer dimension only 119.

4. On the 2-modular Brauer characters for finite classical groups

4.1. Let G be a connected reductive group defined over 𝔽_{q}, with corresponding Frobenius map F. We assume that the root system of G is a sum of root systems of type A_{n}, B_{n}, C_{n} or D_{n}, and F leaves stable each simple component of G. We assume that q is a power of a good prime for G, that the centre of G is connected and that the results in Section 2 are applicable. (This certainly is the case if G is split and q is a large enough power of a large enough prime.)

For each irreducible character χ of G^{F} let χ^{∗} be the irreducible character such that χ^{∗} = ±D_{G}(χ).

4.2. Let {C_{n} | 1 ≤ n ≤ N} be the F -stable unipotent classes of G, ordered such that dim C_{n} < dim C_{m} implies n < m. For each n, let u_{n,r} (1 ≤ r ≤ d_{n}) be representatives of the G^{F} -orbits on C_{n}^{F}. By [Ge, Proposition 5.6], there exist irreducible characters χ_{n,s} of G^{F} (1 ≤ s ≤ d_{n}) with support C_{n} and such that the matrix of scalar products

(χ_{n,s}^{∗}, Γ_{u_{n,r}}^{G})_{G} (1 ≤ r, s ≤ d_{n})

is the identity matrix. The characters χ_{n,s} lie in geometric conjugacy classes corresponding to semisimple elements of 2-power order in the dual group. Hence they lie in the union B_{1} of unipotent 2-blocks of G^{F} (see [BM]). By [Ge, Theorem 2.5], the characters {χ_{n,s} | all n, s} give rise to an ordinary basic set of Brauer characters in B_{1}. The complete matrix of scalar products of these characters with the various GGGR’s satisfies the following conditions.

(χ_{m,s}^{∗}, Γ_{u_{n,r}}^{G})_{G} = \{ \begin{matrix}1 & , if n = m, r = s \\ 0 & , if n > m or n = m and r ≠ s. \end{matrix}

Ordering the pairs (n, r) lexicographically yields that the corresponding matrix of scalar products has a block lower unitriangular shape, with one block for each unipotent class C_{n}. Moreover, the diagonal blocks are identity matrices of sizes d_{1}, …, d_{N}. This implies that the decomposition matrix of B_{1} has a lower unitriangular shape (cf. [Ge]).

Remark 4.3. Assume that G^{F} has a cuspidal unipotent character χ_{0}. Let C_{n} (where n ∈ {1, …, N}) be the unipotent support of χ_{0} = χ_{0}^{∗}. Then the set of characters {χ_{n,s} | 1 ≤ s ≤ d_{n}} can be chosen in such a way that it contains the character χ_{0}.

Proof. Recall the construction of the set {χ_{n,s}} from [Ge]. There is a canonical map Φ from the set of F -stable special conjugacy classes in the dual group G^{∗} to the set of F -stable unipotent classes in G. The special classes in G^{∗} parametrize the families of the irreducible characters of G^{F} in the various geometric conjugacy classes of characters. Now χ_{0} is a unipotent character. Hence the corresponding special class in G^{∗} is a unipotent class, C^{∗} say. Then, by [L3, Theorem 11.2], the class C_{n} is the image of C^{∗} under the map Φ.

Since C^{∗} is special in G^{∗}, the class C_{n} is special in G. Hence it corresponds to a family F of irreducible characters of the Weyl group of G. Then we are done if we can show that (cf. [Ge], Sections 4, 5)

|A_{G}(u)| = |G_{F}| (u ∈ C_{n}^{F}),

where G_{F} is the finite group associated with the family F by the method of [L1, Chapter 4]. Since χ_{0} is cuspidal, the family F is cuspidal, by [L1, (8.5.6)]. The above assertion is contained in Theorem 3.3(c). This completes the proof.

Theorem 4.4. Assume, in addition to (4.1), that G^{F} has a cuspidal unipotent character χ_{0}. Then the restriction of χ_{0} to the 2-regular elements of G^{F} is an irreducible 2-modular Brauer character.

Proof. The idea of the proof is similar as in the proof of [GHM, Theorem 6.10]. For each n we take an F -stable Levi subgroup L_{n} in some F -stable parabolic subgroup of G of minimal possible dimension such that C_{n} ∩ L_{n}^{F} ≠ 0 and A_{L_{n}}(u_{n}) ≅ A_{G}(u_{n}) for some u_{n} ∈ C_{n} ∩ L_{n}^{F}. If we choose u_{n} to be a split element then, by Theorem 3.2, the class C′_{n} = (u_{n})_{L} is cuspidal in L_{n} and the conditions (2.4a), (2.4b) are satisfied for the pair (L_{n}, u_{n}). The representatives u_{n,r} can be chosen in L_{n} where u_{n} = u_{n,1}. By Proposition 2.3 and Corollary 2.7, we may replace each Γ_{u_{n,r}}^{G} by

γ_{n,r} := R_{L_{n}}^{G}(Γ_{u_{n,r}}^{L_{n}}),

and still obtain a lower unitriangular matrix of scalar products, that is, we have

(χ_{m,s}^{∗}, γ_{n,r})_{G} = \{ \begin{matrix}1 & , if n = m, r = s \\ 0 & , if n > m or n = m and r ≠ s. \end{matrix}

Now let m ∈ {1, …, N} such that C_{m} is the unipotent support of χ_{0}. We claim that

(χ_{0}, γ_{n,r})_{G} = (χ_{0}^{∗}, D_{G}(γ_{n,r}))_{G} = 0, unless m = n.

Assume that the scalar product is non-zero. Let x ∈ G^{F} be unipotent. Now the dual of the Harish-Chandra induced GGGR is zero on x unless C_{n} ≤ (x)_{G} (note that R_{L}^{G} ∘ D_{L} = D_{G} ∘ R_{L}^{G}), and χ_{0} is zero on x unless x lies in C_{m} or in a class of strictly smaller dimension. Since the scalar product is non-zero this implies the existence of an x such that both conditions are satisfied (cf. the proof of Corollary 2.7), that is, we have

C_{n} ≤ (x)_{G} and dim(x)_{G} ≤ C_{m}.

This implies that dim C_{n} ≤ dim C_{m} with equality only if C_{n} = C_{m}. Now we use the assumption that χ_{0} is cuspidal. Since we assume that the above scalar product is non-zero, we conclude that L_{n} cannot be a proper subgroup. Hence L_{n} = G and C_{n} is cuspidal. By Theorem 3.2 this forces C_{m} ≤ C_{n}. Combining with the previous inequality of dimensions we conclude that C_{n} = C_{m}, and our claim is proved.

Since χ_{0} has unipotent support C_{m} we may assume, by Remark 4.3, that χ_{0} is one of the characters χ_{m,s} (1 ≤ s ≤ d_{m}). Hence the scalar product

(χ_{0}, γ_{n,r})_{G} where r ∈ {1, …, d_{n}},

is non-zero (and then equals 1) for precisely one value of r. We conclude that the row corresponding to the character χ_{0} in the above matrix of scalar products contains precisely one non-zero entry, which is 1. Since the characters {χ_{m,s} | all m, s} give rise to an ordinary basic set of Brauer characters in the unipotent 2-blocks of G^{F} (see [Ge, Theorem 2.5]) the result follows (cf. the proof of [GHM, Theorem 6.10]).

The following result was pointed out to us by G. Hiss.

Proposition 4.5. We keep the assumptions in (4.1). Then the irreducible Brauer characters in the principal 2-block of G^{F} are rational valued.

Proof. Recall that the irreducible characters of G^{F} are distributed into geometric classes, and these classes correspond to F -stable conjugacy classes of semisimple elements in a group G^{∗} dual to G. If s ∈ G^{∗}^{F} is semisimple and T^{∗} is an F -stable maximal torus in G^{∗} such that s ∈ T^{∗}, we denote by R_{T^{∗},s}^{G} the associated Deligne-Lusztig character. The geometric conjugacy class corresponding to the conjugacy class of s is the set of all irreducible characters of G^{F} which appear with non-zero multiplicity in R_{T^{∗},s}^{G}, for various T^{∗} containing s.

Now fix an irreducible character χ in B_{1}, the union of unipotent 2blocks of G^{F}. Then it lies in a geometric conjugacy class corresponding to a semisimple element s ∈ G^{∗}^{F} whose order is a power of 2 (see [BM]). We show that χ(g) ∈ ℤ for all g ∈ G^{F} of odd order.

Let K (respectively, K′) be the subfield of ℚ̄_{ℓ} generated over ℚ by the roots of unity whose order is a power of 2 (respectively, whose order is odd). Let T^{∗} be an F -stable maximal tori in G^{∗} such that s ∈ T^{∗}. The character formula for R_{T^{∗},s}^{G} (see [Ca, Theorem 7.2.8]) and the rationality of the Green functions (see [Ca, §7.6]) show that R_{T^{∗},s}^{G}(x) ∈ K for all elements x ∈ G^{F}. By [L1, Main Theorem (4.23)] and the explicit shape of the Fourier transform matrices for groups of classical type, the character χ is uniquely determined by its multiplicities as a constituent in the characters R_{T^{∗},s}^{G}, for various T^{∗} containing s. An argument about algebraically conjugate characters (similar to the one in the proof of [FS, (3A)]) shows that χ(x) ∈ K for all x ∈ G^{F}. On the other hand, let g ∈ G^{F} have odd order. Then, clearly, χ(g) ∈ K′. Combined with the previous result this implies that χ(g) ∈ K∩K′ = ℚ. Since character values are algebraic integers, we conclude that χ(g) ∈ ℤ.

We have shown that the restriction of each irreducible character in B_{1} to the set of elements of odd order is rational valued. Since the decomposition matrix of any block has elementary divisors 1 we conclude that the irreducible Brauer characters in B_{1} are rational valued, too.

4.6. In [GHM2, Theorem 4.12] it was shown that for groups of type A_{n} there exists a strong connection between cuspidal unipotent classes as defined above and cuspidal unipotent 2-modular Brauer characters. It is very likely that such a connection continues to hold for arbitrary classical groups G as above with ℓ = 2. We close this section with some remarks and open questions.

By [Ge] the irreducible Brauer characters φ in the union of 2-blocks B_{1} of G^{F} are in natural bijection with the unipotent conjugacy classes of G^{F}: φ = φ_{(u)} for some unipotent class (u)_{G^{F}}. (This bijection is given by the GGGR’s and the unitriangular shape of the decomposition matrix.)

Assume that u ∈ G^{F} is unipotent such that (u)_{G} is a non-cuspidal class. Then φ_{(u)} has non-zero inner product with the Harish-Chandra induction of a GGGR from a proper Levi subgroup. By [Hi, Lemma 4.1(c)] this implies that φ_{(u)} is a non-cuspidal Brauer character. One can see easily that the converse is not true in general (e.g., in G^{F} = SO_{5}(q)). To overcome this gap, we propose to call a unipotent class (u)_{G^{F}} in G^{F} cuspidal, if it has empty intersection with all F -stable Levi complements in all proper F -stable parabolic subgroups of G. The remarks in (3.1) show that, if (u)_{G^{F}} is cuspidal in G^{F} then (u)_{G} is a cuspidal class of (G, F).

(1) Is it true that φ_{(u)} is cuspidal if and only if (u)_{G^{F}} is cuspidal in this sense ?

In fact we suspect that the precise subdivision of the Brauer characters into 2-modular Harish-Chandra series (as introduced by Hiss [Hi]) should be described by the corresponding unipotent classes, as is the case for type A_{n} [GHM2, Theorem 4.12]. More pecisely, we pose the following question.

(2) Is it true that φ_{(u)} lies in the Harish-Chandra series determined by the pair (L, ψ), where L is minimal such that L^{F} ∩ (u)_{G^{F}} ≠ ∅ and ψ = ψ_{(v)} is a cuspidal Brauer character of L^{F} with v ∈ L^{F} ∩ (u)_{G^{F}} as in Theorem 3.2?

We hope to pursue this issue in a subsequent paper.

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M.G.: Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, 52062 Aachen, Germany

G.M.: IWR der Universität Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany