ON THE EXISTENCE OF A UNIPOTENT

SUPPORT FOR THE IRREDUCIBLE CHARACTERS

OF A FINITE GROUP OF LIE TYPE

Meinolf Geck and Gunter Malle

Abstract. In 1980, Lusztig posed the problem of showing the existence of a unipotent support for the irreducible characters of a finite group of Lie type. This problem was solved by Lusztig [12] in the case where the characteristic of the field over which the group is defined is large enough. The first named author [2] extended this to the case where the characteristic is good. It is the purpose of this paper to remove this condition as well, so that the existence of unipotent supports is established in complete generality.

1. Introduction

Let G be a connected reductive group defined over the finite field with q elements, and let F: G → G be the corresponding Frobenius map. Let Irr(G^{F}) be the set of (complex) irreducible characters of the finite reductive group G^{F}. Let ρ ∈ Irr(G^{F}) and C be an F -stable unipotent class in G. Following Lusztig [12] we define an average value of ρ on C^{F} as follows: let u_{1}, …, u_{r} ∈ C^{F} be representatives for the G^{F} -classes contained in C^{F}. For each j let A_{G}(u_{j}) be the group of components of the centralizer of u_{j} in G. Then our average value is defined as

AV(C, ρ) := ∑_{j=1}^{r}[A_{G}(u_{j}) : A_{G}(u_{j})^{F}] ρ(u_{j}).

It will turn out that this form of average values is more accessible to explicit computations than just averaging over the elements in C^{F} themselves.

The class C is called the unipotent support of ρ if C has maximal dimension such that AV(C, ρ) ≠ 0 and if C is unique among all classes with this property.

Our main result asserts that unipotent supports always exist; furthermore, we give an explicit formula for the average value of an irreducible character on its unipotent support, see Theorem 3.7. Our method will be to reduce the case of small characteristic to that in large characteristic, which is known by Lusztig’s work [12].

It has already been shown in [2, Prop 1.3] that average values as above can be expressed in terms of the Green functions of G^{F}. (In fact, this is essentially the only result from [2] that we will use here.) The exact computation of these Green

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 20C33, 20G40.

The second author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft

functions in turn involves the knowledge of the Springer correspondence for G. In good characteristic, the unipotent classes and the Springer correspondence are independent of the characteristic. In bad characteristic, the situation is rather different. We use the fact, due to Spaltenstein [19], that the partially ordered set of unipotent classes of a group in good characteristic can be embedded in that of a group of the same type but in bad characteristic. Then our main tool is a comparison result about the Springer correspondence in good and bad characteristic, see Theorem 2.1. This is derived from the explicit knowledge of the generalized Springer correspondence, as computed by Lusztig–Spaltenstein [14] and Spaltenstein [20]. This might also be of independent interest.

Our main result shows that the unipotent support of an irreducible character is ‘independent of q’. A precise formulation of this statement can be given using Lusztig’s canonical map from irreducible characters to unipotent classes, whose definition was already given in [7] in good characteristic. Our above comparison result shows that the analogously defined map in bad characteristic shares the same formal properties as the original map in good characteristic, see Proposition 3.6.

The explicit formula for the average value of an irreducible character ρ on its unipotent support C_{ρ} also allows us to give ‘elementary’ characterizations for some invariants attached to ρ. Namely, in the framework of [7, (4.23)–(4.26)], it is possible to give a precise sense to the statement that the degree of ρ is given by a polynomial in q. Thus, there exist well-defined integers n_{ρ} > 0 and a_{ρ} ≥ 0 such that

±n_{ρ}ρ(1) = q^{a_{ρ}} + integral linear combination of higher powers of q.

The formula in Theorem 3.7 recovers n_{ρ} and a_{ρ} from the value AV(C_{ρ}, ρ), the cardinality of A_{G}(u) (for u ∈ C_{ρ}), and the dimension of the variety of Borel subgroups containing u.

In Section 4, we discuss some applications of our results. We show that the notion of unipotent support behaves well with respect to twisted induction of characters from Levi subgroups and induction of unipotent classes as defined by Lusztig– Spaltenstein [13]. More generally, we show that there is a similar compatibility with respect to Lusztig’s Jordan decomposition of characters and Spaltenstein’s generalized induction of classes [19, III.12]. Furthermore, we show that the unipotent support of a cuspidal character must be a cuspidal class in the sense defined by the authors in [3]. Finally, we give some examples and extend our results to the Suzuki and Ree groups.

2. Springer correspondence in good and bad characteristic

In this section we assume that G is any connected reductive group defined over an algebraically closed field k of characteristic p ≥ 0. Let W be the Weyl group of G with respect to a fixed maximal torus T ⊆ G, and S ⊆ W be the set of simple reflections determined by a fixed Borel subgroup B ⊆ G containing T.

Let X_{G} be the partially ordered set of unipotent classes in G. For each C ∈ X_{G} we fix once and for all an element u ∈ C and let A_{G}(u) be the group of components of the centralizer of u in G. We then define N_{G} to be the set of all pairs (C, ψ) where C ∈ X_{G} and ψ ∈ Irr(A_{G}(u)) (for the chosen element u ∈ C). The Springer correspondence gives an injective map i_{G}: Irr(W) → N_{G}. Note that, in the framework of [8], this works without any restriction on G or p.

We want to compare the Springer correspondence for G (or, at least, certain properties of it) with that of another group of the same type as G but possibly defined over a field of characteristic different from p. This can be done in terms of results obtained by Spaltenstein [19], by relating everything to a group in characteristic 0.

Let G′ be a connected reductive group over ℂ and T′ ⊆ G′ be a maximal torus such that the root data of G and G′ with respect to T and T′, respectively, are isomorphic. We also fix a Borel subgroup B′ ⊆ G′ containing T′. Then the pair (W, S) can be canonically identified with the corresponding pair in G′ defined with respect to T′ ⊆ B′.

Let X_{G′} be the set of unipotent classes in G′. For each C ∈ X_{G′} we choose an element u ∈ C and let A_{0}(u) be the group of components of the centralizer of u in G′. We define N_{G′} in an analogous way as the set N_{G} was defined for G. Then, the Springer correspondence for G′ gives an injective map i_{0}: Irr(W) → N_{G′}.

Let π_{G}: X_{G′} → X_{G} be Spaltenstein’s map, see [19, Théorème III.5.2]. This map is uniquely characterized by the following three properties:

• it respects the usual partial orderings on X_{G′} and on X_{G},

• it preserves the dimensions of classes, and

• it satisfies certain compatibility conditions with respect to parabolic sub-

groups in G and G′ containing B and B′, respectively. Moreover, π_{G} is an isomorphism of partially ordered sets onto its image.

If p is a good prime then π_{G} is an isomorphism of partially ordered sets. In this case, we can also canonically identify the groups A_{G′}(u′) and A_{G}(u) (where u′ ∈ C for some C ∈ X_{G′} and u ∈ π_{G}(C)), and then the Springer correspondence for G′ coincides with that for G. This follows from the explicit results obtained by Springer, Hotta–Springer, Shoji, and Alvis–Lusztig (see [16] and the references there). If p is a bad prime the situation may be different.

Using the map π_{G}, we can now state the following result.

Theorem 2.1. Let G′ be a connected reductive group over ℂ and of the same type as G, as above. Let i_{G}: Irr(W) → N_{G} and i_{0}: Irr(W) → N_{G′} be the Springer correspondences for G and G′, respectively. Let E ∈ Irr(W) and i_{G}(E) = (C, ψ) ∈ N_{G}, i_{0}(E) = (C_{0}, ψ_{0}) ∈ N_{G′}. Then the following hold.

(1) If ψ_{0} is the trivial character then C = π_{G}(C_{0}) and ψ also is the trivial character.

(2) We always have C ≤ π_{G}(C_{0}). If p > 0 is a good prime then C = π_{G}(C_{0}).

(Note that the pairs in N_{G} of the form (C, 1), where 1 stands for the trivial character, always lie in the image of i_{G}; see, for example, [2, Remark 3.2].)

We also consider the following, somewhat related result concerning special characters of W. Note that such a character always corresponds to a pair in N_{G} of the form (C, 1) under the Springer correspondence (see [7, 13.1.1] in good characteristic and then use (1) of the above theorem for the general case.)

Proposition 2.2. Let E_{1} ∈ Irr(W) be a special character and i_{G}(E_{1}) = (C, 1) ∈ N_{G}. If E′ ∈ Irr(W) is in the same family as E_{1} and i_{G}(E′) = (C′, ψ′) ∈ N_{G} then C′ ≤ C.

By standard reduction arguments (cf. [8, (10.1)]) it suffices to prove these results in the case where G is simple. Note also that the Springer correspondence is insensitive to the isogeny type of G. Taking into account the remarks preceding Theorem 2.1 we see that, in order to prove this theorem, we only need to consider a group G over a field of bad characteristic p. Once this is established it suffices to prove Proposition 2.2 only for a group G′ of the same type as G but over ℂ.

If G is of type A_{n} then all primes are good and all characters of W are special. Hence we are done in this case. If G is of exceptional type G_{2}, F_{4}, E_{6}, E_{7} or E_{8}, the statements in Theorem 2.1 and Proposition 2.2 are easily read off the tables in [20, 5.4], and those in [7, Chap. 4] concerning families of characters.

In the remainder of this section we will consider a group G of classical type B_{n}, C_{n} or D_{n} over a field of characteristic 2. As before let G′ be the corresponding group of the same type over ℂ. The statements in Theorem 2.1 and Proposition 2.2 will follow by comparing several combinatorial maps defined in [8, 14, 19]. More precisely, we describe combinatorially the map i_{0}: Irr(W) → N_{G′} for G′ and the map i_{G}: Irr(W) → N_{G} for G. Furthermore, let p_{0}: N_{G′} → X_{G′} and p_{G}: N_{G} → X_{G} denote the projection maps onto the first factor. These have natural sections s_{0}: X_{G′} → N_{G′} and s_{G}: X_{G} → N_{G} defined by C ↦ (C, 1). We will sometimes identify elements of X with their image in N. We have to study the resulting diagram of maps

(2.3)

Irr(W) ⟶^{i_{0}} N_{G′}  ⟶^{p_{0}} X_{G′}
↓^{id}  ↓^{π_{G}}
Irr(W) ⟶^{i_{G}} N_{G} ⟶^{p_{G}} X_{G}

A. Some combinatorics.

In order to write down the Springer correspondence and the map π_{G} for groups of classical type we first have to recall the definition of certain combinatorial objects, so-called generalized symbols. These were first introduced by Lusztig and Spaltenstein.

Let n, r, s be positive integers and d ∈ {0, 1}. Following [14, §1] we define X̃_{n,d}^{r,s} to be the set of ordered pairs (A, B) of finite sequences A = (a_{1}, …, a_{m+d}), B = (b_{1}, …, b_{m}) with

\begin{matrix}(1) & a_{i} − a_{i−1} ≥ r + s for 1 < i ≤ m + d, \\ (2) & b_{i} − b_{i−1} ≥ r + s for 1 < i ≤ m, \\ (3) & b_{1} ≥ s, \\ (4) & ∑_{i=1}^{m+d} a_{i} + ∑_{i=1}^{m} b_{i} = n + rm(m + d − 1) + sm(m + d). \end{matrix}

On X̃_{n,d}^{r,s} there is a shift operation X̃_{n,d}^{r,s} → X̃_{n,d}^{r,s} defined by

(A, B) ↦ ((0, a_{1} + r + s, …, a_{m+d} + r + s), (s, b_{1} + r + s, …, b_{m} + r + s)).

The quotient of X̃_{n,d}^{r,s} by this shift is denoted by X_{n,d}^{r,s}, its elements are called symbols and will also be denoted by representatives (A, B). The set X_{0,d}^{r,s} consists of a single element Λ_{0,d}^{r,s}: if d = 0 then Λ_{0,0}^{r,s} := (−, −), if d = 1 then Λ_{0,1}^{r,s} := ((0), −). There exists an addition

X_{n,d}^{r,s} × X_{n′,d}^{r′,s′} ⟶ X_{n+n′,d}^{r+r′,s+s′}

defined by component-wise addition of representatives of the same length. An element Λ = (A, B) ∈ X_{n,d}^{r,s} is called distinguished if a_{1} ≤ b_{1} ≤ a_{2} ≤ … ≤ b_{m} (and b_{m} ≤ a_{m+1} if d = 1). The subset of X_{n,d}^{r,s} of distinguished symbols is denoted by D_{n,d}^{r,s}. Two (classes of) symbols are called similar if the multisets of the entries of suitable representatives coincide. Thus for any symbol Λ ∈ X_{n,d}^{r,s} there exists a unique distinguished symbol D(Λ) in X_{n,d}^{r,s} which is similar to Λ.

Finally, if s = 0, we denote by Y_{n,0}^{r} the quotient of X_{n,0}^{r,0} by the operation of swapping the two rows of a symbol, and we write Y_{n,1}^{r} for X_{n,1}^{r,0}.

We introduce a sorting procedure R^{r,s}: X_{n,d}^{r,s} → X_{n,d}^{r,s} on symbols Λ ∈ X_{n,d}^{r,s} as follows. If Λ = (A, B) write

(c_{1}, c_{2}, c_{3}, …) := (a_{1}, b_{1}, a_{2}, b_{2}, …).

If c_{i} ≤ c_{i+1} for all i then R^{r,s}(Λ) := Λ. Otherwise, let j be the minimal index with c_{j} > c_{j+1}, and let l be maximal such that c_{j+2k} > c_{j+2k+1} for all 0 ≤ k ≤ l. Define R^{r,s}(Λ) := (c̃_{1}, c̃_{2}, …) where

c̃_{i} := \{ \begin{matrix}c_{i+1} & if i = j + 2k, 0 ≤ k ≤ l, \\ c_{i−1} & if i = j + 2k + 1, 0 ≤ k ≤ l, \\ c_{i} & otherwise. \end{matrix}

Since Λ ∈ X_{n,d}^{r,s} we have c_{i} ≤ c_{i+2} − r − s for all i. Now

\begin{matrix}c̃_{j} = c_{j+1} ≥ c_{j−1} + r + s ≥ c_{j−2} + r + s = c̃_{j−2} + r + s, \\ c̃_{j+1} = c_{j} > c_{j+1} ≥ c_{j−1} + r + s = c̃_{j−1} + r + s, \end{matrix}

by the choice of j, and

\begin{matrix}c̃_{j+2l} = c_{j+2l+1} < c_{j+2l} ≤ c_{j+2l+2} − r − s = c̃_{j+2l+2} − r − s, \\ c̃_{j+2l+1} = c_{j+2l} ≤ c_{j+2l+2} − r − s ≤ c_{j+2l+3} − r − s = c̃_{j+2l+3} − r − s, \end{matrix}

by the choice of l, so R^{r,s}(Λ) ∈ X_{n,d}^{r,s} as well, and it is similar to Λ. Moreover the smallest index i such that c̃_{i} > c̃_{i+1} is strictly larger than j. Thus there exists an m ≥ 0 such that (R^{r,s})^{m}(Λ) ∈ D_{n,d}^{r,s} is distinguished. We also write (R^{r,s})^{∞}(Λ) for this distinguished symbol (R^{r,s})^{m}(Λ).

For Λ′ ∈ X_{n,d}^{r+r′,s+s′} we define R^{r,s}(Λ′) := R^{r,s}(Λ′ − Λ_{0,d}^{r′,s′}) + Λ_{0,d}^{r′,s′}, so its entries (c̃′_{1}, c̃′_{2}, …) are obtained from the entries (c′_{1}, c′_{2}, …) of Λ′ via

c̃′_{j+2k} = c′_{j+2k+1} − s′, c̃′_{j+2k+1} = c′_{j+2k} + s′ if j is odd,
c̃′_{j+2k} = c′_{j+2k+1} − r′, c̃′_{j+2k+1} = c′_{j+2k} + r′ if j is even,

for 0 ≤ k ≤ l, and c̃′_{i} = c′_{i} otherwise. Clearly, both operations carry over to Y_{n,d}^{r} and Y_{n,d}^{r+r′}.

B. The symplectic groups.

Assume that G, G′ are of adjoint type C_{n}. Let W_{n} be the Weyl group of G of type C_{n}. The irreducible characters of W_{n} can be parametrized by pairs of partitions (α, β) of n, i.e., α = (α_{1} ≤ α_{2} ≤ … ≤ α_{m′}), β = (β_{1} ≤ β_{2} ≤ … ≤ β_{m}) with ∑ α_{i} + ∑ β_{i} = n. By adding zeros we may normalize (α, β) so that m′ = m + 1, so this yields a natural parametrization

Irr(W_{n}) ⟶^{∼} X_{n,1}^{0,0}, (α, β) ↦ (α, β).

Via the addition defined above we also get bijections

ψ_{G′}: Irr(W_{n}) ≅ X_{n,1}^{0,0} ⟶^{∼} X_{n,1}^{1,1}, (α, β) ↦ (α, β) + Λ_{0,1}^{1,1},

and

(2.4)  ψ_{G}: Irr(W_{n}) ≅ X_{n,1}^{0,0} ⟶^{∼} X_{n,1}^{2,2}, (α, β) ↦ (α, β) + Λ_{0,1}^{2,2}.

So in particular

(2.5)  ψ_{G}((α, β)) = ψ_{G′}((α, β)) + Λ_{0,1}^{1,1}.

We will henceforth replace Irr(W_{n}) in the above diagram (2.3) by X_{n,1}^{1,1} respectively X_{n,1}^{2,2} (see also [8, 12.2.4] and [14, 1.3]).

We now define the inverse of i_{0} on X_{G′} ⊂ N_{G′}. The unipotent classes of G′ are parametrized by partitions λ of 2n such that any odd part occurs an even number of times. Let λ be such a partition. By adding zeros we may arrange so that λ_{1} = 0 and the number of parts of λ is odd. It can easily be checked that the map described in [8, 11.6] can be given as follows. Partition the sequence (λ_{1}, λ_{2}, …, λ_{r}) into blocks of lengths one or two such that all even λ_{i} lie in a block of length 1 and the odd λ_{i} lie in blocks of length 2. Define a sequence (c′_{1}, …, c′_{r}) by

(2.6)

c′_{i} := λ_{i}/2 + (i − 1) if {λ_{i}} is a block,
\begin{matrix}c′_{i} := (λ_{i} + 1)/2 + i − 1, \\ c′_{i+1} := (λ_{i+1} + 1)/2 + i − 1 = c′_{i} \end{matrix} \} if \{λ_{i}, λ_{i+1}\} lie in one block.

Then by [8, Thm. 12.3] i_{0}^{−1}((λ, 1)) is the unique distinguished symbol in X_{n,1}^{1,1} with entries c′_{1}, …, c′_{r}. Conversely, it can easily be seen that every distinguished symbol can be obtained this way, so i_{0} restricts to a bijection i_{0} : D_{n,1}^{1,1} →^{∼} X_{G′}. Furthermore, the image i_{0}^{−1}((λ, ψ)) for arbitrary ψ is a symbol similar to i_{0}^{−1}((λ, 1)).

The unipotent classes of G are parametrized by pairs (λ, ϵ) where λ is as before and ϵ: {λ_{1}, …, λ_{r}} → {0, 1, ω} is a certain sequence of signs, such that ϵ_{λ_{i}} = ω on parts λ_{i} of odd length (see [19, I.2.5, I.2.6]). The generalized Springer correspondence i_{G} is described in [14, 2.1]. Let (λ, ϵ) be the label of a unipotent class. Assume again that λ_{1} = 0 and the number r of parts of λ is odd. Furthermore, partition the parts of λ into blocks of lengths 1 or 2 such that λ_{i} lies in a block of length 1 if and only if ϵ_{λ_{i}} = 1. Then integers c_{i} are defined as follows:

(2.7)

c_{i} := λ_{i}/2 + 2(i − 1) if {λ_{i}} is a block,
\begin{matrix}c_{i} := (λ_{i} + 1)/2 + 2(i − 1), \\ c_{i+1} := (λ_{i} − 1)/2 + 2i = c_{i} + 1 \end{matrix} \} if \{λ_{i}, λ_{i+1}\} is a block, ϵ_{λ_{i}} ≠ 0,

\begin{matrix}c_{i} := (λ_{i} + 2)/2 + 2(i − 1), \\ c_{i+1} := (λ_{i} − 2)/2 + 2i = c_{i} \end{matrix} \} if \{λ_{i}, λ_{i+1}\} is a block, ϵ_{λ_{i}} = 0.

The image of (λ, ϵ) under i_{G}^{−1} is then the unique distinguished symbol with entries c_{1}, …, c_{r}. Thus again i_{G} restricts to a bijection i_{G}: D_{n,1}^{2,2} →^{∼} X_{G}.

Finally, according to [19, III.6.1] the map π_{G} is given by

π_{G}: N_{G′} ⟶ N_{G}, λ ↦ (λ, ϵ_{1}),

where ϵ_{1} is defined by the property that no entry of ϵ_{1} is equal to 0; in particular (λ, ϵ_{1}) is the class of largest dimension among all classes parametrized by pairs (λ, ϵ) (loc. cit.).

We have now given combinatorial descriptions for all maps in (2.3).

Proposition 2.8. Theorem 2.1 and Proposition 2.2 hold for groups of type Sp_{2n}.

Proof. By what we introduced above, Theorem 2.1(1) is equivalent to the statement that

π_{G} ∘ i_{0} = i_{G} ∘ π_{1,1}^{2,2} on D_{n,1}^{1,1},

where π_{1,1}^{2,2}: D_{n,1}^{1,1} → D_{n,1}^{2,2} is the bijection Λ ↦ Λ + Λ_{0,1}^{1,1}. But by (2.5) this amounts to showing c_{i} = c′_{i} +(i−1) for the integers defined in (2.6) and (2.7), which is clear. (Note that the third case in (2.7) can not occur by the definition of π_{G}.)

For part (2) of Theorem 2.1 let E ∈ Irr(W_{n}) and c_{1}, c_{2}, …, c_{r} be the entries of Λ := ψ_{G}(E) ∈ X_{n,1}^{2,2}. Then the entries of Λ′ := ψ_{G′}(E) ∈ X_{n,1}^{1,1} are (c_{1}, c_{2} − 1, …, c_{r} −r +1). We apply the sorting procedure R^{1,1}. Namely, R^{1,1}(Λ′) is similar to Λ′, thus the unipotent class corresponding to it under the generalized Springer map i_{0} remains the same. On the other hand, the class C̃ := i_{G}(R^{1,1}(Λ)) is larger than the class C := i_{G}(Λ) by the description of the partial order on X_{G} in [19, I.2.10]: if we had c_{j} = c_{j+1} then the label for C̃ is obtained from the one for C by changing ϵ_{λ_{i}} from 0 to 1, and otherwise the corresponding partition is replaced by a strictly smaller one. Thus we have

i_{0}((R^{1,1})^{∞}(Λ′)) = i_{0}(Λ′), i_{G}((R^{1,1})^{∞}(Λ)) ≥ i_{G}(Λ),

and π_{G}(i_{0}((R^{1,1})^{∞}(Λ′))) = i_{G}((R^{1,1})^{∞}(Λ)) by the first part, so i_{G}(Λ) ≤ π_{G}(i_{0}(Λ′)), which proves Theorem 2.1(2).

We now prove Proposition 2.2 (for G′). Let now E′ = (α, β) ∈ Irr(W_{n}) and Λ′ := (α, β) + Λ_{0,1}^{1,0} ∈ X_{n,1}^{1,0} be the corresponding sequences of β-numbers. By [7, 4.5] it is known that E′ is special if and only if Λ′ is distinguished, and moreover two characters in the same family have similar symbols. Thus (R^{1,0})^{∞}(Λ′) is the symbol for the special character in the family of E′. Let Λ = ψ_{G′}(E′) = (α, β)+Λ_{0,1}^{1,1} ∈ X_{n,1}^{1,1} be the symbol for the Springer correspondent of E′. It is easily checked using [19, I.2.10] that i_{0}(R^{1,0}(Λ)) ≥ i_{0}(Λ) in the partial order relation. Thus repeated application of R^{1,0} gives the assertion of Proposition 2.2. □

Remark. It was pointed out to us by Frank Lübeck that the description of the generalized Springer correspondence in [8, 14] for G a symplectic group is not quite correct. Nevertheless, according to Shoji it remains true that the images of (u, ψ), (u′, ψ′) are similar symbols if and only if u = u′. Moreover the image of (u, 1) is as described above (see [18, Remark 5.8] for the case of odd characteristic). These are the only facts we need in our argument.

C. The odd-dimensional orthogonal groups.

Assume now that G, G′ are of adjoint type B_{n}. Then there exists a bijective isogeny from G to a simple group of adjoint type C_{n}. To check Theorem 2.1 for SO_{2n+1} we will compare the classes and the Springer correspondence in good characteristic with those of Sp_{2n} in characteristic 2. As above, let W_{n} be the Weyl group of type C_{n}. We have to describe the maps in diagram (2.3). First, Irr(W_{n}) is identified with Y_{n,1}^{2} via

(2.9)  ψ_{G′}: Irr(W_{n}) ≅ X_{n,1}^{0,0} ⟶^{∼} Y_{n,1}^{2},  (α, β) ↦ (α, β) + Λ_{0,1}^{2,0},

where again (α, β) is normalized such that α has one part more than β. We will henceforth replace Irr(W_{n}) by Y_{n,1}^{2} (respectively X_{n,1}^{2,2} via (2.4)).

The unipotent classes of G′ are parametrized by partitions λ of 2n + 1 such that any even part occurs an even number of times. Let λ be such a partition. Note that λ has an odd number of parts. It can easily be checked that the map described in [8, 11.7] can be given as follows. Partition the sequence (λ_{1}, λ_{2}, …, λ_{r}) into blocks of lengths one or two such that all odd λ_{i} lie in a block of length 1 and the even λ_{i} lie in blocks of length 2. Then define

(2.10)

c′_{i} := (λ_{i} − 1)/2 + i − 1 if {λ_{i}} is a block,
\begin{matrix}c′_{i} := λ_{i}/2 + i − 1, \\ c′_{i+1} := λ_{i+1}/2 + i − 1 = c′_{i} \end{matrix} \} if \{λ_{i}, λ_{i+1}\} lie in one block.

Then by [8, Thm. 13.3] i_{0}^{−1}((λ, 1)) is the unique distinguished symbol in Y_{n,1}^{2} with entries c′_{1}, …, c′_{r}. It is easily seen that i_{0} restricts to a bijection i_{0}: D_{n,1}^{2} →^{∼} X_{G′}. Furthermore, i_{0}^{−1}((λ, ψ)) is similar to i_{0}^{−1}((λ, 1)) for any ψ.

We have already given the generalized Springer correspondence for Sp_{2n} in Section B. It remains to describe the map π_{G} from the partially ordered set of unipotent classes X_{G′} of G′ to the partially ordered set of unipotent classes of Sp_{2n} in characteristic 2. Let λ′ be the label of a unipotent class of G′. Then π_{G}(λ′) has the label (λ, ϵ) where

(2.11)  λ_{i} := \{ \begin{matrix}λ′_{i} − 1 & if λ′_{i} and i are odd, λ′_{i−1} < λ′_{i}, \\ λ′_{i} + 1 & if λ′_{i} is odd, i is even, λ′_{i} < λ′_{i+1}, \\ λ′_{i} & else, \end{matrix}

and ϵ_{λ_{i}} = 0 if and only if both λ′_{i} and i are even (see [19, III.8.2]). We have now given combinatorial descriptions for all maps in (2.3).

Proposition 2.12. Theorem 2.1 and Proposition 2.2 hold for type SO_{2n+1}.

Proof. We first consider the statements in Theorem 2.1. Note that if ψ_{G′}(E) is distinguished for some E ∈ Irr(W_{n}), then also ψ_{G}(E) is distinguished by (2.4) and (2.9). Let (c′_{1}, …, c′_{r}) be the sequence of entries of the distinguished symbol ψ_{G′}(E), where E is labeled by (α, β). We subdivide the c′_{i} into blocks of lengths 1 or 2, such that {c′_{i}} is a block if and only if c′_{i} ≠ c′_{i−1}, c′_{i} ≠ c′_{i+1}, and c′_{i} ≠ c′_{i+1} − 1 if i is even. Then by (2.10) the parts of the partition λ′ = i_{0}(E) are seen to be given by the formula

(2.13) λ′_{i} := 2α_{k} + 1 if i = 2k − 1, β_{k−1} < α_{k} < β_{k},

λ′_{i} := 2β_{k} − 1 if i = 2k, α_{k} < β_{k} < α_{k+1} + 1,

\begin{matrix}λ′_{i} := 2α_{k}, \\ λ′_{i+1} := 2β_{k} = λ′_{i} \end{matrix} \} if i = 2k − 1, β_{k} = α_{k},

\begin{matrix}λ′_{i} := 2β_{k} − 1, \\ λ′_{i+1} := 2α_{k+1} + 1 = λ′_{i} \end{matrix} \} if i = 2k, β_{k} = α_{k+1} + 1,

\begin{matrix}λ′_{i} := 2β_{k} − 2, \\ λ′_{i+1} := 2α_{k+1} + 2 = λ′_{i} \end{matrix} \} if i = 2k, β_{k} = α_{k+1} + 2.

(The fact that ψ_{G′}(E) is distinguished shows that the above covers all cases that can arise.) On the other hand, the generalized Springer correspondence for Sp_{2n} described in (2.7) yields

(2.14) λ_{i} := 2α_{k} if i = 2k − 1, β_{k−1} < α_{k} < β_{k},

λ_{i} := 2β_{k} if i = 2k, α_{k} < β_{k} < α_{k+1} + 1,

\begin{matrix}λ_{i} := 2α_{k}, \\ λ_{i+1} := 2β_{k} = λ_{i} \end{matrix} \} if i = 2k − 1, β_{k} = α_{k},

\begin{matrix}λ_{i} := 2β_{k} − 1, \\ λ_{i+1} := 2α_{k+1} + 1 = λ_{i} \end{matrix} \} if i = 2k, β_{k} = α_{k+1} + 1,

\begin{matrix}λ_{i} := 2β_{k} − 2, \\ λ_{i+1} := 2α_{k+1} + 2 = λ_{i} \end{matrix} \} if i = 2k, β_{k} = α_{k+1} + 2,

and ϵ_{λ_{i}} = 0 only in the last case, for the image (λ, ϵ) = i_{G}(E). A comparison with (2.11) now shows that λ in (2.14) can be obtained from λ′ in (2.13) via π_{G}, which is assertion (1).

For (2) let E ∈ Irr(W_{n}) with label (α, β). Similar to the proof of Proposition 2.8 we apply R^{2,0} to change E in such a way that the class i_{0}(E) remains the same, and the class of i_{G}(E) increases. It only remains to check that i_{G}(R^{2,0}(Λ)) ≥ i_{G}(Λ) for Λ := ψ_{G}(E), which follows from (2.14) by [19, I.2.10]. This proves (2), and hence completes the proof of Theorem 2.1.

Proposition 2.2 is proved precisely as in the case of Sp_{2n} using the map R^{1,0} and the characterization of families and special characters in [7, 4.5]. □

D. The even-dimensional orthogonal groups.

Assume that G, G′ are of type SO_{2n}. We write Ĝ, Ĝ′ for the extension of G, G′ by the graph automorphism of order 2, i.e., for the general orthogonal group GO_{2n}. In [8, 14] the generalized Springer correspondence is only given for orbits under this graph-automorphism. So here let N_{G′}, N_{G} be the set of pairs (C, ψ) consisting of a unipotent element of G′ (resp. G) and a character of the group of connected components of its centralizer, up to conjugation in Ĝ′, Ĝ, and X_{G′}, X_{G} the partially ordered sets of unipotent conjugacy classes of Ĝ′, Ĝ contained in G′, G respectively.

Let W′_{n} be the Weyl group of type D_{n} considered as a subgroup of W_{n}, Irr(W_{n})′ the set of orbits of irreducible characters of W′_{n} under the action of W_{n}. Then Irr(W_{n})′ is parametrized by unordered pairs of partitions (α, β) of n. Following [8, 13.2.5] we first identify Irr(W_{n})′ with Y_{n,0}^{2} via

(2.15)  ψ_{G′}: Irr(W_{n})′ ≅ Y_{n,0}^{0} ⟶^{∼} Y_{n,0}^{2},  (α, β) ↦ (α, β) + Λ_{0,0}^{2,0},

where (α, β) is normalized such that α and β have the same number of parts. In characteristic 2, we define

(2.16)  ψ_{G}: Irr(W_{n})′ ≅ Y_{n,0}^{0} ⟶^{∼} Y_{n,0}^{4},  (α, β) ↦ (α, β) + Λ_{0,0}^{4,0},

where (α, β) is normalized as before (see [14, 3.3]). We will henceforth replace Irr(W_{n})′ by Y_{n,0}^{2} respectively Y_{n,0}^{4}.

The unipotent classes of Ĝ′ contained in G′ are parametrized by partitions λ of 2n such that any even part occurs an even number of times [19, I.2.5]. Let λ be such a partition. Note that λ has an even number of parts. Partition the sequence (λ_{1}, λ_{2}, …, λ_{r}) into blocks of lengths one or two such that all odd λ_{i} lie in a block of length 1 and the even λ_{i} lie in blocks of length 2. It can easily be checked that the map described in [8, 11.7] is again given by (2.10). Thus by [8, Thm. 13.3] i_{0}^{−1}((λ, 1)) is the unique distinguished symbol in Y_{n,0}^{2} with entries c′_{1}, …, c′_{r}. So i_{0} restricts to a bijection i_{0}: D_{n,0}^{2} →^{∼} X_{G′}.

The unipotent classes of Ĝ contained in G are parametrized by pairs (λ, ϵ) where λ is a partition of 2n in which every odd part occurs an even number of times, and ϵ: {λ_{1}, …, λ_{r}} → {0, 1, ω} is a certain sequence of signs, which is equal to ω on parts of odd length (see [19, I.2.6]). The generalized Springer correspondence is described in [14, 3.1]. Let (λ, ϵ) be the parameter of a unipotent class. Assume that the number r of parts of λ is even. Furthermore, partition the entries of λ into blocks of lengths 1 or 2 such that λ_{i} lies in a block of length 1 if and only if ϵ_{λ_{i}} = 1. Then integers c_{i} are defined as follows:

(2.17)

c_{i} := (λ_{i} − 2)/2 + 2(i − 1) if {λ_{i}} is a block,
\begin{matrix}c_{i} := (λ_{i} − 1)/2 + 2(i − 1), \\ c_{i+1} := (λ_{i} − 3)/2 + 2i = c_{i} + 1 \end{matrix} \} if \{λ_{i}, λ_{i+1}\} is a block, ϵ_{λ_{i}} ≠ 0,

\begin{matrix}c_{i} := λ_{i}/2 + 2(i − 1), \\ c_{i+1} := (λ_{i} − 4)/2 + 2i = c_{i} \end{matrix} \} if \{λ_{i}, λ_{i+1}\} is a block, ϵ_{λ_{i}} = 0.

The image of (λ, ϵ) under i_{G}^{−1} is then the unique distinguished symbol with entries c_{1}, …, c_{r}. Thus again i_{G} restricts to a bijection i_{G}: D_{n,0}^{4} →^{∼} X_{G}.

Finally, the map π_{G} from the partially ordered set of unipotent classes X_{G′} of Ĝ′ in G′ to the partially ordered set of unipotent classes of Ĝ contained in G can be described as follows. Let λ′ be the label of a unipotent class in G′. Then π_{G}(λ′) has the label (λ, ϵ) where

(2.18)  λ_{i} := \{ \begin{matrix}λ′_{i} − 1 & if λ′_{i} is odd, i is even, λ′_{i−1} < λ′_{i}, \\ λ′_{i} + 1 & if λ′_{i} and i are odd, λ′_{i} < λ′_{i+1}, \\ λ′_{i} & else, \end{matrix}

and ϵ_{λ_{i}} = 0 if and only if λ′_{i} is even and i is odd (see [19, III.7.2]). We have now given combinatorial descriptions for all maps in (2.3).

Proposition 2.19. Theorem 2.1 and Proposition 2.2 hold for groups of type SO_{2n}.

Proof. We first check the assertions for the sets of unipotent classes of Ĝ, Ĝ′. We start with the statements in Theorem 2.1. By the description of ψ_{G′} and ψ_{G} in (2.15) and (2.16), if ψ_{G′}(E) is distinguished for some E ∈ Irr(W_{n})′, then so is ψ_{G}(E). Denoting the entries of ψ_{G′}(E) by (c′_{1}, …, c′_{r}) we see from (2.10) as in the previous proof that the parts of the partition λ′ = i_{0}(E) are given by

(2.20) λ′_{i} := 2α_{k} + 1 if i = 2k − 1, β_{k−1} < α_{k} < β_{k} − 1,

λ′_{i} := 2β_{k} − 1 if i = 2k, α_{k} < β_{k} < α_{k+1} + 2,

\begin{matrix}λ′_{i} := 2β_{k} − 2, \\ λ′_{i+1} := 2α_{k+1} + 2 = λ′_{i} \end{matrix} \} if i = 2k, β_{k} = α_{k+1} + 2,

\begin{matrix}λ′_{i} := 2α_{k} + 1, \\ λ′_{i+1} := 2β_{k} − 1 = λ′_{i} \end{matrix} \} if i = 2k − 1, β_{k} = α_{k} + 1,

\begin{matrix}λ′_{i} := 2α_{k}, \\ λ′_{i+1} := 2β_{k} = λ′_{i} \end{matrix} \} if i = 2k − 1, β_{k} = α_{k}.

For (λ, ϵ) = i_{G}(E) we obtain from (2.17)

(2.21) λ_{i} := 2α_{k} + 2 if i = 2k − 1, β_{k−1} < α_{k} < β_{k} − 1,

λ_{i} := 2β_{k} − 2 if i = 2k, α_{k} < β_{k} < α_{k+1} + 2,

\begin{matrix}λ_{i} := 2β_{k} − 2, \\ λ_{i+1} := 2α_{k+1} + 2 = λ_{i} \end{matrix} \} if i = 2k, β_{k} = α_{k+1} + 2,

\begin{matrix}λ_{i} := 2α_{k} + 1, \\ λ_{i+1} := 2β_{k} − 1 = λ_{i} \end{matrix} \} if i = 2k − 1, β_{k} = α_{k} + 1,

\begin{matrix}λ_{i} := 2α_{k}, \\ λ_{i+1} := 2β_{k} = λ_{i} \end{matrix} \} if i = 2k − 1, β_{k} = α_{k},

and ϵ_{λ_{i}} = 0 only in the last case. The description of π_{G} in (2.18) now allows to conclude that π_{G}(i_{0}(E)) = i_{G}(E) in the cases considered. This proves part (1).

For part (2) the arguments are essentially the same as in the previous proof. Let E ∈ Irr(W_{n})′ with label (α, β). We apply R^{2,0} repeatedly to transform both Λ′ := ψ_{G′}(E) and Λ := ψ_{G}(E) into distinguished symbols. Again, it is routine to check that the class corresponding to R^{2,0}(Λ) is bigger than the class of Λ in the partial ordering [19, I.2.10]. This proves (2).

To prove Proposition 2.2 for G′ we can use R^{1,0} and the description of families and special characters in [7, 4.6 and 4.7].

Finally, we descend to the unipotent classes of G, G′. If a class of Ĝ′ splits into two G′-classes, these have the same dimension, so the validity of 2.1(2) is not affected by this splitting. As for 2.1(1), it suffices to note that all splitting classes are induced in the sense of Lusztig and Spaltenstein. Thus by the property [20, 4.1] of the generalized Springer correspondence the Springer correspondent can be determined by Macdonald’s j-induction in the Weyl group. In particular, it does not depend on the characteristic. On the other hand, the image of the map π_{G} is determined by property [19, III.5.2(1)] by fusion of classes. But now by [19, II.7.6] the induced and the fused class are related in the same way, regardless of the characteristic. This shows that in fact property 2.1(1) also holds for the splitting unipotent classes of G′, G.

By [7, 4.6 and 4.7] those characters in Irr(W′_{n}) which do not extend to W_{n} all lie in one-element families. Thus Proposition 2.2 is trivially satisfied for these characters. This completes the descend to the connected group G′. □

E. A combinatorial formula for dim B_{u}.

Let Λ ∈ X_{n,d}^{r,s} for d ∈ {0, 1} be a symbol. We define

(2.22) a(Λ) := \{ \begin{matrix}∑_{\{c,c′\}} min(c, c′) − r\binom{m}{2} \frac{4m+1}{3} − s\binom{m+1}{2} \frac{4m−1}{3}, & if d = 1, \\ ∑_{\{c,c′\}} min(c, c′) − r\binom{m}{2} \frac{4m−5}{3} − s\binom{m}{2} \frac{4m+1}{3}, & if d = 0, \end{matrix}

where the sum runs over all unordered pairs of entries of Λ and m ≥ 0 is the number of coefficients in the second component of Λ. First note that a(Λ) is indeed well-defined on symbols. Furthermore, it is clear that a is constant on similarity classes of symbols.

By the results of Lusztig and Spaltenstein quoted above, the sets of unipotent conjugacy classes of classical groups can be naturally parametrized by distinguished symbols in X_{n,1}^{1,1}, X_{n,1}^{2,2}, Y_{n,0}^{2}, Y_{n,0}^{4}, according to the cases. It thus makes sense to speak of the symbol Λ_{u} associated to a unipotent class C ∋ u of a classical group.

Proposition 2.23. Let C be a unipotent conjugacy class of a classical group, u ∈ C and Λ_{u} be the associated distinguished symbol. Then

dim B_{u} = a(Λ_{u}).

Proof. This is a combinatorial exercise, using Spaltenstein’s formulae for dim B_{u}. We give some details for the case of Sp_{2n} in odd characteristic, where the corresponding distinguished symbols lie in X_{n,1}^{1,1}. First, the class of regular unipotent elements has dim B_{u} = 0 and the associated symbol is ((n), −), so the formula is correct in this case. Furthermore, ((n), −) is equivalent to the symbol Λ := ((0, 2, 4, …, 2n, 3n), (1, 3, 5, …, 2n + 1)), and by our above remark the avalue is constant on equivalence classes. Now every distinguished symbol in X_{n,1}^{1,1} can be obtained from Λ by a sequence of steps in each of which only two adjacent entries are changed by plus and minus 1. On the other hand, for λ parametrizing a unipotent class of Sp_{2n}, the value of dim B_{u} is given by

dim B_{u} = \frac{1}{4} ∑_{i≥1}(λ_{2i−1}^{∗}^{2} + λ_{2i}^{∗}(λ_{2i}^{∗} − 2)),

where λ^{∗} denotes the partition dual to λ [19, II.6.3]. It remains to verify that the elementary steps described above give the same change in both formulae, using the connection (2.6) between λ and the associated symbol. We omit further details. □

Note that the sets X_{n,1}^{1,0} and Y_{n,0}^{1} have been used by Lusztig to parametrize unipotent characters of groups of classical type. In that context the value a(Λ) gives the precise order of vanishing of the character degree, considered as a polynomial in q, at q = 0.

3. A formula for the average value

We assume from now on that p > 0 and that G is defined over the finite subfield of k with q elements, with corresponding Frobenius map F: G → G. We also assume that the chosen maximal torus T and the chosen Borel subgroup B ⊇ T are F -stable.

Let ρ ∈ Irr(G^{F}) and ρ_{unif} be the uniform projection of ρ. By [2, 1.3], we have AV(C, ρ) = AV(C, ρ_{unif}) for all F -stable unipotent classes C in G. The values of ρ_{unif} at unipotent elements can be expressed in terms of Green functions, as follows.

For any w ∈ W let T_{w} ⊆ G be an F -stable maximal torus obtained by twisting T with w and let Q_{w} be the Green function corresponding to T_{w}. For E ∈ Irr(W)^{F} let Q_{E} = |W |^{−1} ∑_{w∈W} Trace(F w, E)Q_{w} (cf. [16, (5.5)]), defined with respect to the ‘preferred extension’ [9, (17.2)] of E to a module for the semi-direct product of W with the cyclic subgroup of Aut(W) generated by F. Since the Green functions are linearly independent there exist unique constants b(ρ, E) ∈ ℂ such that

ρ_{unif}(g) = ∑_{E∈Irr(W)^{F}} b(ρ, E)Q_{E}(g) for all unipotent g ∈ G^{F} .

Our strategy will be to get hold on the coefficients b(ρ, E) and the average values of the corresponding Green function Q_{E}. From now until (3.6) we will assume that the center of G is connected. In the main result, Theorem 3.7, this condition will be removed using regular embeddings.

A. Green functions and Springer correspondence.

The Frobenius map F acts on the set N_{G}. With each i = (C, ψ) ∈ N_{G}^{F} we can associate a class function Y_{i} on G^{F} defined as follows. Since i is F -stable we have F (C) = C; furthermore, if V is an A_{G}(u)-module affording the character ψ there exists an automorphism α_{V}: V → V of finite order such that α_{V} ∘ y = F (y) ∘ α_{V} for all y ∈ A_{G}(u). Then Y_{i}(g) = 0 for g ∈ G^{F} \ C^{F} and Y_{i}(u_{y}) = Trace(α_{V} ∘ y, V) for y ∈ A_{G}(u). Note that this determines Y_{i} only up to multiplication by a root of unity. The functions Y_{i} (for the various i ∈ N_{G}^{F}) form a basis of the space of class functions on G^{F} supported on unipotent elements (cf. [10, (24.2.7)]).

The Springer correspondence for G is compatible with the action of F on Irr(W) and on N_{G}. Hence it restricts to an injective map i_{G}: Irr(W)^{F} → N_{G}^{F}. Let E ∈ Irr(W)^{F} and i_{G}(E) = (C, ψ) ∈ N_{G}^{F}. For u ∈ C we denote by B_{u} the variety of Borel subgroups containing u, and let d := dim B_{u}. Then, for a suitable normalization of the Y_{i}, we have that

(3.1) Q_{E} = q^{d}Y_{i_{G}(E)} + linear combination of Y_{(C′,ψ′)} where dim C′ < dim C.

For large enough p, this is contained in [16]. In general, it follows by combining the methods in [10, Section 24], with [17, Theorem 5.5] and [2, Prop. 3.5]. Since the values of all Green functions are rational integers, we deduce from equation (3.1) that Y_{i_{G}(E)} is also rational valued. This implies that, for any F -stable unipotent class C there exists a sign ε_{C} = ±1 such that

(3.2)  Y_{(C,1)}(g) = ε_{C} for all g ∈ C^{F} .

Indeed, the pair i_{0} := (C, 1) ∈ N_{G} always lies in the image of the Springer map Irr(W) → N_{G}. By definition, the corresponding function Y_{i_{0}} is a scalar multiple of the characteristic function on C^{F}. The fact that the values of this function are rational integers therefore implies (3.2).

In the following three results (3.3)–(3.5), the condition ‘p and q are large enough’ means that the results in [12] are applicable; in particular, the unipotent support of an irreducible character exists in this case.

Proposition 3.3. Assume that p and q are large enough. Let ρ ∈ Irr(G^{F}) and C_{ρ} be its unipotent support. Let E ∈ Irr(W)^{F} and i_{G}(E) = (C, ψ) ∈ N_{G}^{F}. If b(ρ, E) ≠ 0 then dim C ≤ dim C_{ρ} with equality only if C = C_{ρ}.

Proof. Let C be an F -stable unipotent class such that i_{G}(E) = (C, ψ) for some E, ψ and such that b(ρ, E) ≠ 0. Assume that C has maximal possible dimension with this property.

We claim that then ρ_{unif}(g) ≠ 0 for some g ∈ C^{F}. To see this, consider any E′ ∈ Irr(W)^{F} such that b(ρ, E′) ≠ 0. Let i_{G}(E′) = (C′, ψ′). By the maximality condition on C we have dim C′ ≤ dim C. Using formula (3.1) we see that if dim C′ < dim C then Q_{E′} does not give any contribution to the restriction of ρ_{unif} to C^{F}. On the other hand, if C′ = C then the restriction of Q_{E′} to C^{F} is a non-zero scalar times Y_{i} where i = i_{G}(E′). We conclude that the restriction of ρ_{unif} to C^{F} is a non-zero linear combination of the functions Y_{i}, where i = (C, ψ′) for some ψ′ and i = i_{G}(E′) for some E′ ∈ Irr(W)^{F} with b(ρ, E′) ≠ 0. Since the Y -functions are linearly independent we conclude that that restriction cannot be identically zero, and our claim is established.

Now the restriction of ρ to C^{F} is a linear combination of the restriction of ρ_{unif} to C^{F} and functions Y_{i} where i ∈ N_{G}^{F} is not in the image of the map i_{G}: Irr(W)^{F} → N_{G}^{F} (cf. [2, (3.6)]). Using once more that the Y -functions are linearly independent we conclude that neither the restriction of ρ to C^{F} can be identically zero. Hence there exists some g ∈ C^{F} such that ρ(g) ≠ 0

We can now use [12, Theorem 11.2] to conclude that dim C ≤ dim C_{ρ}, with equality only for C = C_{ρ}. This completes the proof. □

Corollary 3.4. Assume that p and q are large enough. Let ρ ∈ Irr(G^{F}) and C_{ρ} be its unipotent support. Let E_{0} ∈ Irr(W)^{F} such that i_{G}(E_{0}) = (C_{ρ}, 1) ∈ N_{G}^{F}. Then AV(C_{ρ}, ρ) = ±q^{d}|A_{G}(u)|b(ρ, E_{0}), where u ∈ C_{ρ}^{F} and d = dim B_{u}.

Proof. We have AV(C_{ρ}, ρ) = AV(C_{ρ}, ρ_{unif}), see [2, Prop 1.3]. As above, let us write ρ_{unif}(g) = ∑_{E} b(ρ, E)Q_{E}(g), for g ∈ G^{F} unipotent. By Proposition 3.3 we see that

AV(C_{ρ}, ρ) = ∑_{E} b(ρ, E)AV(C_{ρ}, Q_{E})

where the sum is over all E ∈ Irr(W)^{F} such that i_{G}(E) = (C_{ρ}, ψ) for some ψ. The formula (3.1) shows that if i_{G}(E) = (C_{ρ}, ψ) then the restriction of Q_{E} to C_{ρ}^{F} equals q^{d}Y_{i_{G}(E)}. In [2] the average value of any class function on a unipotent class C is expressed in terms of the scalar product of that class function with a certain function f′_{C}. Applying this to C = C_{ρ} and the class function Y_{i}, and using the formulae in [2, (2.1)] we see that the average value AV(C_{ρ}, Y_{i}) is zero unless i = (C_{ρ}, 1) in which case it equals ±|A_{G}(u)| (cf. equation 3.2). Thus, we can conclude that AV(C_{ρ}, ρ) = ±q^{d}|A_{G}(u)|b(ρ, E_{0}), as desired. □

B. Series and families of irreducible characters.

In order to get hold on b(ρ, E_{0}) we now have to recall some facts from [7, Chap. 4] concerning the classification of irreducible characters of G^{F}. Let G^{∗} be a group dual to G. Then G^{∗} is also defined over 𝔽_{q} and we denote again by F the corresponding Frobenius map. Let T^{∗} ⊆ G^{∗} be an F -stable maximal torus in duality with T. The Weyl group of G^{∗} with respect to T^{∗} can be canonically identified with W, where the actions of the Frobenius maps of G and G^{∗} on W are inverse to each other.

Let s ∈ T^{∗} be an element such that the G^{∗}-conjugacy class of s is F -stable, and let W_{s} ⊆ W be the Weyl group of C_{G^{∗}}(s) with respect to T^{∗}. The Frobenius map F induces an automorphism γ: W_{s} → W_{s}. Let Irr(W_{s})^{γ} be the set of all γ-invariant irreducible characters of W_{s}. For each E ∈ Irr(W_{s})^{γ} we have a corresponding almost character R_{E}^{s}, which is defined as a certain rational linear combination of Deligne-Lusztig generalized characters. (Note that almost characters are only welldefined up to sign, see [7, (3.7)]; a definitive choice of these signs is given in terms of the ‘preferred extensions’ in [9, (17.2)].)

Let ρ be an irreducible character of G^{F}. Then there exists some s ∈ T^{∗} and E ∈ Irr(W_{s})^{γ} as above such that the scalar product between ρ and R_{E}^{s} is non-zero; if s′ ∈ T^{∗} is another such element then s, s′ are conjugate (see [7, (8.4.4)]). Thus, we can associate with ρ a unique F -stable conjugacy class of semisimple elements in G^{∗}. In this way, we have series of irreducible characters corresponding to the various s ∈ T^{∗} such that the class of s in G^{∗} is F -stable.

Assume that ρ ∈ Irr(G^{F}) lies in the series defined by s ∈ T^{∗}. If E, E′ ∈ Irr(W_{s})^{γ} are such that ρ has non-zero scalar product with R_{E}^{s} and with R_{E′}^{s} then E, E′ belong to the same family of characters of W_{s} (see [7, (5.25) and (6.17)]). Thus, we can associate with ρ a unique γ-stable family of characters of W_{s}. Let E_{1} be the unique special character in this family and D_{E_{1}} ∈ ℚ[X] its ‘generic degree’ (where X is an indeterminate). As in [7, (4.1)], we define integers f_{E_{1}} ≠ 0 and a_{E_{1}} ≥ 0 by the condition that

D_{E_{1}}(X) = f_{E_{1}}^{−1}X^{a_{E_{1}}} + higher powers of X.

The number a_{E_{1}} is canonically associated with ρ. In order to indicate this, we shall also denote it by a_{ρ} and call it the generic q-part of ρ(1). Finally, there exists a unique integer n_{ρ} ≥ 1 such that the scalar product of ρ and R_{E_{1}}^{s} is given by ±n_{ρ}^{−1}. We call n_{ρ} the generic denominator in ρ(1). These definitions are justified by the fact that we have

ρ(1) = ±n_{ρ}^{−1}q^{a_{ρ}}N

where N is an integer satisfying N ≡ 1 modulo q (see [7, (4.26.3)]).

Proposition 3.5. Assume that p and q are large enough. Let ρ ∈ Irr(G^{F}) and C_{ρ} be its unipotent support. Let E_{0} ∈ Irr(W)^{F} such that i_{G}(E_{0}) = (C_{ρ}, 1) ∈ N_{G}^{F}. Then b(ρ, E_{0}) = ±n_{ρ}^{−1} and a_{ρ} = dim B_{u} (for u ∈ C_{ρ}).

Proof. We compute AV(C_{ρ}, ρ) in two different ways. On one hand, by Corollary 3.4, this equals ±q^{d}|A_{G}(u)|b(ρ, E_{0}), where u ∈ C_{ρ}^{F} and d = dim B_{u}.

On the other hand, let u_{1}, …, u_{r} ∈ C_{ρ}^{F} be representatives for the G^{F} -classes contained in C_{ρ}^{F} and Γ_{0} = ∑_{j=1}^{r}[A_{G}(u_{j}) : A_{G}(u_{j})^{F}]Γ_{u_{j}} where Γ_{u_{j}} denotes the generalized Gelfand-Graev character associated with the unipotent element u_{j} ∈ C_{ρ}^{F}. By [12, (9.11)] and [2, Theorem 1.4] we have d = a_{ρ} and our average value is equal to q^{d} times the scalar product between Γ_{0} and D_{G}(ρ̄) where D_{G} denotes Alvis–Curtis–Kawanaka duality. Since these scalar products are integers, we can work with ρ instead of ρ̄. We conclude that it remains to show that the scalar product between Γ_{0} and D_{G}(ρ) equals n_{ρ}^{−1}|A_{G}(u)|, up to sign. In order to see this, we go through the proof of [12, Theorem 11.2].

Let (s, E_{1}) be associated with ρ as described above. Let L_{0} be an F -stable tame local system on T in duality with s ∈ T^{∗}, and let ε be the sign character of W_{s}. We can associate with L_{0} and E_{1}⊗ε ∈ Irr(W_{s})^{F} an F -stable character sheaf A = A_{E_{1}⊗ε}^{L_{0}} on G and we can choose an isomorphism φ: F^{∗}A → A so that the corresponding characteristic function χ_{A,φ} is a class function on G^{F} of norm 1. Then it is known that χ_{A,φ} coincides with R_{E_{1}⊗ε}^{s}, up to a scalar multiple which is in fact a root of unity (see, for example, [4, Cor. 2.3.2] or [17]).

Equation (h) in the proof of [12, (11.2)] now tells us that the scalar product of Γ_{0} with D_{G}(ρ) is the same as the scalar product of |A_{G}(u)|χ_{A,φ} with D_{G}(ρ), hence it is the same as the scalar product of |A_{G}(u)|R_{E_{1}⊗ε}^{s} with D_{G}(ρ), up to a root of unity. Since these scalar products must be integers, the root of unity is ±1.

Finally, observe that R_{E_{1}⊗ε}^{s} = ±D_{G}(R_{E_{1}}^{s}), see [7, (6.8.6)], and recall that D_{G} is a self-adjoint involution. Hence the scalar product of R_{E_{1}⊗ε}^{s} with D_{G}(ρ) is the same (up to sign) as the scalar product of R_{E_{1}}^{s} with ρ, which is ±n_{ρ}^{−1}. This completes the proof. □

C. Lusztig’s map from irreducible characters to unipotent classes.

Recall the following construction from [12, (10.5)] or [7, (13.3)]. Let Y be the cocharacter group of the maximal torus T. Then Y is naturally a module for W. Let s ∈ T^{∗} and W_{s} ⊆ W be the stabilizer of s in W. By restriction, Y also is a module for W_{s}. Using the generalized Macdonald j-induction (see [13]), we can associate an irreducible character E ∈ Irr(W) with any special character E_{1} ∈ Irr(W_{s}). This is done as follows. Let b_{E_{1}} be the smallest r ≥ 0 for which E_{1} appears with non-zero multiplicity in the r-th symmetric power of Y. Then the multiplicity of E_{1} is in fact equal to 1. Moreover, there exists a unique E ∈ Irr(W) such that E appears with non-zero multiplicity both in the character induced from E_{1} and in the b_{E_{1}}-th symmetric power of the W -module Y. We shall write E = j_{W_{s}}^{W}(E_{1}). Thus, we can define a map

Φ_{G}: {(s, E_{1}) | s ∈ T^{∗} and E_{1} ∈ Irr(W_{s}) special} ⟶ {unipotent classes in G}

by associating with the pair (s, E_{1}) the unipotent class C satisfying i_{G}(E) = (C, ψ) for some ψ. (This coincides with the map described in [7, (13.3)] or [12, (10.5)] in good characteristic.)

Let ρ ∈ Irr(G^{F}) and (s, E_{1}) be associated with ρ as described in (B) above. Let E′_{1} ∈ Irr(W_{s})^{γ} be the unique special character which lies in the same family as E_{1} ⊗ sign. It is easy to see that the class C := Φ_{G}(s, E′_{1}) is F -stable. Thus, we also obtain a map

ξ_{G}: Irr(G^{F}) ⟶ {F -stable unipotent classes in G},

given by associating with ρ the unipotent class Φ_{G}(s, E′_{1}). (This coincides with the map described in [7, (13.4)] or [12, (11.1)] in good characteristic.) Using the results in the previous section we can now prove that the properties stated in [loc. cit.] remain valid in all characteristics.

Proposition 3.6. Let s ∈ T^{∗} and E_{1} ∈ Irr(W_{s}) be special. Furthermore, let C be a unipotent class in G and d := dim B_{u} (for u ∈ C).

(a) Let E = j_{W_{s}}^{W}(E_{1}) ∈ Irr(W). If i_{G}(E) = (C, ψ) ∈ N_{G} then ψ = 1 and d = b_{E_{1}}.

(b) Let ρ, ρ′ ∈ Irr(G^{F}) such that ρ′ = ±D_{G}(ρ). If C = ξ_{G}(ρ) then d = a_{ρ′}.

Proof. If p is large enough, the statement about ψ in (a) holds by [12, Theorem 10.7(iii)]. Using the results in the previous section, this implies that it is also true for any p. More precisely, the argument goes as follows.

Let k̃ be an algebraically closed field, G̃ a connected reductive group over k̃, and T̃ ⊆ G̃ a maximal torus so that the root data of G and G̃ with respect to T and T̃, respectively, are isomorphic. Then we can identify W with the Weyl group of G̃. Assume that the characteristic of k̃ is large enough so that the results in [12] are applicable. We can then also find a semisimple element s̃ in the dual group so that the stabilizer Ws̃ ⊆ W can be identified with W_{s} (see [1, Theorem 2.1]).

Consider also a group G′ of the same type as G but over an algebraically closed field of characteristic 0, as in the previous section.

Now we can argue as follows. Let iG̃(E) = (C̃, ψ̃) ∈ NG̃. By [12, Theorem 10.7(iii)] we have ψ̃ = 1. Since πG̃: X_{G′} → XG̃ is a bijection there is a unique C_{0} ∈ X_{G′} such that πG̃(C_{0}) = C̃. Let E_{0} ∈ Irr(W) such that i_{0}(E_{0}) = (C_{0}, 1). Using Theorem 2.1(1), we conclude that iG̃(E_{0}) = (πG̃(C_{0}), 1) = (C̃, 1) = iG̃(E). Since the Springer correspondence is injective, we therefore have that E_{0} = E. Now we can apply once more Theorem 2.1 to pass from G′ to G and conclude that i_{G}(E) = (C, 1), where C = π_{G}(C_{0}).

Now consider the asserted equality d = b_{E_{1}} in (a). Again, if p is large enough, it holds by the results in [6, (13.1.1) and (13.3)]. By a similar argument as before, it then holds for all p, using the fact that Spaltenstein’s map preserves the dimension of classes.

In order to prove (b), assume that (s, E_{1}) corresponds to ρ′ as in (B). Then (s, E′_{1}) corresponds to ρ where E′_{1} is the special character in the same family as E_{1} ⊗ sign (cf. the last part of the proof of Proposition 3.5). By definitions, we have Φ_{G}(s, E_{1}) = C = ξ_{G}(ρ). Since a_{ρ′} = a_{E_{1}} (see (B)) and a_{E_{1}} = b_{E_{1}} (recall that E_{1} is special), we are done using (a). □

Before we state our main result we extend the above constructions to the case where the center of G is not necessarily connected. In that case, we can embed G ‘regularly’ as a closed subgroup into a group G_{0} over 𝔽_{q} with a connected center and such that G, G_{0} have the same derived subgroup (cf. [11]). Clearly, the embedding G ⊆ G_{0} defines a bijection between the F -stable unipotent classes of G and G_{0}, respectively.

Let ρ ∈ Irr(G^{F}) and ρ_{0} ∈ Irr(G_{0}^{F}) such that ρ appears with non-zero multiplicity in the restriction of ρ_{0}. By [11, Prop. 10] the restriction of every irreducible character of G_{0}^{F} to G^{F} is multiplicity-free. Let m_{ρ} ≥ 1 be the total number of irreducible characters of G^{F} which occur in the restriction of ρ_{0}. (Note that m_{ρ} does not depend on the choice of ρ_{0} and that m_{ρ} is not divisible by p.) Then we let

a_{ρ} := a_{ρ_{0}}, n_{ρ} := m_{ρ}n_{ρ_{0}}, and C_{ρ} := Φ_{G_{0}}(s, E_{1}),

where (s, E_{1}) is associated with ρ_{0} as in (B). Equivalently, we could define C_{ρ} := ξ_{G_{0}}(±D_{G_{0}}(ρ_{0})). Note that C_{ρ} does not depend on the choice of ρ_{0}. It is readily checked that the formula in (B) for ρ(1) remains valid.

Theorem 3.7. Let G be any connected reductive group over 𝔽_{q}, with corresponding Frobenius map F. Let ρ ∈ Irr(G^{F}) and C_{ρ} be the F -stable unipotent class associated with ρ as above. Then C_{ρ} is the unipotent support of ρ and we have

AV(C_{ρ}, ρ) = ±n_{ρ}^{−1}q^{d}|A_{G}(u)|, d = a_{ρ},

where u ∈ C_{ρ}^{F} and d = dim B_{u}.

In particular, this means that every irreducible character of a finite group of Lie type has a unipotent support (see §5 for the Suzuki and Ree groups).

Proof. Let G ⊆ G_{0} be a regular embedding, and ρ_{0} ∈ Irr(G_{0})^{F} such that ρ occurs in the restriction of ρ_{0} to G^{F}. Using the argument in [2, Lemma 5.1] we see that if C is any F -stable unipotent class in G then |A_{G_{0}}(u)|AV(C, ρ_{0}) = m_{ρ}|A_{G}(u)|AV(C, ρ); note that A_{G_{0}}(u) is a quotient of A_{G}(u). In view of the above definitions of C_{ρ}, a_{ρ}, n_{ρ}, this implies that, in order to prove the theorem, we can assume that the center of G is connected.

Recall from [2] that as far as average values on unipotent classes are concerned, we can work with ρ_{unif} instead of ρ. As before let b(ρ, E) ∈ ℚ such that ρ_{unif}(g) = ∑_{E} b(ρ, E)Q_{E}(g) for all unipotent g ∈ G^{F}.

We associate with ρ a semisimple element s ∈ T^{∗} and a special character E_{1} ∈ Irr(W_{s})^{γ} as described in (B) above. Let ρ correspond to the element x̄_{ρ} in the parameter set X̄(W_{s}, γ) by a bijection as in [7, Theorem 4.23]. Then the constants b(ρ, E) only depend on W_{s}, γ, E and x̄_{ρ} (cf. [2, Cor. 3.8]). Hence these constants are ‘independent of q’. This means that we can choose a power q̃ of a large prime and a group G̃ defined over 𝔽q̃ of the same type as G, with the same Weyl group W and the same action of the Frobenius map on W; furthermore, by [1, Theorem 3.3], we can choose q̃ in such a way that there exists a semisimple element s̃ in the dual group such that the stabilizer Ws̃ ⊆ W can be identified with W_{s} and the induced action of the Frobenius map on it is the same as it was for W_{s}. Then we can also identify the corresponding parameter sets X̄(Ws̃, γ̃) and X̄(W_{s}, γ). Take an irreducible character ρ̃ of G̃^{F} in the series defined by s̃ so that x̄ρ̃ corresponds to x̄_{ρ} under this identification. We then have constants b̃(ρ̃, E) defined analogously as before. The fact that these are ‘independent of q’ means that we have

b̃(ρ̃, E) = b(ρ, E) for all E ∈ Irr(W)^{F} .

We can certainly assume that q̃ was chosen large enough so that the results in [12] are applicable. Hence, in particular, the character ρ̃ has a unipotent support, which we denote by C̃ρ̃. By [12, Theorem 11.2], C̃ρ̃ is the class associated with ρ̃ in the same way as C_{ρ} was associated with ρ above.

Now let G′ be a group over ℂ as in Section 2, and let π_{G}: X_{G′} → X_{G} , πG̃: X_{G′} → XG̃ be Spaltenstein’s maps between the respective partially ordered sets of unipotent classes. Since q̃ is a power of a large prime, πG̃ is an isomorphism of partially ordered sets (cf. the remarks preceding Theorem 2.1). In order to simplify notation, we identify X_{G′} with XG̃ and consider π_{G} directly as a map XG̃ → X_{G}. Then the argument in the proof of Proposition 3.6 shows that

(a)  C_{ρ} = π_{G}(C̃ρ̃) ∈ X_{G}.

Since π_{G} preserves dimensions, we have dim C_{ρ} = dim C̃ρ̃.

Next we show that the assertion of Proposition 3.3 remains true. Let C be any F -stable unipotent class in G and E ∈ Irr(W)^{F} such that b(ρ, E) ≠ 0 and i_{G}(E) = (C, ψ) for some ψ. We claim that then

(b)  dim C ≤ dim C_{ρ} with equality only if C = C_{ρ}.

Indeed, let iG̃(E) = (C̃, ψ̃) ∈ N F ; G̃. Since b̃(ρ̃, E) = b(ρ, E) ≠ 0, Proposition 3.3 implies that dim C̃ ≤ dim C̃ρ̃ with equality only for C̃ = C̃ρ̃. By Theorem 2.1(2), we have dim C ≤ dim C̃ with equality only for C = π_{G}(C̃). Combining this with the previous inequality we deduce that

dim C ≤ dim C̃ ≤ dim C̃ρ̃ = dim C_{ρ}.

Moreover, if dim C = dim C_{ρ} then we have equality everywhere. The equality dim C = dim C̃ implies that C = π_{G}(C̃) and the equality dim C̃ = dim C̃ρ̃ implies that C̃ = C̃ρ̃. Using (a), we conclude that C = π_{G}(C̃) = π_{G}(C̃ρ̃) = C_{ρ}. This completes the proof of (b).

Now we show that the assertion of Corollary 3.4 remains valid. Let E_{0} ∈ Irr(W)^{F} such that i_{G}(E_{0}) = (C_{ρ}, 1) ∈ N_{G}^{F}. We claim that

(c)  b(ρ, E_{0}) = ±n_{ρ}^{−1} ≠ 0.

To see this, let Ẽ_{0} ∈ Irr(W)^{F} such that iG̃(Ẽ_{0}) = (C̃ρ̃, 1) ∈ N F ; G̃. Since π_{G}(C̃ρ̃) = C_{ρ} (cf. (a)) we have i_{G}(Ẽ_{0}) = (C_{ρ}, 1) by Theorem 2.1(1), and since the Springer correspondence is injective we conclude that E_{0} = Ẽ_{0}. Since b̃(ρ̃, E_{0}) = b(ρ, E_{0}) and n_{ρ} = nρ̃, Corollary 3.4 implies (c).

Finally, it remains to check that C_{ρ} is the unipotent support of ρ and that AV(C_{ρ}, ρ) has the asserted form.

Indeed, let C′ be any F -stable unipotent class in G such that AV(C′, ρ) ≠ 0. Then there exists some E ∈ Irr(W)^{F} such that b(ρ, E) ≠ 0 and such that AV(C′, Q_{E}) ≠ 0. Let i_{G}(E) = (C, ψ) ∈ N_{G}^{F}. Since Q_{E} is non-zero on some element of C′ we conclude that dim C′ ≤ dim C with equality only for C = C′ (using once more the formula (3.1)). On the other hand, since b(ρ, E) ≠ 0, we conclude using (b) that dim C ≤ dim C_{ρ} with equality only if C = C_{ρ}. Combining these two inequalities we see that dim C′ ≤ dim C_{ρ} with equality only for C′ = C_{ρ}.

We can now transfer the proof of Corollary 3.4, essentially word by word, to conclude that AV(C_{ρ}, ρ) = ±q^{d}|A_{G}(u)|b(ρ, E_{0}) where E_{0} ∈ Irr(W)^{F} is such that i_{G}(E_{0}) = (C_{ρ}, 1). Then (c) yields the desired form for our average value. This completes the proof. □

Remark 3.8. What can we say about the sign in the above expression for AV(C_{ρ}, ρ)?

Assume first that p and q are large enough so that the results in [12] are applicable. Then AV(C_{ρ}, ρ) = q^{d}(Γ_{0}, D_{G}(ρ)) where Γ_{0} is a sum of certain generalized Gelfand-Graev characters and D_{G} denotes Alvis–Curtis–Kawanaka duality (cf. the proof of Proposition 3.5). This implies that AV(C_{ρ}, ρ) is ε_{ρ} times a positive integer where ε_{ρ} = ±1 is such that ε_{ρ}D_{G}(ρ) is an irreducible character. Comparing this with our formula in Theorem 3.7 we deduce that

AV(C_{ρ}, ρ) = ε_{ρ}n_{ρ}^{−1}q^{d}|A_{G}(u)|.

On the other hand, our average value is given in terms of the constant b(ρ, E_{0}) and the average value of Q_{E_{0}} on C_{ρ}, where i_{G}(E_{0}) = (C_{ρ}, 1). In order to compute the latter value we need to determine the sign for the corresponding Y -function in (3.2). If q is a power of a good prime, this problem is solved (by work of Springer, Shoji, and Beynon–Spaltenstein) and it yields that the sign is always 1, see [16] and the references there. Since all the constants b(ρ, E) are ‘independent of q’ we can conclude that the above formula remains valid whenever q is a power of a good prime.

It seems likely that the above formula for AV(C_{ρ}, ρ) remains valid even if q is a power of a bad prime, but at the moment we cannot prove this since the sign in the formula (3.2) does not yet seem to be known in general.

Remark 3.9. Let G′ be a group of the same type as G but defined over an algebraically closed field of characteristic 0. Let π_{G}: X_{G′} → X_{G} be Spaltenstein’s map between the respective partially ordered sets of unipotent classes. Theorem 3.7 implies that the image of the map ρ ↦ C_{ρ} is contained in the image of π_{G}. Hence, in bad characteristic, there can exist F -stable unipotent classes in G which are not the unipotent support of any irreducible character of G^{F} (see Sp_{4} over a field of characteristic 2, for example). This is in contrast to the situation in good characteristic, where every F -stable unipotent class is the unipotent support of some irreducible character, see [7, (13.4.2)].

4. Some applications

Throughout this section, let G be any connected reductive group defined over 𝔽_{q}, with corresponding Frobenius map F. We will now discuss some consequences of the existence of unipotent supports, and give examples.

A. Induction of classes.

Let L ⊆ G be a closed subgroup which is the Levi complement in some parabolic subgroup of G. Following Lusztig–Spaltenstein [13] we have an operation of induction of unipotent classes from L to G, defined as follows. Choose a parabolic subgroup P ⊆ G such that P = U_{P} L where U_{P} is the unipotent radical of P. Given any unipotent class C in L, there exists a unique unipotent class Ĉ in G such that Ĉ ∩ CU_{P} is dense in CU_{P}. By [13, Theorem 2.2] the class Ĉ does not depend on the chosen parabolic subgroup P containing L. Hence we can write Ĉ = Ind_{L}^{G}(C).

Now assume that L is F -stable. If, as above, Ĉ is induced from C using the parabolic subgroup P, then F (Ĉ) is induced from F (C) using the parabolic subgroup F (P). So, if C is F -stable then Ĉ must be F -stable, too. Thus, the induction takes F -stable classes in L to F -stable classes in G.

Denote by R_{L}^{G} the operation of twisted induction, see [5]; it maps virtual characters of L^{F} to virtual characters of G^{F}. This operation preserves series of characters, in the following sense. The dual group L^{∗} can be identified with a closed subgroup in G^{∗}. Let s ∈ L^{∗} be semisimple and F -stable. Then s defines a series of characters of L^{F} and of G^{F}, respectively. If ρ ∈ Irr(L^{F}) lies in the series defined by s, then R_{L}^{G}(ρ) is an integral linear combination of irreducible characters in the series of G^{F} defined by s (see [5, Cor. 6]).

Assume furthermore that C_{G^{∗}}(s) ⊆ L^{∗}. Then there exists a sign ε such that εR_{L}^{G} defines a bijection between the irreducible characters in the series of L^{F} and G^{F} defined by s, respectively (see [5, Prop. 10]). The following result shows that, in this situation, the notion of unipotent support is compatible with twisted induction of characters and induction of unipotent classes.

Proposition 4.1. In the above set-up, assume that F (s) = s and that C_{G^{∗}}(s) ⊆ L^{∗}. Let ρ ∈ Irr(L^{F}) be in the series defined by s and let ρ̂ := ±R_{L}^{G}(ρ) ∈ Irr(G^{F}). If C is the unipotent support of ρ then Ind_{L}^{G}(C) is the unipotent support of ρ̂.

Proof. Since induction of classes and twisted induction of characters are compatible with regular embeddings (cf. [11]), we may assume that the center of G is connected. It will be convenient to use a slightly different set-up for the description of the R_{L}^{G}- map. This can be done as follows, see [7, (6.21)]. Let P ⊆ G be a parabolic subgroup with Levi decomposition P = U_{P} L. There exists some g ∈ G and a subset J ⊆ S such that gP g^{−1} = P_{J} and gLg^{−1} = L_{J}, where P_{J} and L_{J} denote the standard parabolic subgroup and the standard Levi complement defined by J, respectively. Then g^{−1}F (g) ∈ N_{G}(T); denote this element by ẇ_{2} and let w_{2} ∈ W be its image under the canonical map N_{G}(T) → W. Then F (w_{2}Jw_{2}^{−1}) = J and the map F′: l ↦ F (ẇ_{2}lẇ_{2}^{−1}) can be considered as a Frobenius map on L_{J} such that the finite groups L^{F} and L_{J}^{F′} are isomorphic. We can therefore regard R_{L}^{G} as a map from virtual characters of L_{J}^{F′} to virtual characters of G^{F}. This description has the advantage that now we can assume that s ∈ T^{∗} and the L_{J}^{∗}-conjugacy class of s is F′-stable. Let W_{J} ⊆ W be the parabolic subgroup corresponding to J ⊆ S. Under the identification of W with the Weyl group of G^{∗}, the group W_{J} is the Weyl group of L_{J}^{∗}. Our assumption that C_{G^{∗}}(s) ⊆ L^{∗} now translates to the condition that

W_{s} ⊆ W_{J} ⊆ W.

Let γ′: W_{s} → W_{s} be the automorphism induced by F′ in the same way as γ: W_{s} → W_{s} was induced by F. As remarked in [7, p.214] we have γ = γ′. By Section 3(B), we can associate with ρ a pair (s, E_{1}) where E_{1} ∈ Irr(W_{s})^{γ} is a special character. We claim that (s, E_{1}) is also the pair associated with ρ̂ = ±R_{L}^{G}(ρ).

Indeed, for any E ∈ Irr(W_{s})^{γ}, let R_{E}^{s,J} and R_{E}^{s} be the corresponding almost characters of L_{J}^{F′} and G^{F}, respectively. Using the formulae in [7, (6.21.3) and (6.21.4)] we see that

R_{L}^{G}(R_{E}^{s,J}) = R_{E}^{s}.

Moreover, ρ appears with non-zero multiplicity in R_{E}^{s,J} if and only if ρ̂ appears with non-zero multiplicity in R_{E}^{s}. (The multiplicities are even the same up to a sign, by [5, Prop. 10].) Applying this to the special character E_{1} and recalling the description in Section 3(B), the claim is proved.

Now we can proceed as follows. We regard ρ as a character of L_{J}^{F′} and let C be its unipotent support in L_{J}. By Theorem 3.7, we have

i_{L_{J}} (E) = (C, 1) where E := j_{W_{s}}^{W_{J}} (E_{1}) ∈ Irr(W_{J}).

On the other hand, let Ĉ be the unipotent support of ρ̂. Then we also have

i_{G}(Ê) = (Ĉ, 1) where Ê := j_{W_{s}}^{W}(E_{1}) ∈ Irr(W).

Since the j-operation is transitive, we conclude that j_{W_{J}}^{W} (E) = Ê. Now we are in the set-up of [20, 4.1]. So we can deduce that Ĉ = Ind_{L}^{G}(C), and the proof is complete. □

B. Unipotent characters and the Jordan decomposition.

Recall from [7] that ρ ∈ Irr(G^{F}) is called unipotent if ρ lies in the series defined by s = 1 ∈ T^{∗}. In this case, the description of Lusztig’s map in Section 3(C) becomes slightly simpler. Indeed, let E_{1} be the unique F -stable special character of W such that the scalar product of ρ with the almost character R_{E_{1}}^{1} is non-zero. By Theorem 3.7 and Proposition 3.6 we have i_{G}(E_{1}) = (C, 1) and C is the unipotent support of ρ.

Let X_{G}^{0} be the set of all unipotent classes C in G such that i_{G}(E) = (C, 1) for some special character E ∈ Irr(W). By Theorem 2.1, this set is ‘independent of the characteristic’, that is, if we also define it with respect to a group G′ over ℂ and of the same type as G, then Spaltenstein’s map π_{G} induces a bijection between X_{G′}^{0} and X_{G}^{0}. The classes in X_{G}^{0} are called special. We have just seen that the unipotent support of a unipotent character is special.

The following characterization of families of unipotent characters was already conjectured by Lusztig [6].

Proposition 4.2. Two unipotent characters of G^{F} belong to the same family if and only if they have the same unipotent support.

Proof. Let ρ, ρ′ be unipotent characters, E_{1}, E′_{1} the special characters in Irr(W) associated with ρ, ρ′, and C, C′ be the unipotent supports of ρ, ρ′, respectively. The characters ρ, ρ′ belong to the same family if and only if E_{1} = E′_{1}. On the other hand, as remarked above, we have i_{G}(E_{1}) = (C, 1) and i_{G}(E′_{1}) = (C′, 1). Hence E_{1} = E′_{1} if and only if C = C′. □

Proposition 4.3. Let ρ be a unipotent character of G^{F} and C_{ρ} be its unipotent support. If C is any F -stable unipotent class in G such that AV(C, ρ) ≠ 0 then C ≤ C_{ρ}. (Recall that this means that C is contained in the Zariski closure of C_{ρ}.)

Proof. As above let E_{1} ∈ Irr(W)^{F} be the unique special character such that the scalar product between ρ and R_{E_{1}}^{1} is non-zero; then i_{G}(E_{1}) = (C_{ρ}, 1). Since we are interested in average values on unipotent classes, we can replace ρ by ρ_{unif}. By [7, 4.23], ρ_{unif} is a linear combination of almost characters R_{E}^{1} where E ∈ Irr(W^{F}) is in the same family as E_{1}. Using the formula (3.1) we see that it is sufficient to prove the assertion only for those C such that i_{G}(E) = (C, ψ) and E is in the same family as E_{1}. Hence we are done by Proposition 2.2. □

Remark. It is intriguing to speculate that the assertion of Proposition 4.3 holds for all irreducible characters, and not only for the unipotent ones. In the same way as the above proposition follows from a property of the Springer correspondence (namely, Proposition 2.2), such a more general result would follow from a more general version of that property of the Springer correspondence: namely, it would have to be possible to formulate the assertions in [12, Corollary 10.9] in terms of the usual order relation on unipotent classes, and not in terms of dimensions. We hope to settle this question (in the positive or in the negative) elsewhere.

Assume that the center of G is connected. Recall from [7, p.x] that we then have a Jordan decomposition of characters: let s ∈ G^{∗} be semisimple and F -stable, and G(s) be a group over 𝔽_{q} ‘dual’ to C_{G^{∗}}(s) (see [7, (8.4)]). Then there exists a bijection between the irreducible characters of G^{F} in the series defined by s and the unipotent characters of G(s)^{F}; moreover, the multiplicities of corresponding characters in the almost characters R_{E}^{s}, R_{E}^{1} defined with respect to G and G(s), respectively, are the same (up to a sign depending only on s). If C_{G^{∗}}(s) is the Levi complement of some parabolic subgroup of G^{∗}, then the Jordan decomposition is given by the R_{L}^{G}–map and Proposition 4.1 shows that the unipotent supports of corresponding characters are related by the induction of classes as defined in [13]. In the general case, we have to work with Spaltenstein’s generalized induction [19, III.12].

Lemma 4.4. Let d_{0}: X_{G} → X_{G} be the unique duality operation (in the sense of [19, III.1.2]) for which the image d_{0}(X_{G}) has minimal possible cardinality (cf. [19, III.9.4]). Then X_{G}^{0} = d_{0}(X_{G}).

Proof. By standard reduction arguments we may assume that G is a simple group over ℂ. If G is of exceptional type, the assertion follows from the tables giving the Springer correspondence [20] and the desription of d_{0} in [19, III.9.1].

If G is of type A_{n}, all characters of the Weyl group and all unipotent classes are special and d_{0} is surjective, so the assertion is obvious.

Now let G = Sp_{2n} be of type C_{n}, with Weyl group W_{n}. An irreducible character E ∈ Irr(W_{n}) parametrized by the pair of partitions (α, β) (normalized so that α has one part more than β) is special if and only if Λ′ = (α, β) + Λ_{0,1}^{1,0} ∈ X_{n,1}^{1,0} is distinguished [7, 4.5]. By 2(B) the Springer correspondent of E is the unipotent class with symbol Λ = Λ′ + Λ_{0,1}^{0,1} = (α, β) + Λ_{0,1}^{1,1} ∈ X_{n,1}^{1,1}. Let (c_{1}, …, c_{2m+1}) denote the entries of Λ. Since Λ′ is distinguished, it follows that c_{i} = c_{i+1} implies that i is even. By (2.6) this means that between any two consecutive odd entries of λ := i_{0}(E) and also above the largest odd entry of λ there is an even number of even entries. This in turn shows that the dual partition λ^{∗} also parametrizes a unipotent class of G, and hence λ ∈ d_{0}(X_{G}) by [19, III.3.11]. On the other hand, if λ ∈ d_{0}(X_{G}) then it is easy to see that the distinguished symbol Λ ∈ X_{n,1}^{1,1} with λ = i_{0}(Λ) is such that Λ − Λ_{0,1}^{0,1} is still distinguished, so λ is the image under the Springer map of a special character.

Next, let G = SO_{2n+1} be of type B_{n}. Again, an irreducible character E = (α, β) ∈ Irr(W_{n}) is special if and only if Λ′ = (α, β) + Λ_{0,1}^{1,0} ∈ Y_{n,1}^{1} is distinguished [7, 4.5]. Let Λ = (α, β) + Λ_{0,1}^{2,0} ∈ X_{n,1}^{2} with entries (c_{1}, …, c_{2m+1}) denote the symbol of the Springer correspondent of E. Since Λ′ is distinguished, it follows that c_{i} = c_{i+1} implies that i is odd. By (2.10) this means that between any two consecutive even entries of λ := i_{0}(E) there is an even number of odd entries, and above the largest even entry there is an odd number of odd entries. Thus the dual partition λ^{∗} also parametrizes a unipotent class of G, and hence λ ∈ d_{0}(X_{G}) by [19, III.3.11]. The converse is also clear.

Finally, assume that G = SO_{2n} is of type D_{n} with Weyl group W′_{n}. An irreducible character E = (α, β) ∈ Irr(W′_{n}) (normalized so that α and β have the same number of parts) is special if and only if Λ′ = (α, β) + Λ_{0,0}^{1,0} ∈ Y_{n,0}^{1} is distinguished [7, 4.6 and 4.7]. According to 2(D) let Λ = (α, β) + Λ_{0,0}^{2,0} ∈ Y_{n,0}^{2} with entries (c_{1}, …, c_{2m}) denote the symbol of the Springer correspondent of E. Since Λ′ is distinguished, it follows that c_{i} = c_{i+1} implies that i is odd. By (2.10) this means that between any two consecutive even entries of λ := i_{0}(E) and also above the largest even entry of λ there is an even number of odd entries. It follows from [19, III.3.10] that e(λ) = λ, and hence λ ∈ d_{0}(X_{G}) since e = d_{0}^{2}. The converse is again obvious. This completes the proof. □

Proposition 4.5. In the above set-up, let ρ be an irreducible character of G^{F} in the series defined by s, and ρ_{1} be the corresponding unipotent character of G(s)^{F}. Let C_{1} be the unipotent support of ρ_{1}. Then the unipotent support of ρ is given by j_{G(s),G}(C_{1}) where j_{G(s),G} denotes Spaltenstein’s generalized induction [19, III.12].

Proof. First note that the Weyl group W_{s} of C_{G^{∗}}(s) (together with the action of F on it) can be identified with the Weyl group of G(s). By the remarks preceding Proposition 4.2, we have i_{G(s)}(E_{1}) = (C_{1}, 1) where E_{1} ∈ Irr(W_{s})^{F} is the unique special character such that ρ_{1} has non-zero multiplicity in the almost character R_{E_{1}}^{1} of G(s)^{F}. Furthermore, by the constructions in Section 3(C) and Theorem 3.7, the unipotent support of ρ is the unique unipotent class C such that i_{G}(j_{W_{s}}^{W}(E_{1})) = (C, 1).

We can express this by the equality C = j_{G(s),G}(C_{1}), where j_{G(s),G}: X_{G(s)}^{0} → X_{G} is the map obtained by composing the map Φ_{G} in Section 3(C) with the bijective correspondence (given by the Springer correspondence) between the special unipotent classes of G(s) and the special characters in Irr(W_{s}). Note that this map j_{G(s),G} is ‘independent of the characteristic’ (cf. the remarks preceding Proposition 4.2).

It remains to show that the family of maps {j_{G(s),G} | s ∈ G^{∗} semisimple} satisfies the conditions in [19, III.12.3] for generalized induction maps. By standard reduction arguments we may assume that G is a simple algebraic group; note also that the Frobenius map F does not play a role for this. Moreover, we can assume that we are in good characteristic (or that G is a group over ℂ). By Lemma 4.4, each map j_{G(s),G} is defined on the ‘correct’ subset of unipotent classes of G(s). We now have to check that the five characterizing conditions in [19, III.12.3] hold for j_{G(s),G}.

Condition (I) is clear by Lemma 4.4. Conditions (IV) and (V) hold by the remarks in [7, (13.3)] concerning the map Φ_{G}. Now consider condition (II) and, instead of condition (III), the condition (III’) described in [19, III.12.6]: these are satisfied by the transitivity of the Macdonald j-induction and the fact (already used at the end of the proof of Proposition 4.1) that the j-induction from a Levi subgroup corresponds to the usual induction of unipotent classes. Thus, conditions (I), (II), (III’), (IV) and (V) are satisfied for the maps j_{G(s),G}. If G is a classical group, we are done by [19, III.12.6]. If G is an exceptional group, it remains to check that j_{G(s),G} yields the same results as in the table in [19, p.221]. We have done this using induce/restrict matrices between the characters of the corresponding Weyl groups and the tables for the Springer correspondence in [20]. This completes the proof. □

C. Cuspidal characters and cuspidal classes.

The Frobenius map F: G → G can be decomposed in the form F = F_{q} ∘ F_{0} = F_{0} ∘ F_{q} where F_{q} acts as multiplication by q on the character group of T, while F_{0} acts as an automorphism of finite order leaving the root system invariant.

Now let G′ be a group of the same type as G but over an algebraically closed field of characteristic 0, as in Section 2. Let T′ ⊆ B′ ⊆ G′ correspond to T ⊆ B ⊆ G. Then we also have an automorphism F_{0}: G′ → G′ such that F_{0} and F′_{0} act in the same way on the character groups of T and T′, respectively.

Let π_{G}: X_{G′} → X_{G} be Spaltenstein’s map between the partially ordered sets of unipotent classes in G′ and G, respectively. The maps F′_{0} and F induce bijections F′_{0}: X_{G′} → X_{G′} and F: X_{G} → X_{G}. Then the following diagram is commutative:

X_{G′}  ⟶^{π_{G}} X_{G}
↓^{F′_{0}}  ↓^{F}
X_{G′}  ⟶^{π_{G}} X_{G}

(Indeed, let C ∈ X_{G} and E ∈ Irr(W) such that i_{G}(E) = (C, 1) ∈ N_{G}. Since F_{q} certainly acts trivially on W and the Springer correspondence is compatible with the actions of F on Irr(W) and on N_{G} we conclude that F_{q}(C) = C. This implies the commutativity of the above diagram.) In analogy to [3, Def. 3.1] we say that an F′_{0}-stable unipotent class C ∈ X_{G′} is non-cuspidal (with respect to F′_{0}) if there exists an F′_{0}-stable proper parabolic subgroup P′ ⊂ G′ and an F′_{0}-stable Levi complement L′ ⊆ P′ such that C has non-trivial intersection with L′ and such that, if we take an F′_{0}-fixed element in this intersection, we have A_{G′}(u) ≅ A_{L′}(u). If no such parabolic subgroup exists, we say that C is cuspidal.

Note that if p is a good prime, π_{G} is an isomorphism and we also have A_{G′}(u′) ≅ A_{G}(u) where u′ ∈ C for some C ∈ X_{G′} and u ∈ π_{G}(C). It follows easily, using the above commutative diagram, that π_{G} defines a bijection between the cuspidal unipotent classes of G′ and the cuspidal unipotent classes of G (in the sense of [3]). Thus, in good characteristic, the notions of cuspidal unipotent classes for G′ and G coincide.

Proposition 4.6. Let ρ ∈ Irr(G^{F}) be cuspidal and C_{ρ} be its unipotent support. Then C_{ρ} is the image under π_{G} of a cuspidal class in G′ (with respect to F′_{0}).

Proof. Assume first that p and q are large enough so that the results in [12] are applicable. As in the proof of Proposition 3.5 (see also [12, (9.11)]) we can then express the average value of ρ on C_{ρ} in the form:

AV(C_{ρ}, ρ) = q^{d}(D_{G}(Γ_{0}), ρ),

where Γ_{0} is a sum of certain generalized Gelfand-Graev characters. Assume, if possible, that C_{ρ} is non-cuspidal. Then C_{ρ} has non-trivial intersection with an F - stable Levi subgroup L in some proper F -stable parabolic subgroup of G. Moreover, by [3, Corollary 2.7], we have

(D_{G}(Γ_{0}), ρ) = (R_{L}^{G}(D_{L}(Γ_{0}^{L})), ρ),

where R_{L}^{G} denotes Harish-Chandra induction and Γ_{0}^{L} is a similar sum as above of generalized Gelfand-Graev characters of L^{F}. Since AV(C_{ρ}, ρ) ≠ 0 we conclude that the scalar product on the right hand side is non-zero. By definition, this means that ρ is not cuspidal, a contradiction. This completes the proof in the case where p and q are large enough.

In the general case, let G̃, ρ̃ and C̃ρ̃ be as in the proof of Theorem 3.7. Since ρ and ρ̃ correspond to the same element in the parameter set X̄(W_{s}, γ), the characterization of cuspidal characters in [7, Lemma 6.3] shows that ρ̃ is a cuspidal character of G̃^{F}. The above discussion can be applied to G̃ and hence C̃ρ̃ is cuspidal.

Let π_{G}: X_{G′} → X_{G} and πG̃: X_{G′} → XG̃ be Spaltenstein’s maps. As in the proof of Theorem 3.7, we have π_{G}(C) = C_{ρ} and C̃ρ̃ = πG̃(C) for a unique class C ∈ X_{G′}.

We have seen that C̃ρ̃, and hence also C, are cuspidal classes. So C_{ρ} is the image of a cuspidal class in G′ under π_{G}. □

Remark. We have remarked above that, in good characteristic, Spaltenstein’s map π_{G}: X_{G′} → X_{G} defines a bijection between the cuspidal unipotent classes in G′ and those in G. In bad characteristic, this need no longer be true. Consider the example where G is simple of type E_{8} defined over a field of characteristic 2. Let C ∈ X_{G′} be the unipotent class denoted by D_{7}(a_{1}) in the table for E_{8} in [20]. This class has non-trivial intersection with a Levi subgroup of type D_{7} but, by [3, Proposition 3.6], it is a cuspidal class for G′. On the other hand, the image of C under π_{G} still has non-trivial intersection with a Levi subgroup in G of type D_{7}. But since A_{G}(u) = 1 for u ∈ π_{G}(C) (see the table in [20]), this now implies that π_{G}(C) is non-cuspidal (in the sense of [3]).

D. Some examples.

Example 1. Let G^{F} = Sp_{4}(2). There are 6 unipotent characters; their degrees are given as follows (see [7, p.359]):

polynomial degree value at q = 2 a_{ρ}, n_{ρ}
1  1  0, 1
\frac{1}{2}q(q^{2} + 1)  5  1, 2
\frac{1}{2}q(q + 1)^{2}  9  1, 2
\frac{1}{2}q(q^{2} + 1)  5  1, 2
\frac{1}{2}q(q − 1)^{2}  1  1, 2
q^{4}  16  4, 1

The unique non-trivial character ρ of degree 1 (with a_{ρ} = 1, n_{ρ} = 2) is a cuspidal unipotent character. There are 5 unipotent classes in the underlying algebraic group, with dim B_{u} given by 4, 2, 2, 1, 0, respectively (see [19, p.233]). Without looking at character values, it follows immediately from Theorem 3.7 that the unipotent support of ρ must be the unique class with dim B_{u} = 1. But it is not difficult to show (note that G^{F} ≅ 𝔖_{6}) that ρ has values ±1 on the two classes of regular unipotent elements in G^{F} (the elements with dim B_{u} = 0).

This is the smallest example of a character which has a non-zero value on some element in a unipotent class of strictly bigger dimension than its unipotent support.

Example 2. Let G^{F} = G_{2}(q) where q ≡ −1 mod 3. The underlying algebraic group has 5 unipotent classes, denoted by 1, A_{1}, Ã_{1}, G_{2}(a_{1}), G_{2}, with dim B_{u} given by 6, 3, 2, 1, 0, respectively (see the table in [19, p.148]).

There is a unique family of unipotent characters of G^{F} of size 8, and all characters in this family have a_{ρ} = 1 (see [7, p.372]). Using Theorem 3.7, it follows immediately that all characters in this family have unipotent support G_{2}(a_{1}). In particular, the two cuspidal unipotent characters of G^{F} have this class as their unipotent support.

Since 3|q + 1 there exists a semisimple element s ∈ G^{∗}^{F} (unique up to G^{∗}- conjugacy) such that C_{G^{∗}}(s)^{F} ≅ SU_{3}(q). Since SU_{3}(q) has a cuspidal unipotent character of degree q(q − 1), the series of G^{F} defined by s contains a cuspidal character ρ of degree q(q^{3} − 1)(q − 1) (by the Jordan decomposition of characters). We have a_{ρ} = 1 and n_{ρ} = 1. Hence Theorem 3.7 implies that G_{2}(a_{1}) is also the unipotent support of ρ.

This example shows that there can exist cuspidal irreducible characters which have the same unipotent support but which lie in series corresponding to semisimple elements s, s′ ∈ T^{∗} with W_{s} ≅ W̸_{s′}.

All of the remaining cuspidal characters lie in series corresponding to regular elements in the various anisotropic maximal tori of G^{∗}. Their restriction to the unipotent elements is given by the Green function associated with the corresponding maximal torus. Hence the unipotent support of any such character is the class of regular unipotent elements.

Example 3. Let G^{F} = G_{2}(q) where q is a power of 3. Now G has 6 unipotent classes, denoted by 1, A_{1}, (Ã_{1})_{3}, Ã_{1}, G_{2}(a_{1}), G_{2}, with dim B_{u} given by 6, 3, 3, 2, 1, 0, respectively (see the table in [19, p.148]). Let ρ be a cuspidal unipotent character. As in Example 2, the unipotent support of ρ is G_{2}(a_{1}). We have |A_{G}(u)| = 2 for u ∈ G_{2}(a_{1}), hence the formula in Theorem 3.7 yields that AV(G_{2}(a_{1}), ρ) = ±2n_{ρ}^{−1}q^{d}, where d = dim B_{u} for u ∈ G_{2}(a_{1}). From the table in [7, p.372] we see that n_{ρ} ∈ {2, 3, 6}. Hence, if ρ is such that n_{ρ} = 3 or 6 then n_{ρ} does not divide |A_{G}(u)| (but still divides 2q^{d}, of course). Such a thing cannot happen in good characteristic, see [7, (13.4.3)].

Example 4. Let G be simple modulo its center and of classical type A_{n}, B_{n}, C_{n} or D_{n}. Assume that q is odd and that G^{F} has a cuspidal unipotent character ρ_{0}. Let C_{0} be its unipotent support. By [7, (8.5.6)], ρ_{0} belongs to a (unique) F - stable cuspidal family of characters of W. From the explicit description of cuspidal families in [7, (8.1)] and the form of the Fourier matrices for classical types in [7, Chap. 4] we conclude that n_{ρ_{0}} = |A_{G}(u)| for u ∈ C_{0} (cf. also [3, Theorem 3.3(c)]). Hence, using the formula in Theorem 3.7 we obtain that AV(C_{0}, ρ_{0}) = ±q^{d}, where d = dim B_{u}. In exceptional groups, it is not always true that the average value of a cuspidal unipotent character on its unipotent support is a power of q (see, for example, type G_{2}).

5. The Suzuki and Ree groups

Now let G be a simple group of type B_{2}, G_{2} or F_{4} defined over an algebraically closed field of characteristic p = 2, 3, 2 respectively, and F a non-standard Frobenius morphism such that the finite group of fixed points G^{F} is a Suzuki or Ree group defined over a field 𝔽_{q^{2}} with q^{2} an odd power of p. The irreducible characters of G^{F} were determined by Suzuki, Ward and the second author respectively [15, 21, 22]. It turns out that the results of Lusztig on the classification of Irr(G^{F}) remain valid in these cases. More precisely, for each ρ ∈ Irr(G^{F}) there exists an F -stable conjugacy class of semisimple elements s in the dual group and a γ-invariant family of irreducible characters of W_{s}. We write E_{1} for the unique special character in the family of Irr(W_{s})^{γ} attached to ρ. Then there exists an integer n_{ρ} ≥ 1 such that the scalar product of ρ with R_{E_{1}}^{s} is given by ±n_{ρ}^{−1} (see [7, 14.2] for the case of F_{4}). It then turns out that the character degrees are all of the form

ρ(1) = ±n_{ρ}^{−1}\sqrt{p}^{−ϵ}q^{a_{ρ}}N

where ϵ = 1 if a_{ρ} is odd and ϵ = 0 if a_{ρ} is even, and where N is an integer satisfying N ≡ 1 modulo \sqrt{p}q. Then the analogue of Theorem 3.7 holds:

Proposition 5.1. Assume that G, p, F are as above. Let ρ ∈ Irr(G^{F}).

(1) There exists a unipotent support C_{ρ} for ρ.

(2) Let u ∈ C_{ρ}^{F} and d = dim B_{u}. Then we have

AV(C_{ρ}, ρ) = ±n_{ρ}^{−1}\sqrt{p}^{−ϵ}q^{d}|A_{G}(u)| and a_{ρ} = d,

where ϵ = 1 if a_{ρ} is odd and ϵ = 0 if a_{ρ} is even.

Proof. This can be checked case by case from the known character tables (see [15, 21, 22]), the order relations on the sets of unipotent classes and the values of d (see [19, Théorème II.10.15]). □

We also obtain the following analogue of Corollary 4.2:

Corollary 5.2. Let G, F be as above. Two unipotent characters of G^{F} belong to the same family if and only if they have the same unipotent support.

Finally, we note that the assertion of Proposition 4.3 also remains true: it is sufficient to observe that the partial order on the set of F -stable unipotent classes is in fact a total order in these cases.

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M.G.: U.F.R. de Mathématiques, Université Paris 7, et UMR 9994 du CNRS, 2 Place Jussieu, F–75251 Paris Cedex 05, France

   E-mail address: geck@mathp7.jussieu.fr

G.M.: I.W.R., Im Neuenheimer Feld 368, D–69120 Heidelberg, Germany

   E-mail address: malle@urania.iwr.uni-heidelberg.de