J.reine angew. Math 425(1992), 155–192  Journal für die reine und angewandte Mathematik_{\begin{matrix}c \\ ◯ \end{matrix}Walter de Gruyter Berlin· New York 1992}

The characters of the group of rational points of a reductive group with non–connected centre

By F. Digne at Amiens, G.I. Lehrer at Sydney and J. Michel at Paris^{1}

0. Introduction

Let G be a connected reductive algebraic group defined over the finite field 𝔽_{q} of characteristic p, and let F be the corresponding Frobenius endomorphism of G. Our object is to give an account of the ordinary character theory of the finite group G^{F} of 𝔽_{q}–rational points of G in the case when the centre Z = Z(G) of G is not connected. We shall take as solved the corresponding problem in the case where Z is connected, although in fact the “solution” does not extend so far as to be able to write down explicit general character values. In this work we occupy ourselves with the problem of character values rather than classification, which has been essentially completed (see [Lu2,§14.1], [DM3], [Lu5]).

The character theory in the non–connected centre case is more “arithmetical” than the connected centre case, in that it involves arithmetic in the field 𝔽_{q}, e.g. Gauss sums and Jacobi sums over 𝔽_{q} (cf. [L1]). Thus we regard as a “solution” to the character problem in this case the reduction of the character values to geometric problems such as the evaluation of Green functions, the explicit computation of the Deligne–Lusztig functors R_{L}^{G} (L an F -stable Levi subgroup of G) and their duals, and geometric questions about the conjugacy classes of G. Unfortunately we are not able to give explicit solutions to all these problems, even in the case of SL(n). However we shall make them explicit below and offer some conjectures as to their solution.

One of the motivations of this work is to give an explicit character table for SL(n, 𝔽_{q}) although many of the results apply more generally. To put the work below into perspective, we set out a program for achieving this aim:

(i) Develop a theory of Gel’fand–Graev characters for groups with non-connected centre.

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1The authors would like to thank the Australian Research Council and the École Normale Supérieure for support which made this collaboration possible

(ii) Determine χ(u) for any character χ and u regular unipotent. More generally,

do this for u distinguished unipotent in G^{F}. (iii) Determine χ(u) for any χ and u any unipotent element. (iv) Determine χ(g) for any regular χ and g ∈ G^{F}.

(v) Determine χ(g) for any χ and g ∈ G^{F}.

Of the steps in this program, (i) has been carried out in general in §2 below, while the first part of (ii) was essentially done in [L1] and has been recast in a more general setting in §3. This entails a discussion of the regular characters of G^{F} — i.e. the constituents of the Gel’fand-Graev characters — and their duals, which are the semisimple characters. This has also been done in §3, where the character values are given in terms of certain trigonometric sums σ_{z} (z ∈ H^{1}(F, Z)) which are computed explicitly using the method of [L1]. Problem (ii) for distinguished elements which are not regular has not been addressed in the current work. This problem is related to the theory of “generalized Gel’fand Graev characters” of Kawanaka [K] and the more general character values are related to the Gauss sums of prehomogeneous vector spaces (see [GK] and the references there). In particular the numbers σ_{z} (z ∈ H^{1}(F, Z)) occuring in §§3, 4 below and first studied in 1978 in [L1] are examples of such sums.

The completion of (iii) depends (given (ii)) on two steps. The first concerns a geometric question about the transformation matrix between two sets of unipotently supported class functions (see (5.9), (7.5)) which has been solved in principle in the case of SL_{n}, but remains open in general. The second step in (iii) is the computation of ^{∗}R_{L}^{G}χ for χ a character of G^{F} and L a rational Levi subgroup of G, where ^{∗}R_{L}^{G} is the Deligne-Lusztig restriction functor. This is carried out for χ regular or semisimple in §6, provided L is G-split (i.e. a Levi subgroup of a rational parabolic subgroup). The G-split case is not sufficient for the completion of (iii) in general (see (7.4)). In any case, in §6 we give an explicit formula (6.5) for the value of a semisimple character on an arbitrary unipotent element which is valid under a set of conditions which apply in the case of SL_{n}. The formula is easily adapted to a similar one for regular characters.

To complete step (iv) given (i),(ii) and (iii) also requires the computation of ^{∗}R_{L}^{G}χ for arbitrary χ and L as above (see (7.1.1)). However here again the case where L is G-split is insufficient. The problem of computing ^{∗}R_{L}^{G}χ for arbitrary L and χ regular or semisimple reduces to the statement that ^{∗}R_{L}^{G} of a Gel’fand-Graev character is a Gel’fand-Graev character, a statement which is true in the G-split case, but open in general. The solution of (v) (given (i)–(iv)) would follow by induction from the solution of all the problems above.

The authors are grateful to R.W. Richardson, who provided the proof of Theorem 5.11.1 below, concerning the intersections of conjugacy classes with parabolic subgroups.

Throughout this paper, we use the term “Levi subgroup of G” to mean a Levi subgroup of some parabolic subgroup of G.

1. The Galois cohomology group H^{1}(F, Z) and the Lang map

The Galois cohomology group H^{1}(F, Z) arises in various contexts in the theory of groups with non–connected centre. In this section we shall give some explicit realisations

Digne, Lehrer and Michel, Reductive groups with non-connected center 157 of H^{1}(F, Z) and discuss some of its subgroups. (1.1) Definition. If H is any algebraic subgroup of G with F (H) ⊂ H, the Lang map L_{H} : H → H is defined by L_{H}(h) = h^{−1}F (h). (1.2) Lemma. (Lang, Steinberg). Suppose the group H of (1.1) is connected. Then L_{H} is surjective. (1.3) Lemma. Let H be an F –stable connected closed subgroup of G such that H ⊇ Z. Then H^{1}(F, Z) ≅ L_{H}^{−1}(Z)/ZH^{F} where L_{H} is the Lang map: H → H. Proof. Consider the short exact sequence

1 → Z → H → H/Z → 1.

The corresponding cohomology long exact sequence is

1 → H^{o}(F, Z) → H^{o}(F, H) → H^{o}(F, H/Z) → H^{1}(F, Z) → H^{1}(F, H) → · · · . Since H is connected, H^{1}(F, H) = 1, whence the above sequence becomes

1 → Z^{F} → H^{F} → (H/Z)^{F} → H^{1}(F, Z) → 1.

But hZ ∈ (H/Z)^{F} ⇔ L(h) ∈ Z, i.e. ⇔ h ∈ L_{H}^{−1}(Z). Thus (H/Z)^{F} = L_{H}^{−1}(Z)/Z and H^{1}(F, Z) ≅ L_{H}^{−1}(Z)/ZH^{F} as stated. □

We shall often apply (1.3) when H is an F –stable maximal torus of G which contains Z. (1.3)′ Corollary. With notation as in (1.3) we have a canonical isomorphism

H^{1}(F, Z) ≅ Z/L_{Z}(Z).

Proof. Consider the sequence

1 → ZH^{F} → L_{H}^{−1}(Z) ⟶^{\overline{L_{H}}} Z/L_{Z}(Z) → 1.

It is easily checked to be exact, whence the result is immediate from (1.3). □

Now suppose that P is a parabolic subgroup of G and that P = LU is a Levi decomposition of P (where U is the unipotent radical and L is an F -stable Levi subgroup). Then the centre Z_{L} of L is characterised as the subgroup of semisimple elements part of the radical of P. We shall need to compare H^{1}(F, Z) and H^{1}(F, Z_{L}). For this, we take H in (1.3) to be a fixed F –stable maximal torus T of G contained in L.

(1.4) Lemma. Let L be a closed, connected F –stable Levi subgroup of G, which contains Z. There is a canonical surjection h_{L} : H^{1}(F, Z) → H^{1}(F, Z_{L}) with ker h_{L} ≅ L_{Z_{L}}^{−1}(Z)/ZZ_{L}^{F} (where Z_{L} = Z(L)).

Proof. Consider the sequence

(1.4.1)  1 → Z → Z_{L} → \overline{Z}_{L} → 1

Since \overline{Z}_{L} is the image of Z_{L} in the adjoint group of G, it is connected. Thus L is surjective on \overline{Z}_{L} = Z_{L}/Z. The long exact sequence corresponding to (1.4.1) is

(1.4.2)  1 → Z^{F} → Z_{L}^{F} → \overline{Z}_{L}^{F} → H^{1}(F, Z) ⟶^{h_{L}} H^{1}(F, Z_{L}) → H^{1}(F, \overline{Z}_{L}) = 1.

This yields the surjective map h_{L}.

Let us identify h_{L} using the canonical isomorphisms (1.3) H^{1}(F, Z) ≅ L_{L}^{−1}(Z)/ZL^{F} and H^{1}(F, Z_{L}) ≅ L_{L}^{−1}(Z_{L})/Z_{L}L^{F}. We have h_{L}(ℓ.ZL^{F}) = ℓ.Z_{L}L^{F}, and the surjectivity of h_{L} follows from the connected nature of Z_{L}/Z as above. Thus

ker h_{L} = (L_{L}^{−1}(Z) ∩ Z_{L}L^{F})/ZL^{F} = L_{Z_{L}}^{−1}(Z)L^{F} /ZL^{F} ≅ L_{Z_{L}}^{−1}(Z)/ZZ_{L}^{F} .

□

(1.5) Lemma. Let L_{1} and L_{2} be two F –stable Levi subgroups of G with L_{1}, L_{2} ⊃ T, so that L_{1} ∩ L_{2} and <L_{1}, L_{2}> are Levi subgroups of G. Write h_{1} = h_{L_{1}}, h_{2} = h_{L_{2}} and h_{12} = h_{L_{1}∩L_{2}}. Then ker h_{12} = ker h_{1}. ker h_{2}.

Proof. We write Z_{1} = Z_{L_{1}}, etc. for brevity. By duality, it is clear that Z_{L_{1}∩L_{2}} = Z_{1}Z_{2}. Thus ker h_{12} = L_{Z_{1},Z_{2}}^{−1}(Z)/(Z_{1}Z_{2})^{F} .Z. But L_{Z_{1}Z_{2}}^{−1}(Z) clearly contains L_{Z_{1}}^{−1}(Z)L_{Z_{2}}^{−1}(Z), whence ker h_{12} ⊇ ker h_{1} ker h_{2}. Conversely, suppose x ∈ Z_{1}Z_{2} and L(x) ∈ Z. Then x = z_{1}z_{2} with z_{i} ∈ Z_{i}, and L(z_{1})L(z_{2}) = L(x) = z ∈ Z.

Thus L(z_{1}) = L(z_{2})^{−1}z ∈ Z_{1} ∩Z_{2} and similarly for L(z_{2}). By the surjective nature of h_{Z_{1}∩Z_{2}}, we have L^{−1}(Z_{1} ∩ Z_{2}) = L^{−1}(Z).Z_{1} ∩ Z_{2}. Hence there exist elements c_{1} and c_{2} such that c_{i} ∈ Z_{1} ∩ Z_{2} and L(z_{i}c_{i}) ∈ Z(i = 1, 2). Moreover since L(z_{1}z_{2}) ∈ Z, we have c_{1}c_{2} ∈ L^{−1}(Z) ∩ Z_{1} ∩ Z_{2} ⊆ L^{−1}(Z) ∩ Z_{i}, (i = 1, 2). Hence x = (z_{1}c_{1}).(z_{2}c_{2}).(c_{1}c_{2})^{−1} = (z_{1}c_{1})(z_{2}c_{2}.(c_{1}c_{2})^{−1}) ∈ L_{Z_{1}}^{−1}(Z)L_{Z_{2}}^{−1}(Z), whence ker h_{12} ⊆ ker h_{1} ker h_{2}. □

We conclude this section with a result which we use implicitly several times below.

(1.6) Lemma. The normalizer N_{G}(G^{F}) is L_{G}^{−1}(Z). Thus L_{G}^{−1}(Z) acts as a group of automorphisms of G^{F}.

Proof. We have x ∈ N_{G}(G^{F}) ⇔ for g ∈ G^{F}, F (xgx^{−1}) = xgx^{−1}, i.e. L(x) centralizes g. Hence x ∈ N_{G}(G^{F}) ⇔ L(x) ∈ C_{G}(G^{F}). But it is well known that C_{G}(G^{F}) is Z. The result follows. □

2. The Gel’fand–Graev characters of G

In this section our goal is to study the “Gel’fand–Graev” characters of G, where Z(G) is not connected. In particular, we compute their duals.

Let B be a fixed F –stable Borel subgroup of G such that B ⊇ T and let U be the unipotent radical of B (whence B = T U). We write Φ, Φ^{+} and Π respectively for the root system of G with respect to T, the positive system and the simple system corresponding to the order on Φ determined by B. It is well–known that the action which F induces on the character group X(T) is given by F = qτ, where τ is induced by a permutation of Π (cf. [L2]).

Write U^{∗} = ∏_{α∈Φ^{+}−Π} U_{α}, where U_{α} is the root subgroup of U defined by α. Then U^{∗}

is the derived subgroup of U and is connected, whence

(2.1). We have U/U^{∗} is abelian and

(U/U^{∗})^{F} ≅ U^{F} /U^{∗F} .

It follows that U^{∗F} contains the derived subgroup of U^{F}. Moreover it is known (cf. [H]) that if G is quasi–simple then (U^{F})′ = U^{∗F} except when G^{F} = B_{2}(𝔽_{2}), F_{4}(𝔽_{2}) or G_{2}(𝔽_{3}). A linear (i.e. degree one) character of U^{F} factorizes through U^{F} /(U^{F})′ and hence except in the 3 cases mentioned above, through U^{F} /U^{∗F}.

(2.2) Lemma. (cf. [L2], p.259). We have

U^{F} /U^{∗F} ≅ ∏_{i∈Π/τ} U_{i}^{F}

where Π/τ is the set of orbits of τ on Π and U_{i} = ∏_{α∈i} U_{α}. Moreover U_{i}^{F} ≅ 𝔽_{q^{|i|}}^{+}.

(2.3) Definition. A linear character ϕ of U^{F} is said to be regular if ϕ is trivial on U^{∗F} and if ϕ is non–trivial on each group U_{i}^{F} of (2.2).

We next give a proof of the following theorem in the context of the present work.

(2.4) Theorem. (Lehrer [L1], 4.5). There is a natural regular permutation action of H^{1}(F, Z) on the set of T^{F} –orbits of regular characters of U^{F}.

Proof. For α ∈ Φ, denote by x_{α} : 𝔽_{q}^{+} → U_{α} an isomorphism satisfying F (x_{α}(a)) = x_{τα}(a^{q}). For each τ–orbit i on Π, take a representative α_{i} ∈ i. Then we have an explicit isomorphism: x_{i} : 𝔽_{q^{|i|}}^{+} → U_{i}^{F} where

(2.4.1)  x_{i}(a) = x_{α_{i}}(a)x_{τα_{i}}(a^{q}) ⋯ x_{τ^{|i|−1}α_{i}}(a^{q^{|i|−1}}) (a ∈ 𝔽_{q}^{+}).

Fix a character χ_{0} of 𝔽_{q^{N}}^{+} , where N is sufficiently large (e.g. ≥ |i|, all i), such that the restriction of χ_{0} to 𝔽_{q}^{+} is non–trivial.

Then any linear character ψ_{i} of U_{i}^{F} is given by

ψ_{i}(x_{i}(a)) = χ_{0}(c_{i}a) (a ∈ 𝔽_{q^{|i|}}).

for some c_{i} ∈ 𝔽_{q^{|i|}}.

Now T acts on U and on U/U^{∗}. The latter action is given (in terms of the isomor-

phism U/U^{∗} ⟶^{∼} ∏_{α∈Π} U_{α}) by

(2.4.3)  t ( ∏_{α∈Π} x_{α}(a_{α})) t^{−1} = ∏_{α∈Π} x_{α}(α(t)a_{α})(a_{α} ∈ \overline{𝔽_{q}}, t ∈ T).

If we call an element ∏_{α∈Π} x_{α}(a_{α}) ∈ U/U^{∗} regular provided a_{α} ≠ 0 for each α ∈ Π,

then we have

(2.4.4). T acts transitively on the set of regular elements of U/U^{∗}

This is a consequence of the fact that the morphism T → ∏_{α∈Π} \overline{𝔽_{q}}^{∗} given by

t ↦ (α_{1}(t), ⋯, α_{ℓ}(t))

(where Π = {α_{1}, ⋯, α_{ℓ}}) is a surjection with kernel Z.

Next we remark that

(2.4.5). The setwise stabilizer of (U/U^{∗})^{F} in T is L_{T}^{−1}(Z).

To see this, observe that t ∈ T stabilizes (U/U^{∗})^{F} if, whenever u ∈ U/U^{∗} satisfies F (u) = u then F (^{t}u) = ^{t}u. Thus F (tut^{−1}) = F (t)uF (t)^{−1} = tut^{−1}, i.e. L(t) centralizes u. But (U/U^{∗})^{F} contains regular elements, whose centralizer in T is Z; this proves (2.4.5). In fact it proves the rather stronger statement

Digne, Lehrer and Michel, Reductive groups with non-connected center 161 (2.4.6). Suppose u is a regular element of (U/U^{∗})^{F}. If t ∈ T, then ^{t}u ∈ (U/U^{∗})^{F} if and only if t ∈ L_{T}^{−1}(Z).

Combining (2.4.6) and (2.4.4) we deduce (2.4.7). The group L_{T}^{−1}(Z) acts transitively on the set of regular elements of (U/U^{∗})^{F}.

The formula (2.4.2) provides a link between regular elements and regular characters. In fact we have (2.4.8). Any linear character ϕ of (U/U^{∗})^{F} is of the form ψ = ∏_{i∈Π/τ} ψ_{i}, where ψ_{i} is of the form in (2.4.2). Moreover the map

ψ ⟼ (c_{1}, c_{2}, ⋯, c_{r}) (r = |Π/τ|).

is an isomorphism: ((U/U^{∗})^{F})^{∧} → ∏_{i∈Π/τ} 𝔽_{q^{|i|}}^{+} and ψ is regular precisely when no c_{i} is zero.

Now L_{T}^{−1}(Z) acts on the set of characters of U^{F} (and U^{F} /U^{∗F}) according to the rule

(2.4.9)  ^{t}ψ(u) = ψ(t^{−1}ut)

where t ∈ L_{T}^{−1}(Z), u ∈ U^{F}, and ψ is a character of U^{F}.

It is clear from (2.4.3) that regular characters are taken to regular characters by this action. Moreover, it follows from (2.4.7) and (2.4.8) that (2.4.10). The group L_{T}^{−1}(Z) acts transitively on the set of regular characters of U^{F}.

Finally, observe that the stabilizer of any regular character is Z, which shows that L_{T}^{−1}(Z)/Z acts regularly on the set of regular characters of U^{F}. Moreover T^{F} ≤ L_{T}^{−1}(Z), whence it follows that L_{T}^{−1}(Z)/ZT^{F} acts regularly on the T^{F} –orbits of regular characters of U^{F}. But L_{T}^{−1}(Z)/ZT^{F} is canonically isomorphic to H^{1}(F, Z) (Lemma (1.3) above) whence the result. □

Denote by Ψ the set of regular characters of U^{F}. Given the choice of χ_{0} ∈ (𝔽_{q^{N}}^{+} )^{∧} above, there is a distinguished element ψ_{0} ∈ Ψ which corresponds to taking c_{i} = 1 for each i ∈ Π/τ in (2.4.8). Explicitly, we have

(2.5)  ψ_{0} ( ∏_{i∈Π/τ} x_{i}(a_{i})) = χ_{0} ( ∑_{i∈Π/τ} a_{i}) .

Theorem (2.4) now gives an explicit parametrization of the T^{F} –orbits on Ψ by H^{1}(F, Z) = L_{T}^{−1}(Z)/ZT^{F}: for each z ∈ H^{1}(F, Z), take a representative t_{z} ∈ L^{−1}(Z), and define ψ_{z} by

(2.6)  ψ_{z} = ^{t_{z}}ψ_{0}

Then define Ψ_{z} to be the T^{F} –orbit of ψ_{z} in Ψ.

(2.7) Definition. For z ∈ H^{1}(F, Z) define the Gel’fand–Graev representation Γ_{z} by

Γ_{z} = Ind_{U^{F}}^{G^{F}} (ψ_{z}) = Γ_{z}^{G}.

We shall use the G in Γ_{z}^{G} only when necessary to avoid ambiguity.

Since all elements of Ψ_{z} are conjugate under T^{F} ≤ G^{F}, clearly Γ_{z} depends only on z. We shall see below that Γ_{z} are all distinct (in fact linearly independent).

(2.8) Theorem. The representations Γ_{z} (z ∈ H^{1}(F, Z)) are multiplicity free.

. The proof is the same as that given by Steinberg [St] in the case where the centre Z(G) is connected, and is therefore omitted. □

We next wish to examine the action of the Lusztig functor R_{L}^{G} and its dual ^{∗}R_{L}^{G} (cf. [DM2]) on the Gel’fand–Graev characters. Recall that the explicit labelling of the Γ_{z} depends on the choice of a non–trivial additive character χ_{0} ∈ (𝔽_{q^{N}}^{+} )^{∧}. Also, if the Levi subgroup L is contained in a rational parabolic subgroup P of G, then Lusztig induction coincides with Harish–Chandra induction, i.e. R_{L}^{G}(ρ) = Ind_{P^{F}}^{G^{F}} (ρ̃), where ρ̃ is the lift of ρ from L^{F} to P^{F} (ρ a character of L^{F}). Similarly the dual ^{∗}R_{L}^{G} of R_{L}^{G} is the truncation functor (see Curtis [C]) from C(G^{F}) to C(L^{F}) (C(H) being the space of complex class functions on H).

Let H be an F -stable closed subgroup of G. We say that H is G-split if there a rational Borel subgroup of G which contains a maximal torus of H. If H is a Levi subgroup, then H is G-split precisely when it is a Levi subgroup of a rational parabolic subgroup of G.

(2.9) Theorem. Suppose L(⊃ T) is a G-split Levi subgroup of G. Assume χ_{0} ∈ (𝔽_{q^{N}}^{+} )^{∧} is fixed (N sufficiently large) and the Gel’fand–Graev characters of G and of L are labelled correspondingly. Then

∗R_{L}^{G}(Γ_{z}^{G}) = Γ_{h_{L}(z)}^{L},

where z ∈ H^{1}(F, Z) and h_{L} : H^{1}(F, Z) → H^{1}(F, Z_{L}) is the canonical surjection of (1.4).

Proof. It remains to show that for each function f ∈ C(L^{F}), we have

(2.9.1)  ⟨ R_{L}^{G}(f), Γ_{z} ⟩_{G^{F}} = ⟨ f, Γ_{h_{L}(z)} ⟩_{L^{F}} = ⟨ f, ψ_{h_{L}(z)} ⟩_{L^{F} ∩U^{F}} .

In view of the above remarks, we have

(2.9.2)  ⟨ R_{L}^{G}(f), Γ_{z} ⟩_{G^{F}} = ⟨ Ind_{P^{F}}^{G^{F}} (f̃), Ind_{U^{F}}^{G^{F}} (ψ_{z}) ⟩.

= ∑_{g∈P^{F} \G^{F} /U^{F}} ⟨ f̃,^{g}ψ_{z} ⟩_{P^{F} ∩^{g}U^{F}}

by the Mackey formula.

If W_{L} is the Weyl group of L, one obtains a transversal for P^{F} \G^{F} /U^{F} by taking the set {ẇ | w ∈ X_{L}} where ẇ is a representative in N_{G}(T)^{F} of w ∈ W and X_{L} is the set of shortest right coset representatives for W_{L} in W. Now for w ∈ X_{L} we have (see e.g. [CL] (1.1),(iv))

(2.9.3)  P^{F} ∩ ^{w}U^{F} = (L^{F} ∩ ^{w}U^{F})(U_{P}^{F} ∩ ^{w}U^{F})

where U_{P} is the unipotent radical of P. Moreover on P^{F} ∩^{w}U^{F}, f̃ = f ⊗ 1 with respect to the factorization (2.9.3). Hence

(2.9.4). For w ∈ X_{L}, we have

⟨ f̃, ẇψ_{z} ⟩_{P^{F} ∩^{w}U^{F}} = ⟨ f, ẇψ_{z} ⟩_{L^{F} ∩^{w}U^{F}} ⟨ 1, ẇψ_{z} ⟩_{U_{P}^{F} ∩^{w}U^{F}} .

But ⟨ 1, ẇψ_{z} ⟩_{U_{P}^{F} ∩^{w}U^{F}} = ⟨ 1, ψ_{z} ⟩_{^{w^{−1}}U_{P}^{F} ∩U^{F}}, which is zero unless ^{w^{−1}}U_{P} ∩ U contains no root subgroup U_{α} with α ∈ Π (since ψ_{z} is regular). Since w ∈ X_{L} this implies that ⟨ 1, ψ_{z} ⟩_{^{w^{−1}}U_{P}^{F} ∩U^{F}} = 0 unless w = w_{0}w_{L}, where w_{0} (resp. w_{L}) is the longest element in W (resp. W_{L}). This is because if w ∈ X_{L} then w^{−1}(Ψ^{+} − Ψ_{L}) ∩ Ψ^{+} = w^{−1}w_{L}Ψ^{+} ∩ Ψ^{+}, which contains a root in Π unless w^{−1}w_{L} = w_{0}.

Thus each summand on the right hand side of (2.9.2) is zero, except the one where g = w_{L}w_{0} which by (2.9.4) is equal to

(2.9.5)  ⟨ f,^{ẇ_{L}ẇ_{0}}ψ_{z} ⟩_{L^{F} ∩^{w_{L}w_{0}}U^{F}} = ⟨ f,^{ẇ_{L}ẇ_{0}}ψ_{z} ⟩_{L^{F} ∩U^{F}}

since L^{F} ∩ ^{ẇ_{0}ẇ_{L}}U^{F} = L^{F} ∩ U^{F}.

Now ψ_{z}/_{L^{F} ∩U^{F}} is T^{F} – conjugate to ψ_{h_{L}(z)}, because ψ_{z} = ^{t}ψ_{0} with t ∈ L_{T}^{−1}(Z) and ψ_{0}/_{L^{F} ∩U^{F}} = ψ_{h_{L}(1)} (see (2.5)). Thus ψ_{z}/_{L^{F} ∩U^{F}} = ^{t}ψ_{h_{L}(1)} = ψ_{h_{L}(z)} by definition of the map h_{L} (see (1.4)). It follows from this argument that it is now sufficient to show that (2.9.6). ^{ẇ_{L}ẇ_{0}}ψ_{0}|_{L^{F} ∩U^{F}} is T^{F} –conjugate to ψ_{0}|_{L^{F} ∩U^{F}}.

But (2.9.6) is clear from the definition (2.5). This completes the proof of (2.9). □

(2.10) Definition. A unipotent element u of G is regular if it satisfies either of the following two equivalent conditions.

(i) u is contained in a unique Borel subgroup of G.

(ii) u is conjugate in G to an element ∏_{α∈Π} x_{α}(c_{α}) (see notation of (2.4)) where c_{α} ≠ 0

for all α.

A proof of the equivalence of the conditions (i) and (ii) may be found in [Ca, 5.1.3].

Recall next the definition of the dual of a class function of G^{F} [C]:

(2.11) Definition. Let f be a class function on G^{F}. The dual Df of f is defined by

Df = ∑_{P ⊇B}(−1)^{n_{F} (P)}R_{L}^{G}^{∗}R_{L}^{G}(f)

where the sum is over the F –stable parabolic subgroups of G which contain the F –stable Borel subgroup B, n_{F} (P) is the number of F –orbits on the simple roots corresponding to P and L is an F -stable Levi subgroup of P.

(2.12) Theorem.

(i) Let z ∈ H^{1}(F, Z). The dual DΓ_{z} vanishes outside the set of regular unipotent elements of G^{F}.

(ii) For z and z′ ∈ H^{1}(F, Z), we have

⟨ DΓ_{z}, Γ_{z′} ⟩ = (−1)^{|Π/τ|}|Z^{F} |δ_{z,z′}

where δ is the Kronecker δ.

Proof. From the definition (2.11) and Theorem (2.9) we have

(2.12.1)  DΓ_{z} = ∑_{P ⊇B}(−1)^{n_{F} (P)}R_{L}^{G}(Γ_{h_{L}(z)})

the sum being over F –stable parabolic subgroups containing B, P = LU_{P}, with L ⊇ T. Moreover

R_{L}^{G}(Γ_{h_{L}(z)}) = Ind_{U^{F}}^{G^{F}} (ψ_{z}|_{U^{F} ∩L_{J}^{F}} ⊗ 1_{U_{P}^{F}}).

Thus, using the standard Levi decomposition of the standard parabolic subgroups, (2.12.1) becomes

(2.12.2)  DΓ_{z} = ∑_{J⊆Π/τ}(−1)^{|J|} Ind_{U^{F}}^{G^{F}} (ψ_{z}|_{U^{F} ∩L_{J}^{F}} ⊗ 1_{U_{J}^{F}})

= Ind_{U^{F}}^{G^{F}} ( ∑_{J⊆Π/τ}(−1)^{|J|}(ψ_{z}|_{U^{F} ∩L_{J}^{F}} ⊗ 1_{U_{J}^{F}}))

where U_{J} is the unipotent radical of P_{J}.

Now for u ∈ U^{F}, ψ_{z}(u) depends only on the projection of u onto U^{F} /U^{∗F} = ∏_{i∈Π/τ} U_{i}^{F}

(see (2.2)). Let ψ_{z,i} be the restriction of ψ_{z} to U_{i}^{F}. Then ψ_{z} = ∏_{i∈Π/τ} ψ_{z,i} and we have

(2.12.3)  ∑_{J⊆Π/τ}(−1)^{|J|} (ψ_{z}|_{U^{F} ∩L_{J}^{F}} ⊗ 1_{U_{J}^{F}} ) = ∑_{J⊆Π/τ}(−1)^{|J|} (∏_{i∈J} ψ_{z,i})

where the right side is regarded as a character of U^{F} which factors through U^{F} /U^{∗F}

(= ∏_{i∈Π/τ} U_{i}^{F}). But

(2.12.4)  ∑_{J⊆Π/τ}(−1)^{|J|} ∏_{i∈J} ψ_{z,i} = ∏_{i∈Π/τ}(1 − ψ_{z,i})

and clearly the character on the right hand side vanishes off regular unipotent elements (i.e. it vanishes on the elements of U^{F} whose projection to U^{F} /U^{∗F} has some U_{i}– component trivial).

Combining this with (2.12.2) completes the proof of (i).

(ii) From (2.12.2) we have DΓ_{z} = Ind_{U^{F}}^{G^{F}} (ϕ), where

ϕ = ∑_{J⊆Π/τ}(−1)^{|J|}(ψ_{z}|_{U^{F} ∩L_{J}^{F}} ⊗ 1_{U_{J}^{F}})

Since the elements of N^{F} = N_{G}(T)^{F} form a transversal for the double cosets U^{F} \G^{F} /U^{F}, we have from Mackey’s formula

(2.12.5)  ⟨ DΓ_{z}, Γ_{z′} ⟩ = ⟨ Ind_{U^{F}}^{G^{F}} (ϕ), Ind_{U^{F}}^{G^{F}} (ψ_{z′}) ⟩

= ∑_{n∈N^{F}} ⟨ ϕ,^{n}ψ_{z′} ⟩_{U^{F} ∩^{n}U^{F}}

But if n ∉ T^{F}, U^{F} ∩ ^{n}U^{F} contains no regular unipotent elements. Moreover ϕ vanishes outside the set of regular unipotent elements by (i) above (or more specifically (2.12.4)).

Thus (2.12.5) reduces to

(2.12.6)  ⟨ DΓ_{z}, Γ_{z′} ⟩ = ∑_{t∈T^{F}} ⟨ ϕ,^{t}ψ_{z′} ⟩_{U^{F}}

= ∑_{t∈T^{F}} (−1)^{|Π/τ|}⟨ ψ_{z},^{t}ψ_{z′} ⟩_{U^{F}} .

Finally, ψ_{z} = ^{t}ψ_{z′} for some t ∈ T^{F} implies that z = z′ in H^{1}(F, Z), and if this condition is satisfied, there are |Z^{F} | values of t ∈ T^{F} such that ^{t}ψ_{z′} = ψ_{z}. Hence ⟨ DΓ_{z}, Γ_{z′} ⟩ = (−1)^{|Π/τ|}|Z^{F} |δ_{z,z′} as required. □

3. Character values on regular unipotent elements

The notation ψ_{z}, Γ_{z}(z ∈ H^{1}(F, Z)) depends on the isomorphisms x_{α} : \overline{𝔽_{q}}^{+} → U_{α} (a ∈ Π) and on the choice of a non–trivial complex character χ_{0} ∈ (𝔽_{q^{N}}^{+} )^{∧} (N sufficiently large) (see (2.4)). We assume this data as given. Then one has also a regular unipotent element u_{0} ∈ U^{F} whose projection to U^{F} /U^{∗F} is given by

(3.1)  \overline{u}_{0} = ∏_{α∈Π} x_{α}(1).

For each z ∈ H^{1}(F, Z) let t ∈ L_{T}^{−1}(Z) be a representative of z. Define the set U_{z} = {x ∈ G^{F} | x ∼_{G^{F}} ^{t}(u_{0}U^{∗})}. This definition is clearly independent of the representative t ∈ z.

(3.2) Proposition.

(i) The sets {U_{z}|z ∈ H^{1}(F, Z)} are disjoint.

(ii) The set of regular unipotent elements of G^{F} is the union of the U_{z}.

(iii) If the characteristic is good for G, then the U_{z} are precisely the regular unipotent classes in G^{F}.

Proof. (i) It is clearly sufficient to show that U_{1} ∩ U_{z} ≠ ϕ implies that z = 0. If U_{1} ∩ U_{z} ≠ ϕ then there are elements g, h ∈ G^{F}, and u_{1}^{∗}, u_{2}^{∗} ∈ U^{∗F} and t ∈ L_{T}^{−1}(Z) with t representing z ∈ H^{1}(F, Z) such that

(3.2.1)  ^{tg}(u_{0}u_{1}^{∗}) = ^{h}(u_{0}u_{2}^{∗}).

Hence ^{(tgt^{−1})t}(u_{0}u_{1}^{∗}) = ^{h}(u_{0}u_{2}^{∗}), so that (3.2.1) may be written

(3.2.2)  ^{t}u_{0}u^{∗} = ^{x}(u_{0}v^{∗})

for some u^{∗}, v^{∗} ∈ U^{∗F} and x ∈ G^{F}.

Since both sides are regular unipotent and lie in B, we see that ^{x}B = B, whence x ∈ B^{F}. Thus x = su_{x} with s ∈ T^{F} and u_{x} ∈ U^{F}. But then comparing the projections of the two sides of (3.2.2) to U/U^{∗} gives immediately that z = 0, completing the proof of (i).

(ii) Each unipotent element x of G^{F} is conjugate in G^{F} to an element u of U^{F}. Moreover x is regular precisely when u has regular projection to U^{F} /U^{∗F} (see (2.4.3) et seq.) by (2.10). But (2.4.7) asserts that L_{T}^{−1}(Z) acts transitively on the regular elements of U^{F} /U^{∗F}; hence u ≡ ^{t}u_{0} ( mod U^{∗F}) for some t ∈ L_{T}^{−1}(Z), whence x ∈ U_{z} for some z ∈ H^{1}(F, Z).

(iii) All the regular unipotent elements of G are conjugate in G. Hence the G^{F} –classes of regular unipotent elements in G^{F} are parametrized by H^{1}(F, A(u_{0})), where A(u_{0}) = C_{G}(u_{0})/C_{G}^{0}(u_{0}). But for u_{0} regular unipotent in B, we have C_{G}(u_{0}) = ZC_{U}(u_{0}) and in case the characteristic is good for G, C_{U}(u_{0}) is connected, whence H^{1}(F, A(u_{0})) = H^{1}(F, Z). Hence the number of regular unipotent classes in G^{F} is |H^{1}(F, Z)|. Since each set U_{z} is a union of regular unipotent classes, and they are disjoint, (iii) follows.□

(3.3) Definition. Let c be a union of conjugacy classes in G^{F}. The class function γ_{c} is defined by

γ_{c}(x) = \{ \begin{matrix}0 & if x ∉ c \\ |G^{F} |/|c| & for x ∈ c. \end{matrix}

If χ is a character of G^{F} and c is a single conjugacy class then χ(c) = ⟨ χ, γ_{c} ⟩_{G^{F}}.

(3.4) Definition. Let z ∈ H^{1}(F, Z). With notation as defined in (2.7) and (3.1), define the complex number σ_{z} by

σ_{z} = ∑_{ψ∈Ψ_{z}} ψ(u_{0}).

These σ_{z} are the same as those in [L1] and we shall give below a recasting of the method given in [L1], §5 for their evaluation. The following result gives the value of characters of G^{F} on regular unipotent elements in terms of the σ_{z}.

(3.5) Theorem. Let χ be an irreducible character of G^{F} and let z ∈ H^{1}(F, Z). Then with notation as above,

|U_{z}|^{−1} ∑_{u∈U_{z}} χ(u) = (−1)^{|Π/τ|} ∑_{z′∈H^{1}(F,Z)} σ_{zz^{′−1}}⟨ Dχ, Γ_{z′} ⟩_{G^{F}} .

Note that in view of (3.2)(iii), the left side of the equation (3.5) reduces to χ(u) when the characteristic is good for G.

Proof of (3.5). It follows from (2.12.2) and (2.12.4) that DΓ_{z} = Ind_{U^{F}}^{G^{F}} (ϕ), where ϕ factors through U^{F} /U^{∗F} and vanishes outside the set of regular elements of U^{F}. Hence DΓ_{z} is constant on the sets U_{z′} (z′ ∈ H^{1}(F, Z)). Write γ_{z′} for γ_{U_{z′}} (in the notation of (3.3)). Then by the above remark, Theorem (2.12)(i) and (3.2) we have

(3.5.1)  DΓ_{z} = ∑_{z′∈H^{1}(F,Z)} c_{z,z′}γ_{z′}

for some coefficient matrix (c_{z,z′}).

Now taking the inner product of both sides of (3.5.1) with Γ_{z′′} and using Theorem (2.12)(ii) shows that the matrix (c_{z,z′}) is invertible, with inverse matrix

(3.5.2)  (−1)^{|Π/τ|}|Z^{F} |^{−1}⟨ γ_{z}, Γ_{z′} ⟩_{G^{F}} .

But Γ_{z′} = Ind_{U^{F}}^{G^{F}} (γ_{z′}) and again γ_{z′} factorizes through U^{F} /U^{∗F}; hence Γ_{z′} is constant on the sets U_{z′′}. Now ⟨ γ_{z}, Γ_{z′} ⟩_{G^{F}} is the average value of Γ_{z′} on U_{z}; hence by the above remark, if t ∈ L_{T}^{−1}(Z) is a representative of z (∈ H^{1}(F, Z) = L_{T}^{−1}(Z)/ZT^{F}), then we have

(3.5.3)  ⟨ γ_{z}, Γ_{z′} ⟩_{G^{F}} = Γ_{z′}(^{t}u_{0}).

Further, Γ_{z′} = Ind_{U^{F}}^{G^{F}} (γ_{z′}) = Ind_{U^{F}}^{G^{F}} (γ_{1} ∘ Ad(t′)^{−1}) = Γ_{1} ∘ Ad(t′)^{−1}, where t′ is a representative in L_{T}^{−1}(Z) of z′ ∈ H^{1}(F, Z). Hence

Γ_{z′}(^{t}u_{0}) = Γ_{1}(^{(t′)^{−1}t}u_{0}) = Γ_{(z′)^{−1}z}(u_{0}) = |U^{F} |^{−1} ∑_{{g∈G^{F} |^{g}u_{0}∈U^{F} }} ψ_{(z′)^{−1}z}(^{g}u_{0}),

by the Frobenius formula for induced characters.

But ^{g}u_{0} ∈ U^{F} implies that g ∈ B^{F} since u_{0} is regular. Hence

(3.5.4)  Γ_{(z′)^{−1}z}(u_{0}) = |U^{F} |^{−1} ∑_{\begin{matrix}t∈T^{F} \\ u∈U^{F} \end{matrix}} ψ_{(z′)^{−1}z}(^{tu}u_{0})

= |Z^{F} | ∑_{ψ∈Ψ(z′)^{−1}z} ψ(u_{0})

= |Z^{F} |σ_{(z′)^{−1}z}.

Using (3.5.2), we see that the matrix c_{z,z′} has inverse matrix whose (z, z′) entry is (−1)^{|Π/τ|}σ_{z(z′)^{−1}}. Hence the relation (3.5.1) may be inverted, obtaining

(3.5.5)  γ_{z} = (−1)^{|Π/τ|} ∑_{z′∈H^{1}(F,Z)} σ_{z(z′)^{−1}}DΓ_{z′}.

Taking inner products with χ and using the fact that D is an involutory isometry completes the proof of (3.5). □

(3.5)′ Scholium. We have, for z ∈ H^{1}(F, Z),

DΓ_{z} = ∑_{z′∈H^{1}(F,Z)} c_{z,z′}γ_{z′}

where the matrix (c_{z,z′}) has inverse (−1)^{|Π/τ|} ((σ_{z(z′)^{−1}})_{z,z′}).

This is the inverse of the relation (3.5.5).

(3.6) Scholium.

(i) The class functions {Γ_{z}|z ∈ H^{1}(F, Z)} are linearly independent and hence a fortiori distinct.

(ii) The linear span of the functions {Γ_{z}|z ∈ H^{1}(F, Z)} is the dual of the space of class functions on G^{F} with regular unipotent support

Both statements are consequences of (3.5.1), the fact that the matrix (c_{z,z′}) of the proof of (3.5) is invertible and the linear independence of the γ_{z}, which follows from the disjoint nature of the sets {U_{z}|z ∈ H^{1}(F, Z)} (3.2(i)).

(3.7) Scholium. Let σ be the matrix whose (z, z′) entry is σ_{zz^{′−1}}(z, z′ ∈ H^{1}(F, Z)). Then (⟨ Γ_{z}, Γ_{z′} ⟩) = \frac{|G^{F} |}{|U_{1}|}(σ^{t}σ)^{−1}.

Proof. We have

(3.7.1)  ⟨ Γ_{z}, Γ_{z′} ⟩ = ⟨ DΓ_{z}, DΓ_{z′} ⟩

= ∑_{z_{1},z_{2}} c_{z,z_{1}}c_{z′,z_{2}}⟨ γ_{z_{1}}, γ_{z_{2}} ⟩

But the sets {U_{z}|z ∈ H^{1}(F, Z)} are disjoint and all have (the same) cardinality |U_{1}|. Hence ⟨ γ_{z_{1}}, γ_{z_{2}} ⟩ = \frac{|G^{F} |}{|U_{1}|}. Further, the matrix with (z, z′) entry c_{z,z′} is (−1)^{|Π/τ|}σ^{−1}. Substituting into (3.7.1) gives (3.7). □

(3.8) Definition. An irreducible character χ of G^{F} is regular if ⟨ χ, Γ_{z} ⟩_{G^{F}} ≠ 0 for some z ∈ H^{1}(F, Z).

To give a parametrization of the regular characters, we introduce the pair (G^{∗}, F^{∗}) where G^{∗} is an 𝔽_{q}-group dual to G and F^{∗} is a Frobenius endomorphism of G^{∗} which is dual to F.

For any semi-simple conjugacy class (s) in G^{∗}^{F^{∗}}, let A(s) = C_{G^{∗}}(s)/C_{G^{∗}}(s)^{∘} and define the class function χ_{(s)} of G^{F} by

(3.9)  χ_{(s)} = |A(s)^{F^{∗}}| ∑_{T^{∗}} ||R_{T^{∗}}^{G}(s)||^{−2}ε_{G}ε_{T^{∗}}R_{T^{∗}}^{G}(s)

where the sum is over the C_{G^{∗}}(s)^{F^{∗}} conjugacy classes of rational maximal tori T^{∗} of C_{G^{∗}}(s) and R_{T^{∗}}^{G}(s) is the Deligne-Lusztig character corresponding to the pair (T^{∗}, s) (which determines a pair (T, θ), θ ∈ (T^{F})^{∧}).

(3.10) Proposition.

(i) χ_{(s)} is a proper character of G^{F}.

(ii) For z ∈ H^{1}(F, Z), we have ⟨ χ_{(s)}, Γ_{z} ⟩_{G^{F}} = 1.

Note that in the case where Z is connected proposition (3.10) is a result of Deligne and Lusztig [DL1, (10.7)].

Proof. Let G̃ ⊃ G be an extension of G by a central torus, such that Z(G̃) is connected (see [L1] for the construction of such a G̃). Since i : G ↪ G̃ is a monomorphism, we have a dual G̃^{∗} such that i^{∗} : G̃^{∗} → G^{∗} is an epimorphism with central kernel, and since centralizers of semisimple elements in G̃^{∗} are connected (the centre of G̃ being connected implies that G̃^{∗} has a simply connected derived group), i^{∗} defines a surjective map (also denoted i^{∗}) from the semi-simple conjugacy classes in G̃^{∗F^{∗}} to those in G^{∗F^{∗}}.

Now using the fact that R_{L}^{G} commutes with restriction from G̃^{F} to G^{F} [DM3, 9.2], it is easy to see that for a semisimple conjugacy class (s̃) of G̃^{∗F^{∗}}, we have

ResG̃^{F} ; G^{F} χ_{(s̃)} = χ_{(i^{∗}(s̃))}.

Hence (i) follows from Deligne-Lusztig (loc. cit.). For (ii), we have:

⟨ χ_{(s)}, Γ_{z} ⟩_{G^{F}} = ⟨ ResG̃^{F} ; G^{F} χ_{(s̃)}, Γ_{z} ⟩_{G^{F}}

= ⟨ χ_{(s̃)}, IndG̃^{F} ; G^{F} Γ_{z} ⟩G̃^{F}

= ⟨ χ_{(s̃)}, Γ ⟩G̃^{F} (Γ = unique Gel’fand-Graev character of G̃^{F}) = 1

□

In view of (3.10) we make the (3.11) Definition. For a semisimple class (s) ⊂ G^{∗F^{∗}} and z ∈ H^{1}(F, Z), define the regular character χ_{(s),z} as the unique common constituent of χ_{(s)} and Γ_{z}.

(3.12) Proposition.

(i) The set {χ_{(s),z} | (s) is a semisimple conjugacy class of G^{∗F^{∗}}, z ∈ H^{1}(F, Z)} constitutes all the regular characters of G^{F}.

(ii) We have χ_{(s),z} = χ_{(s′),z′} if and only if (s) = (s′) and ϕ_{z} = ϕ_{z′}, where ϕ_{z} is a character of A(s)^{F^{∗}} (where A(s) = C_{G^{∗}}(s)/C_{G^{∗}}(s)^{∘}) which is defined in the proof below.

Proof. (i) follows easily from the corresponding result for G̃. If Γ is the (only) Gel’fand-Graev representation of G̃^{F} we have [DL1] Γ = ∑_{(s̃)} χ_{(s̃)}. Hence

ResG̃^{F} ; G^{F} Γ = |Z̃^{F} /Z^{F} | ∑_{z∈H^{1}(F,Z)} Γ_{z} = ∑_{(s̃)} χ_{(i^{∗}(s̃))}

and the result follows.

(ii) (cf. [L1], §3). First note that if (s) ≠ (s′) then χ_{(s)} and χ_{(s′)} are disjoint. This is because if (s̃) and (s̃′) are corresponding classes in G̃^{∗F^{∗}}, χ_{(s)} and χ_{(s′)} are the respective restrictions of χ_{(s̃)} and χ_{(s̃′)}; but by the results of [L1], restrictions of irreducible characters either coincide or are disjoint, the former being the case only when one is transformed into the other by a linear character of G̃^{F} /G^{F}. But linear characters of G̃^{F} /G^{F} correspond to elements of ker i^{∗} ⊂ Z(G̃^{∗F^{∗}}); if z ∈ ker i^{∗}, χ_{(s̃)} is transformed into χ_{s̃z}. But i^{∗}(s̃z) = i^{∗}(s̃), whence s ≠ s′ implies that χ_{(s̃)} and χ_{(s̃′)} have disjoint restrictions to G^{F}, so that χ_{(s),z} ≠ χ_{(s′),z′} unless s = s′.

Next suppose s = s′. Let A be the fundamental group of DG^{∗} (i.e. its weight lattice modulo its root lattice, and where DH is the derived group of H). Define the monomorphism α_{s} : A(s) → A as follows. First identify A with ker i^{∗} ∩ D(G̃^{∗}) which is a subgroup of Z(D(G̃^{∗})). Define α′_{s} : C_{G^{∗}}(s) → A by α′_{s}(g) = [g̃, s̃] (= g̃s̃g̃^{−1}s̃^{−1}), where g̃ and s̃ are mapped onto g and s respectively by i^{∗}. One checks easily that α′_{s} is a homomorphism with kernel C_{G^{∗}}(s)^{∘}. Let α_{s} be the induced map on A(s)(= C_{G^{∗}}(s)/C_{G^{∗}}(s)^{∘}).

Now our assumed data has included a bijection between pairs (T^{∗}, s) and pairs (T, θ) (see (3.9) above) which comes from chosen embeddings: \overline{𝔽_{q}}^{×}⟶^{∼}(ℚ/ℤ)_{p′} ↪ \overline{ℚ}^{×}. This same data yields an isomorphism

(3.12.1)  ω : H^{1}(F, Z)⟶^{∼}(A^{F^{∗}})^{∧}

which comes from the identification G̃^{F} /G^{F} Z̃^{F} ⟶^{∼}H^{1}(F, Z) (cf. [L1]). This is the map ω : z ↦ ϕ_{z} of the statement.

Its dual ω̂ : A^{F^{∗}} → (H^{1}(F, Z))^{∧} is defined as follows: (ker i^{∗})^{F^{∗}} corresponds under duality to characters of G̃^{F} /G^{F}, and (ker i^{∗})^{F^{∗}} ∩D(G̃^{∗}) corresponds to those characters of G̃^{F} /G^{F} which are trivial on Z̃^{F}, hence to characters of G̃^{F} /G^{F} Z̃^{F} ≃ H^{1}(F, Z).

Now to see (ii), let z ∈ H^{1}(F, Z) and let g_{z} ∈ G̃^{F} correspond to z (i.e. g_{z} ∈ coset of G^{F} Z̃^{F}). Then we have ^{g_{z}}Γ_{z′} = Γ_{zz′} whence ^{g_{z}}χ_{(s),z′} = χ_{(s),zz′}, since ^{g_{z}}χ_{(s̃)} = χ_{(s̃)}. Hence we have χ_{(s),z} = χ_{(s),z′} if and only if g_{z^{−1}z′} fixes χ_{(s),z}. We are therefore reduced to showing that g_{z} fixes χ_{(s),z′} if and only if the restriction ϕ_{z} of ω_{z} to A(s)^{F^{∗}} is trivial.

Now write H(s) = {θ ∈ (G̃^{F} /G^{F} Z̃^{F})^{∧} | θχ_{(s̃)} = χ_{(s̃)}} and write K(s) = {z ∈ H^{1}(F, Z) | ^{g_{z}}χ_{(s),z′} = χ_{(s),z′}} (K(s) is independent of z′). Since the restriction of χ_{(s̃)} to G^{F} is multiplicity free (by (3.10)), it follows (e.g. by [L1, §2]) that H(s) and K(s) are in duality, i.e. that K(s) = {z ∈ H^{1}(F, Z) | θ(z) = 1∀θ ∈ H(s)}. Now θ is mapped by the isomorphism ω̂^{−1} (see above) to an element s_{θ} ∈ A^{F^{∗}} ⊂ (ker i^{∗})^{F^{∗}}, and θχ_{(s̃)} = χ_{(s̃)} if and only if (s_{θ}s̃) = (s̃), i.e. if s_{θ}s̃ = g^{−1}s̃g for some g ∈ G̃, i.e. if s_{θ} ∈ A(s). It follows that θ ∈ H(s) ⇔ s_{θ} ∈ A(s), whence z ∈ K(s) ⇔ Res_{A(s)^{F^{∗}}}^{A^{F^{∗}}} ω_{z} = 1. □

(3.13) Scholium. The group (H^{1}(F, Z))^{∧} permutes the geometric conjugacy classes of G^{F}. The class (s) ∈ G^{∗F^{∗}} has isotropy group A(s)^{F^{∗}} (identified as a subgroup of (H^{1}(F, Z))^{∧} via (3.12.1)).

(3.14) Corollary. We have, for each z ∈ H^{1}(F, Z), Γ_{z} = ∑ χ_{(s),z} where the sum is over the semisimple classes (s) in G^{∗F^{∗}}.

We are now able to compute the value of an arbitrary character of G^{F} on regular unipotent classes, at least in good characteristic; in general we give the average value on the sets {U_{z} | z ∈ H^{1}(F, Z)}. Let χ be any irreducible character of G^{F}. For brevity, for the remainder of this section, we write

(3.14.1)  χ_{z} = |U_{z}|^{−1} ∑_{u∈U_{z}} χ(u)

It follows from Theorem (3.5) that χ_{z} = 0 unless ⟨ Dχ, Γ_{z′} ⟩ ≠ 0 for some z′ ∈ H^{1}(F, Z). By (3.12)(i) and using the fact that D permutes the irreducible characters of G^{F} up to sign, we deduce that χ_{z} = 0 unless Dχ = ±χ_{(s),z′} for some semisimple class (s) ⊂ G^{∗}^{F^{∗}} and z′ ∈ H^{1}(F, Z).

(3.15) Theorem.

(i) Let χ be any irreducible character of G^{F}. Then χ_{z} = 0 (notation as in (3.14.1)) for all z ∈ H^{1}(F, Z) unless χ = ±Dχ_{(s),z} for some semi-simple class (s) ⊂ G^{∗}^{F^{∗}} and z ∈ H^{1}(F, Z).

(ii) For each pair ((s), z) as in (i), ρ_{(s),z} = ε_{G}ε_{C_{G^{∗}} (s)}Dχ_{(s),z} is an irreducible character of G^{F}. Moreover ρ_{(s),z} = ρ_{(s′),z′} if and only if χ_{(s),z} = χ_{(s),z′} (see (3.12) (ii)).

(iii) We have

|U_{z′}|^{−1} ∑_{u∈U_{z′}} ρ_{(s),z}(u) = ε_{s} ∑_{a∈(A^{F^{∗}})_{s}^{∧}} σ_{z′z^{−1}a}

where (A^{F^{∗}})_{s}^{∧} = {λ ∈ (A^{F^{∗}})^{∧} | Res_{A(s)^{F^{∗}}}^{A^{F^{∗}}} (λ) = 1}, (A^{F^{∗}})^{∧} is identified with

H^{1}(F, Z) via (3.12.1) and ε_{s} = ε_{C_{G^{∗}} (s)}ε_{Rad(G)}.

Proof. The statement (i) follows from the preamble to the statement above, together with the fact that D is an involution.

(ii) Let (s̃) be a semisimple class of G̃^{∗F^{∗}} which lies over (s) (i.e. i^{∗}(s̃) = s). Then by (3.10) we have

(3.15.1)  ResG̃^{F} ; G^{F} χ_{(s̃)} = ∑_{z} χ_{(s),z}

where the sum is over a complete set of distinct characters χ_{(s),z} (z ∈ H^{1}(F, Z)). Now apply the functor D to both sides of (3.15.1), taking into account that D commutes with ResG̃^{F} ; G^{F} (cf. [DM3, loc. cit.]), to obtain

(3.15.2)  ResG̃^{F} ; G^{F} (εG̃ε_{CG̃^{∗} (s̃)}ρ_{(s̃)}) = ∑_{z} Dχ_{(s),z}

where ρ_{(s̃)} is the (semisimple) character defined in [DL, (10.7)].

It follows that

(3.15.3)  ResG̃^{F} ; G^{F} (ρ_{(s̃)}) = ∑_{z} ε_{G}ε_{C_{G^{∗}} (s)}Dχ_{(s),z}

where the sum is over the same set as (3.15.1). Moreover (3.15.3) is the irreducible decomposition of the left hand side. The statement (ii) follows.

(iii) Apply (3.5): we have

(3.15.4)  |U_{z′}|^{−1} ∑_{u∈U_{z′}} ρ_{(s),z}(u) = ε_{G}ε_{Rad(G)} ∑_{z′′∈H^{1}(F,Z)} σ_{z′z^{′′−1}}⟨ Dρ_{(s),z}, Γ_{z′′} ⟩

Now by (ii), we have Dρ_{(s),z} = ε_{G}ε_{C_{G^{∗}} (s)}χ_{(s),z} (since D^{2} = id). Moreover ⟨ χ_{(s),z}, Γ_{z′′} ⟩_{=}1 if z′′ = za, where a ∈ (A^{F^{∗}})_{s}^{∧}, and is zero otherwise. The formula follows. □

The characters {ρ_{(s),z} | (s) a semisimple class in G^{∗}^{F^{∗}}, z ∈ H^{1}(F, Z)} will be called the semisimple characters of G^{F}. They are precisely the ones which take non-zero values on regular unipotent classes (at least in good characteristic). Observe that χ_{(s),z} is a semisimple character only when (in the notation of (3.15.1) above) Dχ_{(s̃)} = ±χ_{(s̃)} and this occurs precisely when s̃ is regular semisimple (whence s is regular semisimple). Hence the regular semisimple characters are

{χ_{(s),z} | (s) is a regular semisimple class in G^{∗}^{F^{∗}}, z ∈ H^{1}(F, Z)}.

Thus in particular χ_{(s),z} has average value zero on U_{z′} (∀z′ ∈ H^{1}(F, Z)) unless s is regular semisimple. Note that (3.15) gives the value of ρ_{(s),z} on U_{z′} in the case where this is a single conjugacy class,e.g. where the characteristic is good for G.

Note that from (3.12)(ii) and its proof and (3.15) it is easy to deduce the degree of χ_{(s),z} from that of χ_{(s̃)}, which is known (cf. [Ca,(8.4.9)]). The same comment applies to the degree of ρ_{(s),z}.

4. On certain Gaussian sums

Theorem 3.5 gives the value (at least in good characteristic) of a character χ of G^{F} on the regular unipotent classes, assuming that the effect of D on irreducibles is known (this is given in the Howlett–Lehrer [HL]) and that the σ_{z} are known. We outline in this section how to compute the σ_{z}, following essentially the method of [L1], but using the context of the present work.

To evaluate σ_{z} = ∑_{ψ∈Ψ_{z}} ψ(u_{0}), recall (see the proof of (2.4)) that L_{T}^{−1}(Z)/Z acts

regularly on the regular characters of U^{F}. Hence for each regular character ψ ∈ Ψ, there is an element t_{ψ} ∈ L_{T}^{−1}(Z), unique modulo Z, such that ψ = ^{t_{ψ}}ψ_{0} (notation as in (2.5)).

Now recall from (1.3)′ the canonical isomorphism \overline{L} : H^{1}(F, Z) = L_{T}^{−1}(Z)/T^{F} Z → Z/L(Z), given by \overline{L}(tT^{F} Z) = L(t)L(Z). For the evaluation of the σ_{z} it is convenient to think of the elements z of H^{1}(F, Z) as cosets of L(Z) in Z. Thus

(4.1)  σ_{z} = ∑_{t} ^{t}ψ_{0}(u_{0})

where the sum is over \{t ∈ L_{T}^{−1}(Z)/Z|\overline{L}(t) = z\} and we use \overline{L} to denote the induced map on L_{T}^{−1}(Z)/Z.

Using the notation of (2.4.8) and (2.5), (4.1) may be written

(4.2)  σ_{z} = ∑_{t} ∏_{i∈Π/τ} χ_{0}(α_{i}(t))

where the sum is over the same range as (4.1), and the α_{i} form a set of representatives for Π/τ (cf. (2.4)).

Now H^{1}(F, Z) is a finite abelian group, whence there is an isomorphism from H^{1}(F, Z) to a direct product of finite cyclic groups. We express this in

(4.3). There is an isomorphism \overline{γ} : H^{1}(F, Z) → ∏_{i=1}^{r} μ_{d_{i}} where μ_{d} is the group of d^{th}

roots of unity in \overline{𝔽_{q}}.

Write \overline{γ}_{i} for the i^{th} component of the map \overline{γ} of (4.3). Now recall again the canonical isomorphism \overline{L} : H^{1}(F, Z) → Z/L(Z) which induces L̂ : (Z/L(Z))^{∧} → (H^{1}(F, Z))^{∧}. Moreover (Z/L(Z))^{∧} ≅ {ϕ ∈ Ẑ|L(Z) ⊆ ker ϕ} = (Z/Z^{∘})^{∧F} ≅ X(Z/Z^{∘})^{F} ⊆ X(Z) since Z/Z^{∘} is finite.

Next, use the exact sequence

0 → X(T/Z) → X(T) → X(Z) → 0

to lift the characters \overline{γ}_{i} from X(Z) to γ_{i} ∈ X(T).

(4.5) Lemma. Maintaining the above notation, let γ_{1}, …, γ_{r} be arbitrary liftings of \overline{γ}_{1}, …, \overline{γ}_{r}. Then the element t ∈ L_{T}^{−1}(Z)/Z satisfies \overline{L}(t) = z if and only if \overline{γ}_{k}(z) = ((F − 1)γ_{k})(t) for k = 1, ⋯, r.

Notice that the value γ_{i}((F −1)t) is well defined even though t is a coset in L_{T}^{−1}(Z)/Z, since γ_{k}|_{Z} = \overline{γ}_{k} ∈ X(Z)^{F}, and therefore the character γ_{k} ∘ (F − 1) = (F − 1)γ_{k} is trivial on Z.

Proof of (4.5). We have \overline{L}(t) = z ⇔ all characters of Z/(F − 1)Z take the same values on z and \overline{L}(t). This is equivalent to \overline{γ}_{k}(z) = \overline{γ}_{k}(\overline{L}(t))∀k, i.e. \overline{γ}_{k}(z) = γ_{k}((F − 1)t) = ((F − 1)γ_{k})t ∀k. □

Next observe that we have an exact sequence

1 → <Φ> → X(T/Z) → P → 1

where P is a finite p-group. Hence there exists a power p_{0} of p such that for any α ∈ X(T/Z) we have p_{0}α ∈ <Φ>. Moreover since X(Z/Z^{∘})^{F} is a finite abelian p′- group, p_{0} may be chosen so that p_{0}\overline{γ} = \overline{γ} for any \overline{γ} ∈ X(Z/Z^{∘})^{F}. So for any lifting γ ∈ X(T) of \overline{γ} ∈ X(Z/Z^{∘})^{F}, p_{0}γ is another lifting of \overline{γ}.

(4.6) Lemma. The liftings {γ_{k}|k = 1, ⋯, r} above may be chosen so that (F − 1)γ_{k} is of the form

∑_{i∈Π/τ} \frac{c_{i,k}}{d_{k}} (q^{|i|} − 1)α_{i}

where c_{i,k} ∈ ℤ and if c_{i,k} ≠ 0 then d_{k} divides q^{|i|} − 1 (α_{i} as in (2.4)).

Proof. Since \overline{γ}_{k} has order d_{k}, for any lifting γ′_{k}, d_{k}γ′_{k} is trivial on Z and therefore we have γ′_{k} ∈ \frac{1}{d_{k}p_{0}} <Φ>, whence we have another lifting γ_{k} = p_{0}γ′_{k} = ∑_{α∈Π} \frac{c_{α,k}}{d_{k}} α for

certain integers c_{α,k}. Moreover as observed above, (F − 1)γ′_{k} is trivial on Z and so (F − 1)γ_{k} ∈ <Φ>. Hence

(4.6.1)  ∑_{α∈Π} \frac{c_{α,k}}{d_{k}} · q · τ^{−1}(α) − ∑_{α∈Π} \frac{c_{α,k}}{d_{k}} α ∈ Φ

which entails

(4.6.2)  \frac{qc_{τα,k} − c_{α,k}}{d_{k}} ∈ ℤ for α ∈ Π.

Now fix i ∈ Π/τ. Define c′_{τ^{−j}α_{i},k} = q^{j}c_{α_{i},k} (j = 0, 1, ⋯). If γ′_{k} = ∑_{α∈Π} \frac{c′_{α,k}}{d_{k}} α, then we

have

(4.6.3)  γ′_{k} − γ_{k} = ∑_{i∈Π/τ} ∑_{j=0}^{|i|−1} \frac{(q^{j}c_{α_{i},k} − c_{τ^{−j}α_{i},k})}{d_{k}}τ^{−j}α_{i}.

But q^{j}c_{α_{i},k} − c_{τ^{−j}α_{i},k} ≡ 0 mod d_{k} by repeated application of (4.6.2). It follows that γ′_{k} − γ_{k} ∈ <Φ> and γ′_{k} is a lifting of the type stipulated in (4.6), with c_{i,k} = c_{α_{i},k}. □

We now assume liftings γ_{k} chosen in accordance with (4.6). Then for t ∈ T we have

(4.6.4)  ((F − 1)γ_{k})(t) = ∏_{i∈Π/τ} α_{i}(t)^{(q^{|i|−1})c_{i,k}/d_{k}}

Combining (4.2), (4.5) and (4.6) we deduce that

(4.7)  σ_{z} = ∑_{(s_{i})_{i∈Π/τ}} χ_{0} ( ∑_{i∈Π/τ} s_{i})

where the sum is over the elements (s_{i}) ∈ ∏_{i∈Π/τ} 𝔽_{q^{|i|}} which satisfy

(4.7.1)  ∏_{i∈Π/τ} s_{i}^{(q^{|i|}−1)c_{i,k}/d_{k}} = \overline{γ}_{k}(z) ∈ 𝔽_{q^{N}}^{∗}

for k = 1, 2, ⋯, r.

To evaluate the σ_{z} in terms of Gauss sums (in the manner of [L1]) we introduce the “Mellin transforms” of the σ_{z}: (4.8) Definition. For ζ ∈ (H^{1}(F, Z))^{∧}, define σ_{ζ} = ∑_{z∈H^{1}(F,Z)} ζ(z)σ_{z}.

Then clearly σ_{z} = |H^{1}(F, Z)|^{−1} ∑_{ζ∈(H^{1}(F,Z))^{∧}} ζ(z^{−1})σ_{ζ}. Thus knowledge of the σ_{ζ} is equivalent to that of the σ_{z}. (4.9) Definition. For a character ϕ ∈ (𝔽_{q^{N}}^{∗})^{∧}, define the Gauss sum

G(ϕ) = ∑_{x∈𝔽_{q^{N}}^{∗}} ϕ(x)χ_{0}(x)

where χ_{0} is the (fixed) additive character of (2.5).

(4.10) Theorem. With the above notation, we have

σ_{ζ} = ∏_{i∈Π/τ} G (∏_{k} ζ_{i,k}^{c_{i,k}})

where ζ_{i,k} is the multiplicative character of 𝔽_{q^{i}} defined by ζ_{i,k}(x) = ζ(z), where z ∈ μ_{k} ⊆ H^{1}(F, Z) is defined by \overline{γ}_{k}(z) = x^{(q^{|i|}−1)/d_{k}} for each k.

Proof. We have from (4.7) and (4.8)

σ_{ζ} = ∑_{z∈H^{1}(F,Z)} ∑_{(s_{i})_{i∈Π/τ}} ζ(z) ∏_{i∈Π/τ} χ_{0}(s_{i})

where the inner sum is over the (s_{i})_{i∈Π/τ} which satisfy (4.7.1). But for each k, the

condition ∏_{i∈Π/τ} s_{i}^{(q^{|i|}−1)c_{i,k}/d_{k}} = \overline{γ}_{k}(z) implies that ∏_{i∈Π/τ} ζ_{i,k}(s_{i})^{c_{i,k}} = ζ(z). Thus

(4.10.1)  σ_{ζ} = ∑_{(s_{i})_{i∈Π/τ}} ∏_{k} ζ_{i,k}(s_{i})^{c_{i,k}} ∏_{i∈Π/τ} χ_{0}(s_{i})

where the sum is over all (s_{i}) ∈ ∏_{i∈Π/τ} 𝔽_{q^{|i|}}^{∗}. The right hand side of (4.10.1) may be

written

∏_{i∈Π/τ} ( ∑_{s_{i}∈𝔽_{q^{|i|}}^{∗}} (∏_{k} ζ_{i,k}^{c_{i,k}}) (s_{i})χ_{0}(s_{i})) ,

which is equal to ∏_{i∈Π/τ} G(∏_{k} ζ_{i,k}^{c_{i,k}}) as stated. □

The above exposition generalizes that of [L1] somewhat. We therefore treat the example of the special linear groups in the present context.

(4.11) Example. The case G^{F} = SL(n, 𝔽_{q}), q odd. Here Z ≅ μ_{n} is the group of scalar matrices and H^{1}(F, Z) ≅ μ_{n}/μ_{n}^{q−1} ≅ μ_{d}, where d = (q − 1, n) ((a, b) denotes the g.c.d. of the integers a and b). Thus r (above) = 1 and one character γ = γ_{1} suffices. If Π = (α_{1}, ⋯, α_{n−1}) is the usual set of simple roots, i.e. α_{i}(diag(t_{1}, ⋯, t_{n})) = t_{i}t_{i+1}^{−1}, then one may take γ = \frac{1}{d}(α_{1} +2α_{2} +⋯+(n−1)α_{n−1}). So γ(diag(a, a, ⋯, a)) = a^{−n/d}, confirming that \overline{γ} generates (H^{1}(F, Z))^{∧}. Now the characters ζ_{i,k} of (4.10) are all equal (there is only one k, and |i| = 1 ∀i). Given ζ ∈ μ̂_{d}, write ζ̃ for this common value. Then (recalling that χ_{0} ∈ (𝔽_{q^{N}}^{+} )^{∧} is fixed) we have, by 4.10

σ_{ζ} = ∏_{i=1}^{n−1} G(ζ̃^{i}).

Let e be the order of the character ζ̃ then e|d and the following formulae are well– known and may be deduced from the results of §6 in [L1].

σ_{ζ} = \{ \begin{matrix}(−1)^{n/e−1}q^{n(e−1)/2e} & if e is odd \\ (−1)^{n/e−1}((−1)^{(q−1)/4}q)^{n(e−1)/2e}G(ε)^{n/e} & if e is even \end{matrix}

where ε is the unique multiplicative character of 𝔽_{q}, of order 2 (the Legendre symbol).

Thus σ_{ζ} is a function of the order e of ζ, and we may denote it by σ(e).

(4.11.1) Theorem. Let e be the order of z ∈ H^{1}(F, Z). Then

σ_{z} = ∑_{k|d} σ(d/k)μ(e/(k, e))ϕ(d/k)/(dϕ(e/(e, k)))

where μ is the Möbius function and ϕ is the Euler ϕ–function.

Proof. If ζ is a generator of (H^{1}(F, Z))^{∧}, we have

σ_{z} = d^{−1} ∑_{i=1}^{d} σ(d/(i, d))ζ^{i}(z) = d^{−1} ∑_{k|d} σ(d/k) ∑_{\begin{matrix}0<i<d \\ (i,d)=k \end{matrix}} ζ^{i}(z).

Let us write g(m, j) = ∑_{ξ} ξ^{j} where the sum is over the primitive m^{th} roots of unity

and j|m. Then ∑_{\begin{matrix}0<i<d \\ (i,d)=k \end{matrix}} ζ^{i}(k) = g(d/k, d/lcm(k, e)). Finally, to evaluate g(m, j), observe

that the map ζ ↦ ζ^{j} takes the primitive m^{th} roots of unity to the primitive (m/j)^{th} roots and has constant fibres which have cardinality ϕ(m)/ϕ(m/j). Thus g(m, j) = ϕ(m)/ϕ(m/j) · g(m/j, 1), and it is well–known that g(k, 1) = μ(k). This completes the proof of (4.11.1). □

5. On the geometry of unipotent classes

We turn to part (iii) of the program outlined in our introduction, i.e. to compute χ(u) for a character χ and u an arbitrary unipotent element in G^{F}. We shall see that the problems that arise in this connection are of wider applicability than merely to (iii). Henceforth we shall use the following notation: for a group H and x ∈ H, (x)_{H} is the conjugacy class of x in H and γ_{x} is the class function γ_{(x)_{H}} defined in (3.3). To avoid ambiguity we sometimes write γ_{x}^{H} = γ_{x}.

The method we employ is as follows. Suppose v is a unipotent element of G^{F}, and suppose further that we have explicit expressions

(5.1.1)  γ_{v}^{G^{F}} = ∑_{K} R_{K}^{G}(γ_{v_{K}}) · n_{K,v}.

where the sum is over certain F –stable Levi subgroups K, v_{K} is regular (or even distinguished) unipotent in K and the n_{K,v} are integers. Then we have

(5.1.2)  χ(v) = ⟨ χ, γ_{v}^{G^{F}} ⟩ = ∑_{K} n_{K,v}⟨ χ, R_{K}^{G}(γ_{v_{K}}) ⟩_{G^{F}}

= ∑_{K} n_{K,v}⟨^{∗}R_{K}^{G}χ, γ_{v_{K}} ⟩_{K^{F}}

If χ is regular or semisimple (and in some other cases) we have some knowledge of ∗R_{K}^{G}χ (see §6 below). Thus the problem of computing χ(v) for χ regular and v unipotent will be approached by essentially reducing the problem to evaluating characters of smaller groups on distinguished unipotent elements. This reduction amounts to the following

(5.1) Problems.

A. Study the effect of ^{∗}R_{L}^{G} and R_{L}^{G} on γ_{u}^{G^{F}} and γ_{v}^{L^{F}}, where u, v are unipotent (particularly regular, or distinguished) elements of G^{F} and L^{F} respectively. B. For an irreducible character χ of G^{F}, give an explicit expression for ^{∗}R_{L}^{G}χ if L

is an F –stable Levi subgroup of a parabolic subgroup of G.

In this section we concentrate on problem (5.1)(A).

We begin with

(5.2) Conjecture. Let u ∈ G^{F} be regular unipotent and let L be an F –stable Levi subgroup of some parabolic subgroup P of G. Then

∗R_{L⊂P}^{G} γ_{u}^{G^{F}} = γ_{p_{L}(u)}^{L^{F}}

for some regular unipotent element p_{L}(u) ∈ L^{F}.

Note that assuming (5.2) and the fact that D commutes with ^{∗}R_{L}^{G}, the result (2.9) above could be generalized to the case where L is not necessarily G-split (i.e. where P is not necessarily rational). In fact, assuming that D commutes with R_{L}^{G} (this is known to be true for sufficiently large p and q (cf. [DM1, Theorem 2])) (5.2) is equivalent to

(5.2)′ Conjecture. For z ∈ H^{1}(F, Z), we have

∗R_{L}^{G}Γ_{z}^{G} = ε_{G}ε_{L}Γ_{z′}^{L}

for some z′ ∈ H^{1}(F, Z_{L}), where ε_{H} = (−1)^{s(H)}, and s(H) is the 𝔽_{q}-rank of H.

Recall that (5.2)′ has been proved in the case where L is G-split in Theorem (2.9). In that case z′ = h_{L}(z).

(5.3) Proposition. The statement (5.2) is true when L is G-split (i.e. if P is F –stable).

Proof. When L is G-split, we have from the definition of ^{∗}R_{L}^{G},

(5.3.1)  ^{∗}R_{L}^{G}γ_{u}^{G}(v) = |U^{F} |^{−1}|C_{G^{F}} (u)||vU ∩ (u)_{G^{F}} |
= |U^{F} |^{−1}|{g ∈ G^{F} |^{g}u ∈ vU}|

where U is the unipotent radical of P. If v is not regular, neither is any member of vU, whence the above number is 0. Thus we take v to be regular, so that there is a unique Borel subgroup B_{L} of L such that v ∈ B_{L}; by replacing B_{L} by a conjugate if necessary, we may assume that u ∈ B_{L}U = B a Borel subgroup of G. If x satisfies xu ∈ vU ⊆ B, then since u is regular, x ∈ B. Therefore

(5.3.2)  |{x ∈ G^{F} |^{x}u ∈ vU}| = |{b ∈ B^{F} |u ∈ ^{b}vU}|
= |U^{F} ||{b ∈ B_{L}^{F} |u ∈ ^{b}vU}|

where the last equality follows from the decomposition B^{F} = B_{L}^{F} U^{F} and the fact that for u ∈ U^{F} and ℓ ∈ L^{F}, ^{u}ℓU = uℓu^{−1}U = ℓ(ℓ^{−1}uℓ)U = ℓU.

Further, for b ∈ B_{L}^{F}, we have u ∈ ^{b}vU ⇔ p_{L}(u) = ^{b}v, where p_{L} is the projection from P to L with kernel U. Hence the number of b ∈ B^{F} for which u ∈ ^{b}vU is |C_{B_{L}^{F}} (v)| = |C_{L^{F}} (v)| since v is regular in L.

Thus from (5.3.1) we have

∗R_{L}^{G}γ_{u}^{G}(v) = \{ \begin{matrix}0 & if v is not regular \\ |C_{L^{F}} (v)| & if v is regular. \end{matrix}

The proof is completed by remarking that ^{∗}R_{L}^{G}γ_{u}^{G} is supported on the unipotent elements of L^{F}, by the character formula for ^{∗}R_{L}^{G}. □

(5.4) Proposition. The statement (5.2) holds when G has connected centre.

Proof. When G has connected centre, H^{1}(F, Z) = {0}, so that there is a unique regular unipotent class (u)_{G^{F}} in G^{F} and a unique Gel’fand–Graev character Γ^{G}. Moreover D_{G}(γ_{u}) = Γ^{G}, and for any F –stable Levi subgroup L we have (see Digne-Michel, C.R.A.S. 297(1983) p.92)

(5.4.1)  ^{∗}R_{L}^{G}Γ^{G} = ε_{G}ε_{L}Γ^{L}.

Since Γ^{G} is a uniform function on G^{F} (i.e. a linear combination of R_{T}^{G}(θ)’s) we have ^{∗}R_{L}^{G}D_{G}Γ^{G} = ε_{G}ε_{L}D_{L}^{∗}R_{L}^{G}Γ^{G}; hence ^{∗}R_{L}^{G}γ_{u} = ε_{G}ε_{L} · ε_{G}ε_{L}D_{L}Γ_{L} = γ_{v}, where v represents the regular unipotent class of L^{F}. □

(5.5) Definition. A unipotent element u ∈ G^{F} is parabolic if there is an F –stable Levi subgroup L of an F –stable parabolic subgroup P of G such that u is G^{F} –conjugate to a regular unipotent element of L^{F}.

Since many of our results concern the parabolic classes, we give the following parametrization.

(5.6) Proposition. The parabolic unipotent classes of G^{F} are parametrized by the set S/ ∼, where S = P(Π)^{F} × H^{1}(F, Z) (P(Π) is the power set of Π) and the equivalence relation ∼ is given by

(5.6.1)  (J, z) ∼ (J′, z′)if J′ = wJ for some w ∈ W and h_{J}(z) = h_{J}(z′),

where h_{J} = h_{L_{J}} (cf. (1.4) above).

Proof. Using the canonical isomorphism H^{1}(F, Z) ≅ L_{G}^{−1}(Z)/G^{F} Z of (1.3), there is a monomorphism: z ↦ z̃: H^{1}(F, Z) → Out(G^{F}) (in fact L_{G}^{−1}(Z) = N_{G}(G^{F})); moreover z̃ may be considered as an (outer) automorphism of any subgroup H^{F} of G^{F}, where H is a closed connected and F –stable subgroup of G, by (1.3). If P = LU is an F – stable parabolic subgroup, then z̃ defines an inner automorphism of L^{F} (i.e. is the trivial element of Out(L^{F})) precisely when z ∈ ker h_{L} (see (1.4)). As above, take the fixed regular unipotent element u_{0} ∈ G^{F}. Then the regular unipotent classes of G^{F} are represented by {z̃(u_{0})|z ∈ H^{1}(F, Z)}.

For L ⊂ P a G-split Levi subgroup, we have a well–defined regular unipotent class p_{L}(z̃(u_{0})) such that ^{∗}R_{L}^{G}γ_{z̃(u_{0})}^{G^{F}} = γ_{p_{L}(z̃(u_{0}))}^{L^{F}} by (5.3). When L = L_{J} and P = P_{J}, denote the G^{F} –class of p_{L}(z̃(u_{0})) by (J, z). Suppose J′ = wJ for some w ∈ W. Then

Adẇ · γ_{P_{L_{J}} (z̃(u_{0}))}^{L_{J}^{F}} = ^{∗}R_{L_{J′}}^{G} γ_{z̃(u_{0})}^{G^{F}} = γ_{P_{L_{J′}} (z̃(u_{0}))}.

Hence (J, z) and (wJ, z) represent the same class. Moreover since z̃ commutes with ∗R_{L}^{G} (use the character formula and the fact that z̃ fixes unipotent radicals), we have

z̃(p_{L}(u_{0})) = p_{L}(z̃(u_{0})),

whence (J, z_{1}) and (J, z_{2}) represent the same class of G^{F} if z_{1}z_{2}^{−1} ∈ ker h_{J}, i.e. h_{J}(z_{1}) = h_{J}(z_{2}).

Thus (J, z) ∼ (J′, z′) implies that (J, z) and (J′, z′) represent the same unipotent class in G^{F}.

Let u_{J,z} be a representative in L_{J}^{F} of the class (J, z). If u_{J,z} is G^{F} –conjugate to u_{J′,z′} then L_{J} is conjugate to L_{J′}, in G^{F}, because L_{J} is the centralizer of a maximal torus of C_{G}(u_{J,z}). Thus J′ = wJ, for some w ∈ W, and the same argument as that above gives h_{J}(z) = h_{J}(z′).

Finally, any parabolic unipotent class is represented by some (J, z), since it is represented in L_{J}, and all are conjugate under H^{1}(F, Z). □

Next we seek the key expression (5.1.1) in the parabolic case. Our method is to express R_{L}^{G}(γ_{v}^{L^{F}}) (for v a parabolic unipotent element of L^{F}) as a linear combination of γ_{u}^{G^{F}} ’s, and then to invert these relations obtaining equations of the form (5.1.1).

(5.7) Proposition. Let P = LU be an F –stable Levi decomposition of the F –stable parabolic subgroup P of G, and let v be an element of L^{F}. Then we have

R_{L}^{G} (γ_{v}^{L^{F}} ) = ∑_{(u)_{G^{F}}} c_{v,u}γ_{u}^{G^{F}}

where c_{v,u} = |U^{F} |^{−1}|vU ∩ (u)_{G^{F}} | and the sum is over those G^{F} –classes for which c_{v,u} ≠ 0.

Proof. For any class function χ on G^{F}, we have

(5.7.1)  χ = ∑_{(x)_{G^{F}}} |C_{G^{F}} (x)|^{−1}⟨ χ, γ_{x} ⟩γ_{x}

because the γ_{x} form an orthogonal basis of C(G^{F}).

Now

R_{L}^{G}(γ_{v})(u) = ⟨ R_{L}^{G}γ_{v}, γ_{u} ⟩_{G^{F}}
= ⟨ γ_{v},^{∗}R_{L}^{G}γ_{u} ⟩_{L^{F}}
= |U^{F} |^{−1}|C_{G^{F}} (u)||vU^{F} ∩ (u)_{G^{F}} |.

The result follows immediately from (5.7.1) □

If, in (5.7), v is unipotent, then vU consists of unipotent elements, so that the sum in (5.7) may be taken over the unipotent classes in G^{F}.

(5.8) Proposition. (cf. Richardson [R1], (5.17)(b) below) If (v)_{G} and (u)_{G} are conjugacy classes in G, say that (v)_{G} ≤ (u)_{G} if (v)_{G} ⊆ \overline{(u)_{G}}. This defines a partial order on the conjugacy classes of G. If P = LU is a Levi decomposition of a parabolic subgroup, then vU ∩ (u)_{G} = ∅ unless (v)_{G} ≤ (u)_{G}.

We remark that the varieties vU ∩ (u)_{G} also occur in the definition of induced unipotent classes [LS]: (u)_{G} is induced from (v)_{L} if dim(vU ∩ (u)_{G}) is maximal as a function of (u)_{G}. In this case it is maximal also as a function of (v)_{L}.

(5.9) Proposition. Let v be a unipotent element of G^{F}. Then

(i) There is a bijection between the set L_{v} of Levi subgroups L of G which are minimal with respect to the property of containing v and the set M_{v} of maximal tori of C_{G}(v)^{∘}. In this bijection, S ∈ M_{v} corresponds to C_{G}(S) ∈ L_{v}.

(ii) All elements of L_{v} are conjugate under an element of C_{G}(v)^{∘}, and there is an

F -stable element of L_{v}. (iii) Any two F -stable elements of L_{v} whose centre has maximal 𝔽_{q}-rank among the

elements of L_{v} are conjugate under G^{F}. (iv) If there is a G-split element in L_{v}, it belongs to the set referred to in (iii) above.

Proof. (i) Suppose L ∈ L_{v}. Then Z(L)^{∘} is a torus in C_{G}(v)^{∘}. Moreover by the minimality of L, Z(L)^{∘} is a maximal torus of C_{G}(v)^{∘}. Conversely, if S is any maximal torus of C_{G}(v)^{∘}, then clearly C_{G}(S) ∈ L_{v}.

(ii) Since all maximal tori in C_{G}(v)^{∘} are conjugate in C_{G}(v)^{∘}, their centralizers are conjugate under C_{G}(v)^{∘}, whence the conjugacy of elements of L_{v}. Given any L ∈ L_{v}, we have F (L) = cLc^{−1} for some c ∈ C_{G}(v)^{∘}. Applying Lang’s Theorem to the (connected) group C_{G}(v)^{∘} provides an F -stable element of L_{v}.

(iii) The F -stable elements of L_{v} correspond to F -stable maximal tori of C_{G}(v)^{∘} (this gives another proof of the second part of (ii)). Moreover under this correspondence, L ∈ L_{v} has centre of maximal 𝔽_{q}-rank precisely when the corresponding maximal torus is maximally split. But the maximally split rational maximal tori of C_{G}(v)^{∘} are conjugate under C_{G}(v)^{∘}^{F}, and the statement (iii) follows.

(iv) We have to show that if L ∈ L_{v} is G-split, then Z(L)^{∘} contains a maximal split torus of C_{G}(v)^{∘}. Let S be a maximal split torus of Z(L)^{∘} and suppose S ⊆ S′, where S′ is a split torus in C_{G}(v)^{∘}. Then L′ = C_{G}(S′) is a rational Levi subgroup containing v. But by Borel-Tits [BT, (4.16)(b)], since L is G-split, L = C_{G}(s) ⊇ L′. By minimality of L, we have L = L′ and therefore S = S′. □

(5.9.1) Remark. The following properties for a unipotent element u ∈ G are equivalent.

(i) u lies in no proper Levi subgroup of G.

(ii) Every semisimple element in C_{G}(u)^{∘} is contained in Z(G).

A unipotent element u ∈ G which satisfies the above conditions will be called distinguished (cf. [Ca, p.173]). It is clear from the above that for any unipotent element v ∈ G^{F}, v is distinguished in L for any L ∈ L_{v}.

Now for each unipotent class (v) in G^{F} choose an F -stable Levi subgroup L_{v} satisfying the conditions of (5.9)(iii). Then L_{v} is determined up to G^{F} -conjugacy and we have the canonically defined class functions ρ_{v} defined by

(5.9.2)  ρ_{v} = R_{L}^{G}γ_{v}^{L_{v}^{F}}

It follows from the character formula for R_{L}^{G} [DM2, (3.2)] that ρ_{v} is unipotently supported. Hence we have a relation

(5.9.3)  ρ_{v} = ∑_{(u)_{G^{F}}} c_{v,u}γ_{u}

which generalizes (5.7)

If L_{v} is G-split then c_{v,u} = 0 unless (v)_{G} is contained in the closure of (u)_{G}. It is an open question whether this is true in general. In the G-split case, the coefficients c_{v,u} ((v)_{G} ≤ \overline{(u)_{G}}) are given by the formula (5.7).

The program outlined in (5.1.2) is realisable provided that the matrix of coefficients c_{v,u} in (5.9.3) can be inverted (this general comment applies even when the assumption concerning the G–split nature of the Levi subgroup L_{v} is omitted). Thus we concern ourselves now with the invertibility of (c_{v,u}) in the parabolic case i.e.in the case where for each unipotent class (v) in G^{F}, the Levi subgroup L_{v} above is G–split. This matrix is block triangular, with blocks indexed by the unipotent classes of G, and with the rows and columns of each diagonal block indexed by the G^{F} -classes into which the given class of G splits. Clearly (c_{v,u}) is invertible if and only if its diagonal blocks are invertible.

Hence we turn to the study of the varieties vU ∩ (u)_{G}.

(5.10) Proposition. Let P = LU be a Levi decomposition of the parabolic subgroup P of G. The following are equivalent.

(i) For any ℓ ∈ L and u ∈ U, ℓu is G–conjugate to ℓ ⇒ ℓu is U–conjugate to ℓ (ii) For any ℓ ∈ L, {g ∈ G|gℓg^{−1} ∈ ℓU} = UC_{G}(ℓ). (iii) For any x ∈ P, x is in some Levi subgroup of P ⇔ π(x) ∼_{G} x, where π : P → L

is the projection with kernel U.

Proof. (ii) ⇒ (i): if ℓ ∈ L and u ∈ U with ℓu = gℓg^{−1}, then by (ii), g = vc, with v ∈ U and c ∈ C_{G}(ℓ). So ℓu = vℓv^{−1} as required.

(i) ⇒ (iii): if x ∈ P is in a Levi subgroup of P, then x ∈ ^{u}L for some u ∈ U, so that x = uℓu^{−1} for some ℓ ∈ L and π(x) = π(uℓu^{−1}) = ℓ = u^{−1}xu ≅_{G} x. Conversely suppose x ∈ P satisfies π(x) ≅_{G} x; then if x = ℓu (ℓ ∈ L, u ∈ U), we have some g ∈ G such that ℓ = gxg^{−1}. So x = ℓu is G–conjugate to ℓ, whence by (i), ∃v ∈ U such that x = vℓv^{−1} ∈ ^{v}L, which is a Levi subgroup of P.

(iii) ⇒ (ii): Suppose gℓg^{−1} = ℓu with u ∈ U. Take x = ℓu in (iii); then π(x) = ℓ ∼_{G} gℓg^{−1} = x. Hence by (iii) x lies in a Levi subgroup of P, i.e. x = vℓ′v^{−1}, with v ∈ U and ℓ′ ∈ L. Applying π, we see ℓ = ℓ′, so that x = vℓv^{−1} = gℓg^{−1}. Hence v^{−1}g ∈ C_{G}(ℓ), i.e. g ∈ U.C_{L}(ℓ), proving (ii). □

Note that it is clear from the nature of property (iii) that each of (i), (ii) and (iii) is independent of the choice of a Levi decomposition of P.

We are indebted to R.W. Richardson for the proof of the next result.

(5.11) Theorem. The three equivalent statements of (5.10) are true for any parabolic subgroup P ≤ G.

In fact we shall prove the statement (5.10)(i) in the following form.

(5.11.1) Theorem. Let P = LU be a Levi decomposition of the parabolic subgroup P of the reductive group G. Then for ℓ ∈ L, we have

(ℓ)_{G} ∩ ℓU = (ℓ)_{U}

where (ℓ)_{G} is the G–conjugacy class of ℓ in G and (ℓ)_{U} = {^{u}ℓ|u ∈ U}.

It is a result of Lusztig [[Lu1], (2.9)(a)] that the right hand side of (5.11.1) is an irreducible component of the left hand side. The Theorem therefore strengthens Lusztig’s result. Before proving it, we indicate its relevance to the matrix (c_{v,u}) of (5.9).

(5.12) Lemma. With notation as in (5.11.1), if a unipotent element v ∈ L^{F} is G– conjugate to u ∈ vU^{F}, then v is U^{F} –conjugate to u.

Proof. From (5.11) it follows that v is U–conjugate to u. But the U^{F} –orbits on (u)_{U^{F}} are parametrized by H^{1}(F, C_{U}(u)/C_{U}^{0}(u)) (by the usual argument involving Lang’s Theorem, see [Sp-St]). However, by a result of Holt and Spaltenstein [J.Aust.Math.Soc (Series A) 38 (1985), 330–350], C_{U}(u) is connected, whence the result. □

(5.12)′ Corollary. Suppose that L_{v} (in (5.9.1)) is G-split for each v. Then the matrix (c_{v,u}) of (5.9.3) is invertible.

Proof. In view of the above remarks, it suffices to prove the invertibility of the diagonal blocks of the matrix (c_{v,u}).

Suppose v, v′ are unipotent elements of L^{F} and G^{F} respectively, such that v ∼_{G} v′. If vU ∩ (v′)_{G^{F}} ≠ ϕ then let v′′ ∈ vU ∩ (v′)_{G^{F}}. Then v ∈ L^{F} is G–conjugate to v′′ ∈ vU. Hence by (5.12) v is U^{F} –conjugate to v′′, and therefore G^{F} –conjugate to v′. It follows that c_{v,v′} = 0 unless v = v′ (if v ∼_{G} v′). Thus the diagonal blocks are diagonal, with non–zero entries on the diagonal. The result follows. □

We now turn to the proof of Theorem 5.11.1. First, let P^{−} = LU^{−} be the parabolic subgroup of G which is opposite P and corresponds to the Levi subgroup L of P. Here U^{−} is the unipotent radical of P^{−}. It is well–known that ULU^{−} is an open (and hence dense) subset of G, since it is a translation of the complement of a finite union of Schubert varieties \overline{BwB}.

(5.13) Lemma. With notation as in (5.11.1), suppose g ∈ ULU^{−} and ^{g}ℓ ∈ ℓU. Then gℓ = ^{u}ℓ for some u ∈ U.

Proof. Write g = uxv with u ∈ U, x ∈ L, v ∈ U^{−}. Then by hypothesis ℓ^{−1} ^{g}ℓ = ℓ^{−1}gℓg^{−1} ∈ U. But ℓ^{−1}gℓ = u^{ℓ}x^{ℓ}v^{ℓ} ∈ ULU^{−}. Hence u^{ℓ}x^{ℓ}v^{ℓ}v^{−1}x^{−1}u^{−1} ∈ U, whence x^{ℓ}v^{ℓ}v^{−1}x^{−1} ∈ U ∩ LU^{−}L = L ∩ P^{−} = 1. It follows that x^{ℓ}v^{ℓ} = xv ∈ P U^{−}, whence by uniqueness, ℓ commutes with both x and v. Hence ^{g}ℓ = ^{uxv}ℓ = ^{u}ℓ as required. □

Next we recall the notion of the limit of a one–parameter subgroup (1 − P S). For background we refer the reader to Mumford [[M], Ch2,§2], Kraft [[Kr], II.2] or Richardson [[R2], §7]. Write K = \overline{𝔽}_{q} and K^{∗} for the multiplicative group of K. A 1 − P S of G is a morphism of algebraic groups λ : K^{∗} → G. Write Y (G) for the set of 1 − P S of G. If X is an algebraic variety over K and ϕ : K^{∗} → X is a morphism, we say that lim_{t→0} ϕ(t) = x ∈ X if there is a morphism ψ : K → X which extends ϕ and such that

ψ(0) = x.

Now let λ ∈ Y (G); define P (λ) = {g ∈ G| lim_{t→0} λ(t)gλ(t)^{−1} exists}. The following facts may be found in the sources cited above.

(5.14).

(i) Let T be a maximal torus of G which contains λ(K^{∗}). Then P (λ) is the parabolic subgroup of G defined by (in the usual notation)

P (λ) = ⟨T, U_{α}|α ∈ Φ(G, T), ⟨α, λ⟩ ≥ 0⟩

(ii) λ(K^{∗}) ⊆ R(P (λ)) (R denoting the radical) and C_{G}(λ(K^{∗})) = L(λ) is a Levi

subgroup of P (λ). (iii) U(λ) = {g ∈ G| lim_{t→0} λ(t)gλ(t)^{−1} = 1} is the unipotent radical of P (λ).

(iv) If ℓ ∈ L(λ), u ∈ U(λ) then lim_{t→0}(λ(t)ℓuλ(t)^{−1}) = ℓ.

(v) Given a parabolic subgroup P with Levi decomposition P = LU, there exists a 1 − P S λ : K^{∗} → Z(L)^{0} = Z such that P = P (λ), L = L(λ) and U = U(λ) as above.

(5.15) Lemma. Let V = (ℓ)_{G} ∩ ℓU. Then every irreducible component of V contains (ℓ)_{U}.

Proof. Let V_{1} be an irreducible component of V and take v ∈ V_{1} such that v lies in no other irreducible component of V. Since V is invariant under the conjugation action of Z and Z is connected, it follows that V_{1} is invariant under Z. Consider the 1 − P S μ(t) = ^{λ(t)}v (t ∈ K^{∗}) where λ is as in (5.14)(v). From (iv) above, we have lim_{t→0} μ(t) = ℓ, so that there is a morphism ϕ : K → V such that

ϕ(t) = \{ \begin{matrix}^{λ(t)}v & (t ≠ 0). \\ ℓ & (t = 0) \end{matrix}

If ℓ ∉ V_{1} this contradicts the irreducible nature of K. Hence ℓ ∈ V_{1}. Finally, since V is U–invariant and U is connected, U stabilizes V_{1}, whence (ℓ)_{U} ⊆ V_{1}. □

(5.16) Lemma. Let η : G → (ℓ)_{G} denote the orbit map η(g) = gℓg^{−1}. Then η is an open map.

Proof. Let π : G → G/G_{ℓ} be the quotient map, where G_{ℓ} = C_{G}(ℓ). Then η is constant on the fibres of π and therefore induces a morphism ξ : G/G_{ℓ} → (ℓ)_{G} such that ξ∘π = η.

Clearly ξ is bijective and by [[R2], p.7] is a homeomorphism of topological spaces. Since π is an open map, it follows that η is open. □

Proof of Theorem (5.11.1). The set ULU^{−} is a (dense) open subset of G, whence by (5.16) η(ULU^{−}) is open in (ℓ)_{G} = η(G).

Let V_{1} be an irreducible component of V = η(G) ∩ ℓU. Then η(ULU^{−}) ∩ V_{1} is an open (dense) subset of V_{1} (it is non–empty by (5.15)) and η(ULU^{−}) ∩ ℓU ⊆ η(U) by (5.13). But since U is unipotent, the orbit η(U) = (ℓ)_{U} is closed in G and hence in V_{1}. It follows (taking the closure of η(ULU^{−}) ∩ V_{1} in V_{1}) that V_{1} = η(U) = (ℓ)_{U}. Thus (ℓ)_{G} ∩ ℓU = (ℓ)_{U} as required. □

(5.17) Remarks.

(a) Given Lusztig’s result [[Lu1], (2.9)(a)] that (ℓ)_{U} is an irreducible component of (ℓ)_{G} ∩ ℓU, Theorem (5.11) follows immediately from Lemma (5.15). We have included the rest of the proof because we believe it is simpler than Lusztig’s.

(b) The statements in (5.14) furnish a proof of (5.8), since if v is unipotent in L, λ is as in (5.14)(v) and u ∈ vU, then lim_{t→0} (^{λ(t)}u) = v, from which it follows that (v)_{G} ⊆ \overline{(u)_{G}}.

(5.18) Corollary. Let G = SL_{n}. Then the matrix (c_{v,u}) of (5.9) is invertible. Thus there exist coefficients d_{v,u} ∈ ℚ (v, u ranging over the unipotent conjugacy classes of SL(n, q) = G^{F}) such that

γ_{u}^{G^{F}} = ∑_{(v)_{G^{F}} :(v)_{G}≤(u)_{G}} d_{u,v}R_{L}^{G}γ_{v}^{L^{F}}

This follows immediately from (5.12)′.

In view of the parametrization (5.6) of the parabolic unipotent classes of G^{F} and the fact that in type A, all unipotent classes are parabolic, the relation (5.18) may be explicitly written

(5.19)  γ_{u_{(J,z)}} = ∑_{(J′,z′)} d_{(J,z),(J′,z′)}R_{L_{K}}^{G} (γ_{u_{(J′,z′)}}^{L_{J′}})

where (J, z) denotes the equivalence class of (J, z) ∈ P(π) × H^{1}(F, z) defined in (5.6), and the sum is over the (J′, z′) which are smaller than (J, z) in the order determined by conjugacy class closure.

6. On the values of characters at arbitrary unipotent elements

It is clear from §5 that the values of a character χ of G^{F} at an arbitrary unipotent element is known in principle if we have the following ingredients:

(6.1).

(i) A transformation matrix expressing the functions γ_{u} in terms of the ρ_{v} (cf. (5.9.3)).

(ii) An explicit decomposition of ^{∗}R_{L}^{G}χ for the L = L_{v} which occur in (5.9).

(iii) The values ξ(v) for components ξ of ^{∗}R_{L}^{G}χ (v distinguished in L = L_{v}).

In the case of SL_{n}, (5.18) carries (i) out, at least in principle. In this section we obtain partial results on (ii) and (iii) for χ regular or semisimple. Our results are complete (in principle) in the case of SL_{n}.

(6.2) Theorem. Suppose L is a rational Levi subgroup of G for which conjecture (5.2)′ holds (this includes the case where L is G-split). Then with the notation of (3.11) we have

∗R_{L}^{G}χ_{(s),z} = ε_{G}ε_{L} ∑_{(t)} χ_{(t),z′}

where the sum is over the semisimple classes (t) ∈ L^{∗F^{∗}} such that t ∈ (s)_{G^{∗}}_{F^{∗}} and where z′ is as in (5.2)′.

For the proof we require

(6.3) Lemma. Denote by E(G^{F} , (s)) the Lusztig series (see [Lu4]) corresponding to the semi-simple class (s) ⊂ G^{∗}^{F^{∗}}.

(i) For χ ∈ E(G^{F} , (s)) we have

(6.3.1)  ^{∗}R_{L}^{G}χ ∈ ⨁_{(t)} ℤE(L^{F} , (t))

the sum being over the same set as in (6.2). (ii) If ξ ∈ E(L^{F} , (t)) then R_{L}^{G}ξ ∈ ℤE(G^{F} , (t)).

The proof of this lemma is essentially the same as the corresponding fact for geometric series, which was proved by Lusztig [Lu3].

Proof of 6.2. Suppose ξ ∈ Irr(L^{F}) satisfies ⟨^{∗}R_{L}^{G}χ_{(s),z}, ξ ⟩ ≠ 0. Then ⟨ χ_{(s),z}, R_{L}^{G}ξ ⟩ ≠ 0, whence ⟨ Γ_{z}, R_{L}^{G}ξ ⟩ = ⟨^{∗}R_{L}^{G}Γ_{z}, ξ ⟩ ≠ 0. Hence by (5.2)′ ⟨ Γ_{z′}, ξ ⟩ ≠ 0, whence ξ = χ_{(t),z′} for some (t) with (t)_{G^{∗}}_{F^{∗}} = (s)_{G^{∗}}_{F^{∗}} (by (6.3)).

Moreover the above computation shows that for each such (t), ⟨ Γ_{z}, R_{L}^{G}χ_{(t),z′} ⟩ = ε_{G}ε_{L} and the common constituent is χ_{(s),z′} whence ⟨ χ_{(s),z′}, R_{L}^{G}χ_{(t),z′} ⟩ = ε_{G}ε_{L} for such t. The result follows. □

(6.4) Corollary. With notation as in Theorem 6.2, we have, for G-split L

∗R_{L}^{G}ρ_{(s),z} = ε_{C_{G^{∗}} (s)} ∑_{(t)} ε_{C_{L^{∗}} (t)}ρ_{(t),h_{L}(z)}

where the sum is over the same semisimple classes of L^{∗F^{∗}} as that in (6.2).

Proof. Apply D to the equation in (6.2), using the fact that for L G-split, D commutes with ^{∗}R_{L}^{G}. We obtain:

ε_{G}ε_{C_{G^{∗}} (s)}^{∗}R_{L}^{G}ρ_{(s),z} = ε_{G}ε_{L} ∑_{(t)} ε_{C_{L^{∗}} (t)}ε_{L}ρ_{(t),h_{L}(z)}

The corollary follows. □

We remark that one could prove a stronger version of (6.4) analogous to the statement of (6.2), for those L for which (5.2) holds.

(6.5) Corollary. Assume that (i) The characteristic is good for G. (ii) Every unipotent element v of G^{F} is regular in a G-split Levi subgroup L_{v}. (iii) The matrix (c_{u,v}) of (5.9.3) has inverse (d_{u,v}).

Then for any unipotent u ∈ G^{F} and semisimple character ρ_{(s),z} of G^{F} we have

ρ_{(s),z}(u) = ε_{C_{G^{∗}} (s)} ∑_{(v)_{G^{F}}} ∑_{(t)_{L^{∗}_{v}}_{F^{∗}}} ∑_{a∈(A(L_{v})^{F^{∗}})_{t}^{∧}} ε_{Rad(L_{v})}d_{u,v}σ_{z(v)(h_{Lv} (z))^{−1}a}

where the first sum is over the unipotent classes (v)_{G^{F}} in G^{F}, the second is over the semisimple classes (t) in L_{v}^{∗}^{F^{∗}} which intersect (s)_{G^{∗}}_{F^{∗}}, A(L) is the group which is defined for L in analogous fashion to A (for G) (see (3.12), (3.15)) and z(v) ∈ H^{1}(F, Z_{L_{v}}) parametrizes the (regular unipotent) class of v ∈ L_{v} (see (5.3) above).

Proof. We have

ρ_{(s),z}(u) = ⟨ ρ_{(s),z}, γ_{u} ⟩ = ⟨ ρ_{(s),z}, ∑_{(v)_{G^{F}}} d_{u,v}ρ_{v} ⟩

= ∑_{(v)_{G^{F}}} d_{u,v}⟨ ρ_{(s),z}, ρ_{v} ⟩

But

⟨ ρ_{(s),z}, ρ_{v} ⟩ = ⟨ ρ_{(s),z}, R_{L_{v}}^{G}γ_{v}^{L_{v}^{F}} ⟩

= ⟨^{∗}R_{L_{v}}^{G}ρ_{(s),z}, γ_{v}^{L_{v}^{F}} ⟩

= ε_{C_{G^{∗}} (s)} ∑_{(t)} ⟨ ε_{C_{L^{∗}_{v}} (t)}ρ_{(t),h_{Lv} (z)}, γ_{v}^{L_{v}^{F}} ⟩ (by (6.4))

= ε_{C_{G^{∗}} (s)} ∑_{(t)} ε_{C_{L^{∗}_{v}} (t)}ρ_{(t),h_{Lv} (z)}(v)

Now by assumption, v is regular unipotent in L_{v}, so that by (5.3) v corresponds to some element z(v) ∈ H^{1}(F, Z_{L_{v}}). But then by Theorem 3.15 (iii), we have, since the assumption on characteristic implies it is also good for L,

ρ_{(t),h_{Lv} (z)}(v) = ε_{Rad(L_{v})}ε_{C_{L^{∗}_{v}} (t)} ∑_{a∈(A(L_{v})^{F^{∗}})_{t}^{∧}} σ_{z(v)(h_{Lv} (z))^{−1}a}

Substituting into the previous expression yields the required formula. □

It is possible to write down a formula analogous to (6.5) for the regular characters. The only difference from (6.5) would be the introduction of a sign into the summands on the right hand side, and the second sum would be over those semisimple conjugacy classes (t) ∈ L_{v}^{∗}^{F^{∗}} such that (t) intersects (s)_{G^{∗}}_{F^{∗}} and (t) is regular semisimple in L_{v}^{∗} (by the remarks after (3.15)). We leave this as an exercise for the reader. We remark finally that the middle sum in (6.5) (or its analogue for χ_{(s),z}) is what is referred to as the “degeneracy rule” by J. A. Green in [JAG].

7. Concluding remarks and open problems

The relation (5.19) (or (5.18)) is an example of the relation (5.1.1). Thus, assuming we know the numbers d_{(J,z),(J′,z′)} explicitly (which we do not, although in principle it is computable in any particular case), the values {χ(v)|χ an irreducible character, v unipotent in G^{F} } are given by (5.1.2). Hence we have

(7.1) Problem. Determine ^{∗}R_{L}^{G}χ for the irreducible characters of G^{F} and any L.

Problem (7.1) is also instrumental in the passage from a knowledge of χ(u) for u unipotent to χ(x) for all x ∈ G^{F}. This is seen as follows. Write x = x_{s}x_{u} (Jordan decomposition). Then Γ_{x_{s}x_{u}}^{G^{F}} = R_{C_{G}(x_{s})^{∘}}^{G}γ_{x_{s}x_{u}}^{C_{G}(x_{s})^{∘F}} [DM2, (3.7)]. Hence we have

(7.1.1)  χ(x) = ⟨ χ, γ_{x}^{G^{F}} ⟩_{G^{F}} = ⟨^{∗}R_{C_{G}(x_{s})^{∘}}^{G}χ, γ_{x_{s}x_{u}}^{C_{G}(x_{s})^{∘F}} ⟩_{C_{G}(x_{s})^{∘F}}

and since x_{s} is central in C_{G}(x_{s})^{∘F}, the right side of (7.1.1) is known, assuming that ∗R_{L}^{G}χ is known, and that ψ(v) is known for ψ ∈ Irr(C_{G}(x_{s})^{∘F}) and v unipotent in C_{G}(x_{s})^{∘F}. Note that implicit in the formula (7.1.1) is the assumption that C_{G}(x_{s})^{∘F} is a Levi subgroup of G. It is well known that this is true assuming that |x_{s}| is not divisible by any bad prime for G, a condition which is always satisfied in type A.

The next problem is a special case of (6.1)(i) (cf.also (7.5) below).

(7.2) Problem. Determine the matrix D = d_{(J,z),(J′,z′)} or its inverse C = c_{(J′,z′),(J,z)} explicitly.

To obtain results such as those in §6 in full generality, the following problems must be addressed.

(7.3) Problem. Determine χ(v) for χ regular and v a distinguished unipotent element in G^{F}. More generally, do this for any χ.

To extend to χ(u) for any unipotent we have

(7.4) Problem. Given a unipotent element v ∈ G^{F}, determine conditions under which there is a G-split rational Levi subgroup L_{v} ∋ v such that v is distinguished in L_{v}?

The significance of this is that for such v, the coefficients (c_{u,v}) in the relation (5.9.3) are known explicitly .

We note that it is not hard to find examples which show that it is not always true that L_{v} may be chosen G-split. For instance, take a G-conjugacy class of Levi subgroups which contains a rational Levi subgroup L but no G-split ones. An element v which is regular unipotent in L provides a counterexample (e.g. take the class of Levi subgroups of type U_{1} × U_{2} in U_{3}).

More generally, the above phenomenon occurs whenever G is not F –split. However even when G is split, there may be unipotent elements u and v ∈ G^{F} such that u and v are geometrically conjugate but L_{u} is G–split, while L_{v} is not. This occurs in all types except type A. Nevertheless the situation may be manageable using the theory of Shintani descent, due to

(7.4.1) Proposition. Assume G is F –split and let v ∈ G^{F} be unipotent. Then there is an element u ∈ G^{F} such that u is conjugate in G to v, with the property that L_{u} is G–split.

Proof. Let S be a maximally split maximal torus of C_{G}(v)^{o}. Then since G is split, s is conjugate in G to a split torus S′ = gSg^{−1}, some g ∈ G. Moreover we may take L_{v} to be C_{G}(S). Thus L′ = gL_{v}g^{−1} = C_{G}(S′) is a G–split Levi subgroup of G (its centre S′ is split) and L′ ∋ gvg^{−1}. Thus L′ has non–trivial intersection with the G–conjugate class of v. But L′ is a reductive group with Frobenius endomorphism F. Hence F fixes all the unipotent L′–classes of L′, whence it follows that L′ contains an F –stable element u which is conjugate in G to v, and u is distinguished in the G–split Levi subgroup L′. □

(7.5) Problem. If v is unipotent in G^{F} and L_{v} is as in (5.9)(iii), investigate the matrix (c_{u,v}), where

R_{L_{v}}^{G}γ_{v} = ∑_{(u)_{G^{F}}} c_{u,v}γ_{u}

where the sum is over the unipotent classes (u) of G^{F}.

In particular, we have a canonically defined set of unipotently supported class functions {ρ_{v} | (v)_{G^{F}} is a unipotent class in G^{F} } (see (5.9.2)).

A specific question concerning (7.5) is

(7.5.1) Problem. Are the ρ_{v} linearly independent?

We conclude with some facts concerning the “rational submatrix” C (see (7.2)) of the matrix in (7.5).

Recall that (cf. (5.7)) we have

(7.6)  c_{v,u} = |U^{F} |^{−1}|vU ∩ (u)_{G^{F}} |.

Hence c_{(J′,z′),(J,z)} = |U^{F} |^{−1}|v_{J′,z′}U ∩(u_{J,z})_{G^{F}} | (in the notation of the proof of (5.6)). Now H^{1}(F, Z) acts on G^{F} as in the proof of (5.6), and z_{0} ∈ H^{1}(F, z) transforms the set v_{J′,z′}U ∩ (u_{J,z})_{G^{F}} into v_{J′,z′z_{0}}U ∩ (U_{J,zz_{0}})_{G^{F}}. This proves that

(7.7). We have (i) c_{(J′,z′),(J,z)} = c_{(J′,z′z_{0}),(J,zz_{0})} for all z_{0} ∈ H^{1}(F, Z). (ii) c_{(J′,z′),(J,z)} depends only on z′z^{−1} modulo ker h_{J} + ker h_{J′} = ker h_{J∩J′}.

The statement (ii) follows easily from (i) since (J, z) depends only on z mod ker h_{J}. In view of (7.7), we may write

(7.8)  c_{(J′,z′),(J,z)} = c_{J′,J,z′z^{−1}}

where z′z^{−1} is taken mod ker h_{J∩J′}.

We have already computed c_{J,Π,z} in the proof of (5.3)

(7.9) Proposition. We have

c_{J,Π,z} = \{ \begin{matrix}|Z_{L_{J}} ||Z_{G^{F}} |^{−1}q^{|J|−|Π|} & if z ∈ ker h_{J} \\ 0 & if z ∉ ker h_{J}. \end{matrix}

Proof. From (5.3.7) we have c_{J,Π,z} = |C_{G^{F}} (u_{Π,1})|^{−1}^{∗}R_{L_{J}}^{G} γ_{u_{Π,1}}(u_{J,z}).

Moreover in the proof of (5.3) it is shown that ^{∗}R_{L_{J}}^{G} γ_{u_{Π,1}}^{G}(u_{J,z}) = 0 unless u_{J,z} is conjugate to the projection of u_{Π,1} from U to L_{J}, i.e. unless z ∈ ker h_{J}, and if this condition is satisfied, then ^{∗}R_{L_{J}}^{G} γ_{Π,1}^{G}(u_{J,z}) = |C_{L^{F}} (u_{J,z})|. This proves (7.9), using the remark that for u regular unipotent in G, |C_{G^{F}} (u)| = |Z_{G^{F}} |q^{|Π|}. □

(7.10) Proposition. We have

c_{J,J,z} = \{ \begin{matrix}0 & if z ∉ ker h_{J} \\ |C_{U_{J}^{F}} (v_{J,1})|^{−1} & if z ∈ ker h_{J}. \end{matrix}

Proof. We have c_{J,J,z} = |U_{J}^{F} |^{−1}|v_{J,z}U_{J} ∩ (v_{J,1})_{G^{F}} |. By the argument in the proof of (5.12)′, if c_{J,J,z} ≠ 0, then v_{J,z} ∼_{G^{F}} v_{J,1}, so that z ∈ ker h_{J}. For z ∈ ker h_{J}, we have c_{J,J,z} = c_{J,J,1} = |U_{J}^{F} |^{−1}|v_{J,1}U_{J} ∩ (v_{J,1})_{G^{F}} |. Moreover for u ∈ U_{J}, v_{J,1}u ∼_{G^{F}} v_{J,1} ⇔ v_{J,1}u ∼_{U_{J}^{F}} v_{J,1} ⇔ u ∈ [v_{J,1}, U_{J}^{F} ]. Hence for z ∈ ker h_{J}, c_{J,J,z} = |U_{J}^{F} |^{−1}| [v_{J,1}, U_{J}^{F} ] |. But the map u ↦ [v_{J,1}, u] : U^{F} → U^{F} has fibres of cardinality |C_{U_{J}^{F}} (v_{J,1})|. Hence its image has cardinality |U_{J}^{F} ||C_{U_{J}^{F}} (v_{J,1})|^{−1}, which proves (7.10). □

We remark finally that (7.10) gives the diagonal entries of the upper triangular matrix (c_{u,v}).

Digne, Lehrer and Michel, Reductive groups with non-connected center 193

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F. Digne, U.F.R. de mathématiques, Université de Picardie, 33, rue St-Leu 80039 Amiens, France

G. I. Lehrer, Department of Pure Mathematics, University of Sydney, N.S.W. 2006, Australia

J. Michel, Département de mathématiques et d’informatique, École Normale Supérieure, 45, rue d’Ulm Paris 75005, France