On Gel’fand-Graev characters of reductive groups with disconnected centre

By F. Digne at Amiens, G. I. Lehrer at Canberra and J. Michel at Paris

0. Introduction.

Let G be a connected reductive algebraic group over an algebraic closure \overline{𝔽}_{q} of the finite field 𝔽_{q} (q being a power of the prime p). Let F be a Frobenius endomorphism corresponding to some 𝔽_{q}-structure on G. The purpose of this paper is, following the theme of [DLM], to present explicit results on the character theory of the group of rational points G^{F}, particularly in the case when the centre of G is not connected. We refer to [DLM] for an explanation as to how the issues treated here relate to the determination of character values. Here we prove two results. The first is conjecture 5.2 of [DLM]; it states that Deligne-Lusztig restriction (defined as the adjoint of Deligne-Lusztig induction) of a Gel’fand-Graev character is again a Gel’fand-Graev character of a possibly non Gsplit Levi subgroup (L is a G-split Levi subgroup of G if it is a Levi component of some rational parabolic subgroup of G). This was proved in the case where the ambient group has connected centre in [DM1]. The general case is treated here using Lusztig’s character sheaves.

The second main result contributes to the explicit computation of the Fourier transform of generalized Gel’fand-Graev characters (see [Lu5] and [Le]); more precisely we determine explicitly certain of the fourth roots of unity occurring in Lusztig’s treatment [Lu5, 7.2 and 7.3]. Lusztig [loc. cit.] showed that generalized Gel’fand-Graev characters may be written as a sum of terms, one for each “block”, where a block is an equivalence class of pairs (C, ζ) such that C is a unipotent conjugacy class of G and ζ is an irreducible local system on C. The Fourier transform of any of these summands involves a fourth root of unity which depends only on the block (not on the character). We compute this number in case the block concerned contains a pair (C, ζ) with C regular unipotent.

Since we depend on the results of [Lu5] and [Lu4] we will assume throughout that p and q are “large enough”, in particular that p is good for G. However the necessity of this assumption is merely due to the methods employed and we expect the results below to be valid more generally.

1. Preliminaries.

We largely follow the notation of [Lu5]. The set of pairs (C, ζ) where C is a unipotent class of G and ζ is a local system on C has been partitioned in [Lu5, 4.4] into equivalence classes called blocks. In [Lu2] Lusztig defines the notion of “cuspidal” pairs (C, ζ); these turn out to be those pairs which do not arise from others by the induction process of [Lu2]. To each block there corresponds a triple (L, C_{0}, ζ_{L}), unique up to G-conjugacy, where L is a Levi subgroup of G, and ι_{0} = (C_{0}, ζ_{L}) is a cuspidal pair on L. Call this a “cuspidal datum” of the block. If I is the block associated with (L, ι_{0}) we will sometimes write L_{I} for L. In this situation, the elements of I are in bijection with the characters of the Coxeter group W_{G}(L) = N_{G}(L)/L. If ι is the pair (C, ζ), we write C_{ι} for C and refer to C_{ι} as the support of ι. We denote by φ_{ι} the corresponding character of W_{G}(L). The following fact may be checked case by case (using the classification of cuspidal data in the introduction of [Lu2] and in [Lu2, §14 and 15])

The authors would like to thank the Australian Research Council for funding the project of which this work was a part and the Universities of Amiens and Sydney for their support and hospitality.

1.1 Fact. Let L be a Levi subgroup of G which supports a cuspidal datum and let M be an arbitrary Levi subgroup of G; then there is at most one M-conjugacy class of Levi subgroups of M which are conjugate to L in G.

This has the following consequence.

1.2 Proposition. If (L, ι_{0}) and ^{g}(L, ι_{0}) are two conjugate cuspidal data of G contained in a Levi subgroup M of G then they are conjugate in M.

For any element x of a reductive group G, put A(x) = C_{G}(x)/C_{G}(x)^{0}. We will sometimes write A_{G}(x) to avoid ambiguity. Let u ∈ G^{F} be a unipotent element; the G^{F} - classes into which the rational points of the G-class of u split are parametrized by the set H^{1}(F, A(u)) of F -conjugacy classes of A(u). This parametrization is uniquely determined by the choice of the G^{F} -class corresponding to 1 ∈ H^{1}(F, A(u)). Let u_{z} be a representative of the G^{F} -class corresponding to z ∈ H^{1}(F, A(u)); we assume the parametrization is such that u_{1} = u. Thus we have 1.3 Let C be the G-conjugacy class of u. The choice of u ∈ C^{F} determines an explicit bijection from the G^{F} -classes in C^{F} to H^{1}(F, A(u)).

1.4 Some definitions. Let C be the class of u in G and let ζ be an F -stable local system on C; we shall follow Lusztig [Lu3, 24.1–24.2] in making a precise choice of an isomorphism, F^{∗}ζ→^{∼}ζ: let I be the block containing (C, ζ); as it is a rational (i.e. F -stable) block, its cuspidal datum (L, C_{0}, ζ_{0}) can be chosen to be rational and such that L is a Levi subgroup of a rational parabolic subgroup (for this last fact, cf. [Lu4, 9.3 below (e)]). Choose a rational unipotent element u_{0} ∈ C_{0} and an isomorphism σ_{0} : F^{∗}(ζ_{0}) ≃ ζ_{0} such that Trace(σ_{0}, (ζ_{0})_{u_{0}}) = dim(ζ_{0}) where ξ_{x} is the stalk at x of the sheaf ξ. This choice and the choice of an extension φ̃_{ι} of φ_{ι} to W_{G}(L) ⋊ <F > uniquely defines a similar isomorphism σ : F^{∗}ζ → ζ for the local system ζ. We shall choose φ̃ to be the “preferred extension” in [Lu3, 17.2 and 24.2]; in particular the extension of a 1-dimensional character is chosen to be trivial on <F >. The order of σ on the stalks at rational points is then a divisor of |A_{L}(u_{0})| as may be seen from the proof of [Lu3, 24.2.4]; hence we may modify the choice of u_{0} and σ_{0} so that Trace(σ, (ζ)_{u_{1}}) = dim(ζ). Note that u_{0} depends on the local system ζ.

The function z ↦ Trace(σ, (ζ)_{u_{z}}) may be regarded as an A(u)-class function on A(u).F. Given the choices above, the local system ζ may be recovered from this function. There is a unique character of A(u) ⋊ <F > which restricts to the above function on A(u).F. Recall that A(u)-class functions on A(u).F may be identifed with F -class functions on A(u) and hence with functions on H^{1}(F, A(u)). We will abuse notation by referring to the functions on A(u).F and H^{1}(F, A(u)) as well as the corresponding character of A(u) ⋊ <F > as ζ.

Suppose we fix an additive character of 𝔽_{q}; composing it with the trace, we obtain an additive character of any finite extension of 𝔽_{q}. Then, to the G^{F} -class of u_{z}, where z is in H^{1}(F, A(u))), there corresponds a generalized Gel’fand-Graev character Γ_{u_{z}} of G^{F}. With this notation, Lusztig [Lu5, 7.5 (a)] defines a function Γ_{ι} corresponding to ι = (C, ζ) ∈ I^{F} by:

1.5  Γ_{ι} = ∑_{a∈A(u)} ζ(aF)Γ_{u_{a}}

(where ζ has been identified as explained above with the restriction of a character to A(u).F).

Consider, following Lusztig [Lu3, 24.2.8 or Lu5, 6.4], the class function Y_{ι} on G^{F} whose value at x ∈ G^{F} is Trace(σ, (ζ)_{x}) if x ∈ C^{F} and 0 otherwise. Define the class function X_{ι} on G^{F} as the trace of the isomorphism induced by σ on the stalks of the extension by zero of the intersection cohomology complex on \overline{C} which corresponds to ζ. If u is an element of C^{F}, the G^{F} -class of which is parametrized by the z ∈ H^{1}(F, A(u)), we have X_{ι}(u) = ζ(aF) where a is any element of z. The sets {X_{ι}} and {Y_{ι}} are both bases of the space of unipotently supported class functions on G^{F} as I varies over the blocks and ι varies over I^{F}; moreover X_{ι} is orthogonal to X_{κ} if ι and κ do not belong to the same block and similarly for the Y_{ι}. The coefficients of the transition matrix (P_{ι,κ}) defined by X_{ι} = ∑_{κ} P_{κ,ι}Y_{κ} were shown by Lusztig [Lu3, 24.2.9 or Lu5, 6.5] to be polynomials in q. The polynomial P_{ι,ι} is 1 and P_{ι,κ} is 0 unless ι and κ are in the same block and the respective supports satisfy C_{ι} ⊂ \overline{C_{κ}}. Now define

X̃_{ι} = q^{\frac{1}{2} (dim G−dim C_{ι}−dim Z_{L_{I}})}X_{ι}

and

P̃′_{ι,κ}(q) = q^{\frac{1}{2} (dim C_{κ}−dim C_{ι})}P_{ι,κ}(\frac{1}{q}).

Let ζ_{I} be the fourth root of unity which Lusztig [Lu5, 7.2] associates to a rational block I. This depends only on the cuspidal datum (L_{I}, ι_{0}) of I and not on the ambient group. If δ = (−1)^{rank(L_{I})−dim Z_{L_{I}}} (cf. [Lu5, 8.4]), put ζ′_{I} = ζ_{I}^{−1}δ. For ι = (C, ζ) ∈ I^{F}, write a_{ι} for |A(u)| where u is any element of C. With this notation, we have

1.6 Proposition ([Ge, 3.5]). We have ⟨ D_{G}(Γ_{ι}), X̃_{κ} ⟩_{G^{F}} = ζ′_{I}a_{ι}P̃′_{ι,κ} where D_{G} is the duality involution.

Let us fix a rational block I and choose (as we may, cf. [Lu4, 9.3 below (e)]) a rational cuspidal datum (L, ι_{0}) of I such that L is G-split. For g ∈ G, denote by int(g) the automorphism y ↦ gyg^{−1}(g ∈ G). Let w ∈ W_{G}(L) and choose ẇ ∈ N_{G}(L) in the coset w. Choose an element g ∈ G such that g^{−1}F (g) = ẇ; the group L_{w} = int(g)(L) is then F -stable and int(g^{−1}) carries the endomorphism F of L_{w} to ẇF on L. We say that L_{w} is “w-twisted from L”; note that w is determined here only up to F -conjugacy in W_{G}(L).

If ι ∈ I^{F}, we have an associated extension φ̃_{ι} of φ_{ι} to W_{G}(L) ⋊ <F > as explained in 1.4. Note that if ι_{0} is an F -stable cuspidal pair for L, then the whole of N_{G}(L) stabilizes ι_{0} [Lu2, 9.2 (b)]. Hence ι_{0} is ẇF -stable for any w ∈ W_{G}(L). If choices analogous to those in 1.4 are made for L with Frobenius ẇF (or equivalently for L_{w} with Frobenius F), we obtain by “transport of structure” functions X_{ι_{0}}^{L_{w}} on L_{w} (these are denoted by χ_{w} in [Lu5, 6.15 (a)]). It follows from [Lu4, 9.3 (f)] that these choices may be made so that we have (cf. [Lu4, formula (e’) in the proof of 9.5])

1.7  X̃_{ι} = |W_{G}(L)|^{−1} ∑_{w∈W_{G}(L)} φ̃_{ι}(wF)R_{L_{w}}^{G} X̃_{ι_{0}}^{L_{w}}

2. Deligne-Lusztig restriction of Gel’fand-Graev characters. 2.1. Let M be a rational Levi subgroup of G and I_{M} a rational block of M. Our first objective in this section is to give a formula for R_{M}^{G}(X̃_{ι}) where ι ∈ I_{M}^{F}. Now I_{M} corresponds to a cuspidal datum (L′, ι′_{0}) where L′ is an M-split Levi subgroup of M and ι′_{0} is a cuspidal pair on L′. The cuspidal datum (L′, ι′_{0}) determines a block I of G. Conversely, if I is a rational block of G with corresponding cuspidal datum (L, ι_{o}) where L is G-split and conjugate to a subgroup of M, then by 1.2 (L, ι_{0}) determines a unique M-class of cuspidal data of M and hence I determines a unique block I_{M} of M.

Let (L′, ι′_{0}) be a rational cuspidal datum in M, such that L′ is M-split and is Gconjugate to the G-split Levi subgroup L of G.

With the above notation, write L′ = int(g)(L), with g^{−1}F (g) = ẇ ∈ N_{G}(L). Write M_{0} = int(g^{−1})M. Then

(i) The groups L and M_{0} are both stable under ẇF. (ii) The automorphism int(g^{−1}) maps W_{M}(L′) into W_{M_{0}}(L) ⊂ W_{G}(L) and F (on M) to

ẇF (on M_{0}). (iii) If L_{v} is an F -stable v-twisted conjugate of L′ in M, for v ∈ W_{M}(L′), then L′_{v} is

v_{0}w-twisted from L, where v_{0} = int(g^{−1})v ∈ W_{M_{0}}(L).

Let φ̃_{ι} be the analogue for W_{M}(L′) ⋊ <F > of the character appearing in 1.7. Then φ̃_{ι} may be regarded by transport of structure using int(g^{−1}), as a character of W_{M_{0}}(L) ⋊ <wF >, which is a subgroup of W_{G}(L) ⋊ <F >.

2.2 Proposition. Maintain the notation of 2.1. For ι ∈ I_{M}^{F} we have

R_{M}^{G}(X̃_{ι}) = ∑_{κ∈I} ⟨ φ̃_{ι}, Res_{W_{M_{0}} (L).wF}^{W_{G}(L).F} φ̃_{κ} ⟩_{W_{M_{0}} (L).wF} X̃_{κ}.

Proof: Applying equation 1.7 to M, we obtain

X̃_{ι} = |W_{M}(L′)|^{−1} ∑_{v∈W_{M}(L′)} φ̃_{ι}(vF)R_{L′_{v}}^{M} X̃_{ι′_{0}}^{L′_{v}}

With the identification in (iii) above, using int(g^{−1}), this may be rewritten

X̃_{ι} = |W_{M_{0}}(L)|^{−1} ∑_{v_{0}∈W_{M_{0}} (L)} φ̃_{ι}(v_{0}wF)R_{L′_{v}}^{M} X̃_{ι′_{0}}^{L′_{v}}

Inducing from M to G we obtain

R_{M}^{G}(X̃_{ι}) = |W_{M_{0}}(L)|^{−1} ∑_{v_{0}∈W_{M_{0}} (L)} φ̃_{ι}(v_{0}wF)R_{L′_{v}}^{G} X̃_{ι′_{0}}^{L′_{v}}

while using 1.7 in G the right-hand side of the proposition expands to

|W_{G}(L)|^{−1} ∑_{κ∈I,v_{1}∈W_{G}(L)} ⟨ φ̃_{ι}, Res_{W_{M_{0}} (L).wF}^{W_{G}(L).F} φ̃_{κ} ⟩_{W_{M_{0}} (L).wF} φ̃_{κ}(v_{1}F)R_{L_{v_{1}}}^{G} X̃_{ι_{0}}^{L_{v_{1}}}

=|W_{G}(L)|^{−1} ∑_{v_{1}∈W_{G}(L)} Ind_{W_{M_{0}} (L).wF}^{W_{G}(L).F} φ̃_{ι}(v_{1}F)R_{L_{v_{1}}}^{G} X̃_{ι_{0}}^{L_{v_{1}}}
=|W_{G}(L)|^{−1}|W_{M_{0}}(L)|^{−1} ∑_{{v_{1},h∈W_{G}(L)|^{h}(v_{1}F)∈W_{M_{0}} (L).wF }} φ̃_{ι}(^{h}(v_{1}F))R_{L_{v_{1}}}^{G} X̃_{ι_{0}}^{L_{v_{1}}}

But v_{1}, h ∈ W_{G}(L) statisfy ^{h}(v_{1}F) ∈ W_{M_{0}}(L).wF if and only if hv_{1}F (h^{−1}) = v_{0}w for some v_{0} ∈ W_{M_{0}}(L). Since v_{1} is F -conjugate in W_{G}(L) to v_{0}w it follows that R_{L_{v_{1}}}^{G} X̃_{ι_{0}}^{L_{v_{1}}} =

R_{L_{v_{0}w}}^{G} X̃_{ι_{0}}^{L_{v_{0}w}}. Hence the last sum may be written

|W_{M_{0}}(L)|^{−1} ∑_{v_{0}∈W_{M_{0}} (L)} φ̃_{ι}(v_{0}wF)R_{L_{v_{0}w}}^{G} X̃_{ι_{0}}^{L_{v_{0}w}}.

To complete the proof we need to verify that with v and v_{0} as above,

R_{L′_{v}}^{G} X̃_{ι′_{0}}^{L′_{v}} = R_{L_{v_{0}w}}^{G} X̃_{ι_{0}}^{L_{v_{0}w}}.  (1)

For this, observe that we have conjugations as follows:

(L′, v̇F) ⟶^{int g^{−1}} (L, v̇_{0}ẇF)
↓^{int x} ↓^{int y}
(L′_{v}, F) ⟶^{int k^{−1}} (L_{v_{0}w}, F)

where k = xgy^{−1} is such that X̃_{ι_{0}}^{L_{v_{0}w}} ∘ int k = X̃_{ι′_{0}}^{L′_{v}}. Now the equations y^{−1}F (y) = v̇_{0}ẇ, x^{−1}F (x) = v̇, g^{−1}v̇g = v̇_{0} and g^{−1}F (g) = ẇ show that k = F (k), whence it follows that the pairs (L′_{v}, X̃_{ι′_{0}}^{L′_{v}}) and (L_{v_{0}w}, X̃_{ι_{0}}^{L_{v_{0}w}}) are rationally conjugate and (1) follows. This completes the proof of 2.2. □

We now focus on the case when the support C_{ι} is the regular class. We use

2.3 Proposition ([Lu2, 7.3]). Let L be a Levi subgroup of a parabolic subgroup P of G, let C_{0} be a unipotent conjugacy class of L and let C be the induced class in G. Let l be in L and u be in the unipotent radical of P and suppose they satisfy l ∈ C_{0}, lu ∈ C. Then the inclusion C_{P}(lu) ↪ C_{G}(lu) induces an injection A_{P}(lu) ↪ A_{G}(lu) and the projection C_{P}(lu)⟶^{π}C_{L}(l) induces an epimorphism A_{P}(lu)⟶^{π}A_{L}(l).

2.4 Proposition ([Lu2, 7.4 and 9.2]). Let I be a block with cuspidal datum (L, ι_{0}). There is a unique ι ∈ I such that dim C_{ι} is maximal; furthermore C_{ι} is the class induced from C_{ι_{0}} and φ_{ι} ∈ Irr(W_{G}(L)) is the identity character. Moreover, if ι_{0} = (C_{ι_{0}}, ζ) is rational (whence C_{ι} is rational), and π is the projection of proposition 2.3 where l ∈ C_{ι_{0}}^{F} and lu ∈ C_{ι}^{F}, then ξ = Ind_{A_{P}(lu).F}^{A_{G}(lu).F} ζ ∘ π is irreducible and ι = (C_{ι}, ξ).

2.5 Corollary. The following are equivalent for a block I with cuspidal datum (L, ι_{0})

(i) There exists ι ∈ I with regular support. (ii) The pair ι_{0} has regular support.

If either holds then the ι in (i) is unique and φ_{ι} = Id.

Proof: This follows from propositions 2.3 and 2.4 and the fact that a class and its induced class have centralizers of the same dimension. □

2.6 Corollary. Suppose ι = (C_{ι}, ξ), where C_{ι} is regular. In the notation of 2.4 we then have A_{G}(lu) = A_{P}(lu), so that ξ = ζ ∘ π.

Proof: This follows from the fact that when lu is regular C_{G}(lu) ⊂ P. □

We shall henceforth denote by C_{M} the regular unipotent class of a Levi subgroup M of G. If L and M are two Levi subgroups such that L ⊂ M, then C_{M} is the class induced from C_{L}. With this notation we have

2.7 Proposition. Let I be a rational block of G, let ι ∈ I^{F} be a pair with support the regular class C_{G} of G and let (L, ι_{0}) be a cuspidal datum of I. For any rational Levi subgroup M of G we have ^{∗}R_{M}^{G}D_{G}Γ_{ι}^{G} = 0 if M does not contain a conjugate of L. Otherwise, if I_{M} denotes the block of M with cuspidal datum ι_{0} (cf. 2.1), we have ∗R_{M}^{G}D_{G}Γ_{ι}^{G} = \frac{a_{ι}}{a_{μ}} D_{M}Γ_{μ}^{M}, where μ is the unique element of the block I_{M} with support C_{M}.

Proof: From proposition 1.6 we obtain, using the relations P_{ι,κ} = 0 if C_{ι} ⊄ \overline{C}_{κ} and P_{ι,ι} = 1

⟨ D_{G}(Γ_{ι}), X̃_{κ} ⟩ = ζ′_{I}a_{ι}δ_{ι,κ}.  (1)

We write ζ′_{I} = ζ′_{ι_{0}} to indicate that this root of unity depends only on the cuspidal datum and not on the ambient group. From (1) and proposition 2.2 we see that if k is not in a block of M with the same (up to G-conjugacy) cuspidal datum as I, then ⟨ D_{G}(Γ_{ι}), R_{M}^{G}X̃_{λ} ⟩ = 0, since R_{M}^{G}X̃_{λ} is a linear combination of the X̃_{κ} with κ in a block with same cuspidal datum as X̃_{λ}. In particular, if M does not contain such a block then ^{∗}R_{M}^{G}D_{G}Γ_{ι} = 0; otherwise if λ is in the block I_{M} of M with same cuspidal datum as I,

⟨^{∗}R_{M}^{G}D_{G}(Γ_{ι}), X̃_{λ} ⟩ = ⟨ D_{G}(Γ_{ι}), R_{M}^{G}X̃_{λ} ⟩
= ⟨ φ̃_{λ}, Res_{W_{M_{0}} (L).wF}^{W_{G}(L).F} φ̃_{ι} ⟩_{W_{M_{0}} (L).wF} δ_{Id,φ_{λ}}ζ′_{ι_{0}}a_{ι}
= δ_{Id,φ_{λ}}ζ′_{ι_{0}}a_{ι},

(2) where the second equality follows from 2.2 and the last equality follows by Frobenius reciprocity, since by 2.5, φ_{ι} = Id and by our choice of extension (1.4), φ̃_{ι} = Id. If ι_{M} = (C_{M}, ζ_{M}) is the element of I_{M} with support C_{M}, we may also write δ_{ι_{M},λ} for δ_{Id,φ_{λ}}. Comparing equation (2) with the analogue of (1) for the group M, we obtain the result as the X̃_{λ} form a basis of the space of unipotently supported class functions on M^{F}. □

A similar result for the Lusztig restrictions of the Γ_{ι} would follow from a commutation formula between R_{M}^{G} and D_{G}. The next result gives such a formula for unipotently supported functions.

2.8 Proposition. If f is a unipotently supported function on M^{F} then

D_{G}(R_{M}^{G}(f)) = ε_{G}ε_{M}R_{M}^{G}(D_{M}(f)),

where ε_{G} = (−1)^{𝔽_{q}- rank(G)}.

Proof: We know from 1.7 that the R_{L}^{M}(X_{ι}^{L}) where (L, ι) is a cuspidal datum span the space of unipotently supported functions on M^{F}. So we have only to prove the formula for these functions. But, by [Lu5, 8.2], if f is the characteristic function of a cuspidal character sheaf, we have D_{G}(R_{L}^{G}(f)) = ε_{G}ε_{L}R_{L}^{G}(D_{L}(f)). Applying this formula once in G, once in M and using the transitivity of Lusztig induction we get for such an f:

D_{G}(R_{M}^{G}(R_{L}^{M}(f))) = D_{G}(R_{L}^{G}(f)) = ε_{G}ε_{L}R_{L}^{G}(D_{L}(f))
= ε_{G}ε_{L}R_{M}^{G}(R_{L}^{M}(D_{L}(f)) = ε_{G}ε_{M}R_{M}^{G}(D_{M}(R_{L}^{M}(f))).

The same formula applies with f replaced by X_{ι}^{L} because X_{ι}^{L} is the restriction to unipotent elements of the characteristic function of a cuspidal character sheaf and because Lusztig induction and duality commute with the restriction to unipotent elements. □

From the above proposition and 2.7 we get

2.9 Proposition. Let ι, I, L, M and I_{M} be as in 2.7; then ^{∗}R_{M}^{G}Γ_{ι}^{G} = 0 if M does not contain a conjugate of L. If M contains a conjugate L, we have ^{∗}R_{M}^{G}Γ_{ι}^{G} = \frac{a_{ι}}{a_{μ}} ε_{G}ε_{M}Γ_{μ}^{M}, where μ is the unique element of I_{M} with support C_{M}.

From 2.3 there is an epimorphism A_{G}(u) → A_{M}(v), where u ∈ C_{G}^{F}, v ∈ C_{M}^{F}. We will denote this epimorphism by h_{M} (or by h_{M}^{G} to avoid ambiguity); since the characteristic is good for G, the group A_{G}(u) may be canonically identified with Z_{G}/Z_{G}^{0} and the epimorphism h_{M} is the canonical map Z_{G}/Z_{G}^{0} → Z_{M}/Z_{M}^{0} and in particular does not depend on the parabolic subgroup used in 2.3. Being compatible with F, h_{M} extends naturally to a map A_{G}(u) ⋊ <F > → A_{M}(v) ⋊ <F > which we will again denote by h_{M}.

From the preceding discussion and 2.6 we see that (C_{G}, ζ) is cuspidal if and only if ζ does not factor through any morphism h_{M} for M a proper Levi subgroup of G. In general ζ determines a Levi subgroup L and a local system ζ_{L} on C_{L} such that (C_{L}, ζ_{L}) is cuspidal and ζ = ζ_{L} ∘ h_{L}. If M contains L then ker h_{M} ⊂ ker h_{L} and h_{L}^{G} = h_{L}^{M} ∘ h_{M}^{G}.

We are now in a position to compute the Deligne-Lusztig restriction of a Gel’fand-Graev character. Recall that we have fixed a rational element u ∈ C_{G}^{F} and that this choice determines a bijection between the G^{F} -classes in C_{G}^{F} and H^{1}(F, A(u)). For any z ∈ A(u) we have chosen a representative u_{z} of the rational class parametrized by the F -class of z in such a way that u_{1} = u. The following result shows that conjectures [DLM, (5.2) and (5.2)′] are true.

2.10 Theorem. Let u be an F -stable regular unipotent element of G. Let M be any rational Levi subgroup of G. Then a regular unipotent element v in M^{F} may be chosen so that under the consequent parametrization (1.3) of the regular unipotent classes of M^{F} by elements of H^{1}(F, Z_{M}) we have for any z ∈ A_{G}(u)

∗R_{M}^{G}Γ_{u_{z}}^{G} = ε_{G}ε_{M}Γ_{v_{h_{M}(z)}}^{M}.

Proof: Inverting the relation (1.5) we have

Γ_{u_{z}}^{G} = |A_{G}(u)|^{−1} ∑_{ζ∈A(u)^{∧}} \overline{ζ(zF)}Γ_{(C_{G},ζ)} (z ∈ H^{1}(F, A(u)))

because, as A(u) is abelian, we may identify H^{1}(F, A(u))^{∧} with the set of F -invariant characters of A(u). Similarly, if we make an arbitrary choice of v in C_{M}^{F} and parametrize the rational regular classes by elements of H^{1}(F, Z_{M}) with respect to this v, we have

Γ_{v_{z}}^{M} = |A_{M}(v)|^{−1} ∑_{ζ∈H^{1}(F,A(v))^{∧}} \overline{a_{ζ}ζ(zF)}Γ_{(C_{M},ζ)} (z ∈ H^{1}(F, A(v)))  (1)

where a_{ζ} is a root of unity arising from the fact that the isomorphism F^{∗}(ζ) ≃ ζ, chosen as explained below 1.6, differs by a root of unity from the isomorphism whose trace at v is 1. We now apply 2.9. The Deligne-Lusztig restriction ^{∗}R_{M}^{G}Γ_{(C_{G},ζ)} is non zero only if there exists a pair μ for M with same cuspidal datum as (C_{G}, ζ). From the discussion preceding the statement of this theorem this implies that ζ is in H^{1}(F, A_{M}(v))^{∧} with the above identification of H^{1}(F, A(u))^{∧} with the set of F -invariant characters of A(u), and that μ = (C_{M}, ζ). Hence

∗R_{M}^{G}Γ_{u_{z}}^{G} = ε_{G}ε_{M}|A_{M}(v)|^{−1} ∑_{ζ∈H^{1}(F,A(v))^{∧}} \overline{ζ(zF)}Γ_{(C_{M},ζ)}

= ε_{G}ε_{M}|A_{M}(v)|^{−1} ∑_{ζ∈H^{1}(F,A_{M}(v))^{∧}} ∑_{z′∈A_{M}(v)} a_{ζ} \overline{ζ(zF)}ζ(z′F)Γ_{v_{z′}}^{M} ,

where the last line follows by inverting (1). As we know that for each z′ there exists an irreducible character which has multiplicity 1 in Γ_{v_{z′}}^{M} and multiplicity 0 in the other Gel’fand-Graev characters of M^{F} (cf. [DLM, 3.10 and 3.12]), we see that for any z′ the expression

ε_{G}ε_{M}|H^{1}(F, A_{M}(v))|^{−1} ∑_{ζ∈H^{1}(F,A_{M}(v))^{∧}} a_{ζ} \overline{ζ(zF)}ζ(z′F)

is an integer. But the average of any finite set of roots of unity is an integer if and only if this average is zero or ±1. Moreover in the latter case all the roots of unity are equal. The sum cannot be 0 for all z′, because otherwise the a_{ζ} would be zero; so there exists z′ = f(z) ∈ H^{1}(F, A_{M}(v)) such that the sum is not 0 and we must have a_{ζ} \overline{ζ(zF)} = ε_{z}\overline{ζ(f(z)F)}, where ε_{z} = ±1. Taking for ζ the identity character in the above formula, we see that ε_{z} is equal to a_{1} for all z, so is independent of z, whence ζ(z^{−1}f(z)) is independent of z for any ζ. Hence f(z) = zz_{0} for some z_{0} ∈ H^{1}(F, A_{M}(v)). Writing ε_{z} = ε (independent of z), we obtain a_{ζ} = ε\overline{ζ(z_{0})} and

∗R_{M}^{G}Γ_{u_{z}}^{G} = ε\frac{ε_{G}ε_{M}}{|A_{M}(v)|} ∑_{ζ∈H^{1}(F,A(u))^{∧}} ∑_{z′∈H^{1}(F,A_{M}(v))} \overline{ζ(z_{0}zF)}ζ(z′F)Γ_{v_{z′}}^{M} = εε_{G}ε_{M}Γ_{v_{z_{0}z}}^{M}.

As we know by restriction from a group with connected centre and same derived group that ∑_{z∈H^{1}(F,Z_{G})} ^{∗}R_{M}^{G}Γ_{u_{z}}^{G} = ε_{G}ε_{M}|H^{1}(F, Z_{G})|/|H^{1}(F, Z_{M})| ∑_{z′∈H^{1}(F,Z_{M})} Γ_{v_{z′}}^{M} (cf. [DLM, formula in the proof of 3.12 and 5.4.1]) we must have ε = 1 and the result follows. □

3. The value of ζ_{I}.

We maintain the notation of §1. However, we assume only that the characteristic p is sufficiently large for the results of [Lu5] to apply. In particular, we make no further assumption on q. Let I be a rational block which contains a pair ι with regular support. In this section, we compute the value of the fourth root of unity ζ_{I}. Let (L, ι_{0}) be a cuspidal datum for I.

In order to state the result, we recall the following notation from [DLM]. For z ∈ H^{1}(F, Z_{G}) let Ψ_{z} be the orbit under T^{F} of the additive character of U^{F} used in the definition of the Gel’fand-Graev characters Γ_{u_{z}}; put σ_{z} = ∑_{ψ∈Ψ_{z}} ψ(u) and for any character ζ of H^{1}(F, Z_{G}) define σ_{ζ} = ∑_{z∈H^{1}(F,Z_{G})} ζ(z)σ_{z}. Recall that σ_{ζ} may be expressed as a product of Gaussian sums (cf. [DLM, 4.10]). We then have

3.1 Theorem. We have ζ_{I} = η_{G}(−1)^{rank(L)−dim Z_{L}}q^{1/2(dim C_{G}−dim G+dim Z_{L})}σ_{ζ}

where η_{G} = (−1)^{semi-simple 𝔽_{q}-rank(G)}.

Proof: Let ι = (C_{G}, ζ) be the pair in I with regular unipotent support. As in the previous section we fix u ∈ C_{G}^{F} and parametrize the rational conjugacy classes in C_{G}^{F} by H^{1}(F, A(u)) = H^{1}(F, Z_{G}). As above denote by u_{z} a representative of the rational class parametrized by z ∈ H^{1}(F, A(u)). We view ζ as a character of H^{1}(F, Z_{G}).

By proposition 1.6, we have

⟨ D_{G}(Γ_{ι}), X_{ι} ⟩_{G^{F}} = q^{1/2(dim C_{G}−dim G+dim Z_{L})}ζ′_{I}|A(u)|,  (1)

because P̃′_{ι,ι} = 1 and a_{ι} = |A(u)| by definition. The left hand side of (1) is equal to

|G^{F} |^{−1} ∑_{z′∈H^{1}(F,Z_{G})}(|G^{F} |/|C_{G}(u_{z′})^{F} |)(D_{G}Γ_{ι})(u_{z′})\overline{ζ(z′)},

as X_{ι}(u_{z′}) = ζ(z′) (cf. the narrative immediately preceding 1.6). Using 1.5 we have

∑_{z,z′∈H^{1}(F,Z_{G})} |A(u)||H^{1}(F, Z_{G})|^{−1}|C_{G}(u_{z′})^{F} |^{−1}\overline{ζ(z′)ζ(z)(D_{G}Γ_{u_{z}})(u_{z′})}.

By [DLM, (3.5)′] we have D_{G}Γ_{u_{z}}(u_{z′}) = c_{z,z′}|C_{G}(u_{z′})^{F} |, where (c_{z,z′}) is the inverse matrix of η_{G}(σ_{z(z′)^{−1}})_{z,z′}. From this we obtain

⟨ D_{G}(Γ_{ι}), X_{ι} ⟩_{G^{F}} = ∑_{z,z′∈H^{1}(F,Z_{G})} |A(u)||H^{1}(F, Z_{G})|^{−1}\overline{ζ(z′)}ζ(z)c_{z,z′}.  (2)

We shall require the following facts about the coefficients c_{z,z′} Lemma 3.2.

(i) The coefficient c_{z,z′} depends only on zz′^{−1}. (ii) If we put c_{z} = c_{z,1} and if, for any character ζ of H^{1}(F, Z_{G}), we put

c_{ζ} = ∑_{z∈H^{1}(F,Z_{G})} ζ(z)c_{z} we have c_{ζ} = η_{G}σ_{ζ}^{−1} where σ_{ζ} is as in the statement of the theorem.

Proof of the lemma: For any finite group Z, let A(Z) be the set of complex matrices (m_{z,z′})_{z,z′∈Z} which satisfy m_{zz_{1},z′z_{1}} = m_{z,z′} for any z_{1} ∈ Z. Clearly A(Z) is a linear subspace of the algebra of |Z| × |Z| complex matrices and since all entries of a matrix in A(Z) are determined by the first column, we have dim_{ℂ} A(Z) ≤ |Z|. The matrices representing the elements of the group algebra ℂZ with respect to the standard basis of the left regular representation of Z form a subalgebra C(Z) of the algebra of |Z| × |Z| matrices and clearly lie in A(Z). Since dim C(Z) = |Z|, we have A(Z) = C(Z). In particular A(Z) is an algebra of matrices and if m ∈ A(Z) is invertible, then mA(Z) = A(Z), so that m^{−1} ∈ A(Z), since the identity matrix is in A(Z).

Since η_{G}(σ_{z(z′)^{−1}})_{z,z′} ∈ A(H^{1}(F, Z_{G})), and (c_{z,z′}) is its inverse, we obtain (i). For (ii), observe that the map (m_{z,z′}) ↦ ∑_{z} m_{z,1}z defines an algebra isomorphism A(Z) → ℂZ and therefore that the map (m_{z,z′}) ↦ ∑_{z} m_{z,1}ζ(z) is an algebra homomorphism: A(Z) → ℂ. The equation (ii) now follows by applying this homomorphism to equation (c_{z,z′})(σ_{z(z′)^{−1}})_{z,z′} = η_{G}. □

We now return to the proof of the theorem. Using 3.2(i) we obtain from (2)

⟨ D_{G}(Γ_{ι}), X_{ι} ⟩_{G^{F}} = ∑_{z,z′∈H^{1}(F,Z_{G})} |A(u)||H^{1}(F, Z_{G})|^{−1}\overline{ζ(z′)}ζ(z)c_{zz^{′−1}}

= ∑_{z_{1},z′∈H^{1}(F,Z_{G})} |A(u)||H^{1}(F, Z_{G})|^{−1}\overline{ζ(z′)}ζ(z′)ζ(z_{1})c_{z_{1}}

= |A(u)| ∑_{z_{1}∈H^{1}(F,Z_{G})} ζ(z_{1})c_{z_{1}} = |A(u)|c_{ζ}.

Applying 3.2(ii) to the right hand side of the above formula and replacing the left hand side by its value as given by equation (1) we obtain

q^{1/2(dim C_{G}−dim G+dim Z_{L})}ζ′_{I}|A(u)| = |A(u)|η_{G}σ_{ζ}^{−1}.

Since ζ′_{I} = ζ_{I}^{−1}(−1)^{rank(L)−dim Z_{L}}, this completes the proof of the theorem. □

We shall now apply the above theorem to the case G = SL_{n}. In that case the block I is defined by a character ζ_{I} of Z_{G} such that ker ζ_{I} = ker h_{L}, so ζ_{I} factorizes through Z_{L}/Z_{L}^{∘} and the resulting injective character gives the cuspidal datum of the block. Let e be the order of ζ (both n and q − 1 are divisible by e); the Levi subgroup L is a group of block diagonal matrices with n/e diagonal blocks of size e and W_{I} is a symmetric group on n/e letters.

The value of σ_{ζ} may be found (up to some misprints) in [DLM, 4.11] and using the corrected the values, the formula for ζ_{I} becomes

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

where G(ε) is a Gauss sum corresponding to the character ε of order 2 of 𝔽_{q}. We know (cf. [Gauss, VII.356]) that, if q = p^{s} and if the additive character used in the definition of the Gaussian sums is given by the trace composed with the character of the prime field which maps 1 to e^{2iπ/p}, we have

G(ε) = q^{1/2}(−1)^{s−1} \{ \begin{matrix}1 & if p ≡ 1 (mod 4), \\ i^{s} & if p ≡ 3 (mod 4). \end{matrix}

It follows that, if the orbit Ψ_{1} (1 ∈ H^{1}(F, Z_{G})) is chosen as in [DLM, 2.5], with the additive character of 𝔽_{q} as above, we obtain

3.3 Proposition. If G is SL_{n}, if q = p^{s} and if the block I is defined by a character of order e of Z_{G}, we have

ζ_{I} = \{ \begin{matrix}1 & if e is odd, \\ (−1)^{n/e(s−1+(q−1)(e−2)/2e)} & if e is even and p ≡ 1 (mod 4), \\ (−1)^{n/e(s−1+(q−1)(e−2)/2e)}.i^{sn/e} & if e is even and p ≡ 3 (mod 4). \end{matrix}

References

[DLM] F. Digne, G. Lehrer and J. Michel, “The characters of the group of rational points of a reductive group with non-connected centre”, Crelle 425 (1992), 155–192.

[DM1] F. Digne and J. Michel, “Foncteurs de Lusztig et fonctions de Green généralisées”, C.R.A.S. 297 (1983), 89–92.

[DM2] F. Digne and J. Michel “Representations of finite groups of Lie type”, London Math.

Soc. Student texts 21, Cambridge university press (1991).

[Ge] M. Geck, “Basic sets of Brauer characters of finite groups of Lie type, III”, Manuscripta Mathematica 85 (1994), 195–216.

[Le] G. Lehrer, “ Fourier transforms, nilpotent orbits, Hall polynomials and Green func-

tions”, (1995) Luminy proceedings.

[Gauss] C.-F. Gauss, “Disquisitiones Arithmeticae” (1802).

[Lu1] G. Lusztig, “Coxeter orbits and eigenspaces of Frobenius”, Inventiones Math. 38 (1976), 101–159.

[Lu2] G. Lusztig, “Intersection cohomology complexes on a reductive group”, Inventiones Math.75 (1984), 205–272.

[Lu3] G. Lusztig, “Character sheaves V”, Advances in Math. 61 (1986) 103–155.

[Lu4] G. Lusztig, “Green functions and character sheaves”, Annals of Math. 131 (1990), 355–408.

[Lu5] G. Lusztig, “A unipotent support for irreducible representations”, Advances in Math. 94 (1992), 139–179.

L.A.M.F.A., Université de Picardie, 33, rue St-Leu 80039 Amiens, France C.M.A., Australian National University, Canberra ACT 0200 Australia

U.F.R. de Mathématiques, Université Denis Diderot, 2, place Jussieu 75005 Paris, France