Lusztig’s parametrization of characters

On Lusztig’s parametrization of characters of finite groups of Lie type

François Digne et Jean Michel

This paper has three parts. In the first part, we extend Lusztig’s results of [11] about the parametrization of characters of finite reductive groups with a connected center, including [11, theorem 4.23] about multiplicities of irreducible characters in the Deligne-Lusztig characters, to the case of groups with non-connected center. We use mostly a method sketched in chapter 14 of [11] in the case of a cyclic center, based on Clifford theory and a result about the unicity of the parametrisation of characters constructed in [11] which we prove in section 6 (part II). This construction has been carried out by Lusztig in [13] but we need more information than he gets there, in order to get the results of section 5 and of part III.

In sections 1 and 2 we state the results we need from Clifford theory, from [11] and about non-connected groups. We also need a result about the commutation of Lusztig twisted induction with isogenies, whose proof is given in section 9 (part III) using Shintani descent. We then apply these results to the parametrization of characters in section 3, where we need the results of part II. Finally section 4 and section 5 describe the multiplicities of irreducible characters in Deligne-Lusztig characters using Lusztig “families”, presented here from a simplified combinatorial viewpoint using the “Mellin transform”.

Part II describes under which conditions Lusztig’s parametrization of irreducible characters in [11] is unique; section 6 deals with families and Weyl groups, and section 7 gives the main theorem.

F.Digne, J.Michel

Part III studies Shintani descent in groups with non-connected center. We want to show how Shintani descent relates to the parametrization introduced in part I. Section 8 recalls facts about Shintani descent and “F′-twisted induction”. In section 9 we prove a result about the commutation of F′-twisted induction with isogenies and deduce the analogous result for Lusztig’s twisted induction. In section 10, we first extend to F -class functions the parametrization of section 5 (when the center is not connected we have to make assumptions that we cannot yet prove in all cases). Finally, we give a formula for Shintani descent of principal series characters using the Fourier transform on families (using section 5 of part I). For this last result we have to quote heavily from [7].

This paper * has been prompted by discussions with B. Srinivasan, and Shoji’s papers [14] and [15] where he gets a complete description of Shintani descent Sh_{F^{m}/F} for m sufficiently divisible and for a group with connected center (Shoji himself uses results of Asai [1] which deal with the case m = 1); this paper also has been prompted by the absence of a convenient written description dealing with groups with non-connected center.

0. Background.

In this section we recall some results from Clifford theory and the theory of F -class functions.

We denote by Irr(G) the set of irreducible characters of the finite group G (over an algebraically closed field of characteristic 0). We now give a general proposition which states basic (well known) results from Clifford theory. Most of these are easy consequences of Mackey formula and Frobenius reciprocity (see also [8, 2.1]).

0.1 Proposition (Clifford theory). Let G be a normal subgroup of a finite group G̃ such that the quotient G̃/G is abelian; let Z̃ be the center of G̃. For ρ ∈ Irr(G̃), we put A(ρ) = {ζ ∈ Irr(G̃/GZ̃) | ρ⊗ζ = ρ} (note that Z̃ is in the kernel of any ζ ∈ Irr(G̃/G) such that ρ ⊗ ζ = ρ). If μ ∈ Irr(G) is a component of ResG̃ ; G ρ, we note G̃(ρ) for the inertia group of μ in G̃ (it depends only on ρ (not on μ)). Then we have:

(i) Ker(A(ρ)) ⊂ G̃(ρ). (ii) There exists μ̃ ∈ Irr(Ker(A(ρ))), ρ̃ ∈ Irr(G̃(ρ)) and a positive inte-

ger e such that:

Ind_{G}^{Ker(A(ρ))}(μ) = ∑_{α∈Irr(Ker(A(ρ))/G)} μ̃ ⊗ α,Res_{G}^{Ker(A(ρ))}(μ̃) = μ,
Ind_{Ker(A(ρ))}^{G̃(ρ)} μ̃ = eρ̃,  Res_{Ker(A(ρ))}^{G̃(ρ)} ρ̃ = eμ̃,
IndG̃ ; G̃(ρ)(ρ̃) = ρ,  ResG̃ ; G̃(ρ)(ρ) = ∑_{x∈G̃/G̃(ρ)} ^{x}ρ̃,

(iii) The quotient group G̃(ρ)/Ker(A(ρ)) has cardinality e^{2} and we have

|A(ρ)| = ⟨ ResG̃ ; G(ρ), ResG̃ ; G(ρ) ⟩_{G}.

* part of this work was done during the authors visit at Essen university If G̃(ρ)/Ker(A(ρ)) is cyclic, then e = 1.

The following result, which is proved in [7, 6.1] gives e = 1 in a general setting for Weyl groups:

0.2 Lemma. Assume that G is a Weyl group and G̃ is the semi-direct product of G by a group A of diagram automorphisms of G, then for any character ρ ∈ Irr(G̃) we have e = 1.

The proof of this lemma requires the following result (cf. [7, 6.2]) that we will need below in the proof of 5.5

0.3 Lemma. Let G be a finite group of the form G_{1} × … × G_{l} and A be a finite group of automorphisms of G acting by permutation of the G_{i}. Let μ = μ_{1} ⊗ … ⊗ μ_{l} be an irreducible character of G. Let A_{i} be the subgroup of Stab_{A}(μ) normalizing G_{i} (and so μ_{i}). If for each i, the character μ_{i} has an extension to G_{i} >< A_{i}, then μ has an extension to G >< Stab_{A}(μ) (i.e., e = 1 for the character μ).

0.4 F -class functions. If G is a finite group and if <F > is a group generated by an element F and acting on G, we denote by C(G/F) the space of complex valued F -class functions on G, i.e. functions φ which verify φ(x.^{F}y) = φ(yx) for any x and y in G (note that the group <F > can be infinite). We may identify C(G/F) with the space of restrictions to the set G.F of class functions on the semi-direct product G ><<F >. This space admits as a basis the set of restrictions to G.F of an arbitrarily chosen extension to G ><<F > of each F-invariant irreducible character of G. If φ_{1} and φ_{2} are elements of C(G/F), we put

⟨ φ_{1}, φ_{2} ⟩_{G.F} = |G|^{−1} ∑_{x∈G.F} \overline{φ_{1}(x)}φ_{2}(x).

We recall that if φ_{1} and φ_{2} are characters of G ><<F > whose restrictions to G are irreducible, then

⟨ φ_{1}, φ_{2} ⟩_{G.F} = \{ \begin{matrix}0, & if Res_{G}^{G ><<F >} φ_{1} ≠ Res_{G}^{G ><<F >} φ_{2} \\ 1, & if φ_{1} = φ_{2} \end{matrix}

(if φ_{1} and φ_{2} have equal restrictions to G, then they differ by multiplication by a linear character of <F >).

If H is a subgroup of G stabilized by F, we denote by Res_{H.F}^{G.F} the restriction of F -class functions, and we define induction of F -class function by

Ind_{H.F}^{G.F} (f)(gF) = |H|^{−1} ∑_{{ γ∈G|^{γ}(gF)∈H.F }} f(^{γ}(gF)).

Induction and restriction are adjoint with respect to the above scalar product.

I

1. Disconnected groups.

In this section we extend the definition of Deligne-Lusztig characters to non connected reductive groups. We begin with a proposition which gives the relation between the Weyl group of a reductive group and that of its connected component.

1.1 Proposition. Let H be a reductive algebraic group, and let T be a maximal torus of H; then we may find representatives of H/H^{∘} in N_{H}(T). We set W = N_{H}(T)/T and W^{∘} = N_{H^{∘}}(T)/T. Let B be a Borel subgroup containing T. We put A = {w ∈ W | ^{w}Φ^{+} = Φ^{+}} where Φ is the root system of H^{∘} and + denotes the order on Φ corresponding to B. Then we have

(i) W = W^{∘} >< A and A ≃ H/H^{∘}. (ii) If H is defined over IF_{q}, with corresponding Frobenius F, and F

stabilizes T and B above, then F stabilizes W, W^{∘} and A.

In the following we consider a (not necessarily connected) reductive algebraic group H defined over IF_{q}, and denote by F the corresponding Frobenius endomorphism. We fix a pair T ⊂ B of an F -stable maximal torus in H included in an F -stable Borel subgroup. Let H^{∘}^{∗} be a group dual to H^{∘} containing a given torus T^{∗} dual to T. Finally we fix a Frobenius endomorphism F^{∗} dual to F. We may identify W^{∘} with N_{H^{∘}}^{∗}(T^{∗})/T^{∗} by mapping w to the dual isogeny w^{∗}, but note that this map is an anti-isomorphism. For any v ∈ W^{∘} we choose a representative v̇^{∗} ∈ N_{H^{∘}}^{∗}(T^{∗}) of v^{∗} and for any representative ȧ ∈ N_{H}(T) of an element a ∈ A, we choose an isogeny (ȧF)^{∗} dual to ȧF. For w ∈ W, if w is in the coset W^{∘}a, we write (ẇȧF)^{∗} for (ȧF)^{∗}ẇ^{∗}.

1.2 Definition. For s ∈ T^{∗}, we put W_{F} (s) = {w ∈ W | ^{(wF)^{∗}}s = s}. (Note that W_{F} (s) depends only on F (not on the choices of F^{∗} and of (ȧF)^{∗}) since two isogenies dual to the same one differ by ad t for some t ∈ T^{∗}).

We will assume given an isomorphism \overline{IF}_{q}^{×}⟶^{∼}(Q/ZZ)_{p′} and an embedding (Q/ZZ)_{p′} ↪ \overline{Q}_{l}^{×}; these choices give for any w ∈ W an identification Irr(T^{wF}) ≃ (T^{∗})^{(wF)^{∗}}.

We now generalize the definition of Deligne-Lusztig characters to non connected groups. The idea is to consider together the various rational forms of H that we get if we take as Frobenius endomorphisms hF, with h ∈ H. If h and h′ are in the same F -class of H, the groups H^{hF} and H^{h′F} are isomorphic. As the set H^{1}(F, H) of F -classes in H is isomorphic to H^{1}(F, A), we may choose a set of representatives in H of H^{1}(F, H) which are representatives of elements of A. Generalized Deligne-Lusztig characters will be constructed for each of the groups H^{ȧF} with ȧ such a representative.

Let U be the unipotent radical of B. For any w ∈ W, let us write wF = b^{−1}vaF b with a, b ∈ A and v ∈ W^{∘}, where a represents the F -class of w as above. Let v̇ ∈ N_{H^{∘}}(T) be a representative of v; we consider the variety Y_{v̇,ȧ} = { x ∈ H | x^{−1}^{ȧF}x ∈ v̇U }. Then H^{ȧF} acts on Y_{v̇,ȧ} by left multiplication, and t ∈ T^{wF} acts by right multiplication by ^{b}t. These two actions commute, so taking the alternating sum of ladic cohomology groups, we get a virtual representation of H^{ȧF} × T^{wF}, and for each character θ of T^{wF} we get a representation of H^{ȧF} on the part of the cohomology on which T^{wF} acts by θ. If s ∈ (T^{∗})^{(wF)^{∗}} corresponds to θ, we will denote by R_{T^{wF}}^{H^{ȧF}} (s) the (generalized) Deligne-Lusztig character that we have just defined. Note that we have defined Deligne-Lusztig characters of the groups H^{ȧF} when ȧ runs over a chosen set of representatives of H^{1}(F, A). We have the following properties of (generalized) Deligne-Lusztig characters:

1.3 Proposition.

(i) Let wF = b^{−1}vaF b as above, then R_{T^{wF}}^{H^{ȧF}} (s) = Ind_{H^{∘}^{ȧF}}^{H^{ȧF}} (ad ḃ ∘

R_{T^{wF}}^{H^{∘}^{ḃ^{−1}ȧF ḃ}}(s)). (ii) The union when ȧ runs over representatives of H^{1}(F, A) of the

Deligne-Lusztig characters of H^{ȧF} can be parametrized by pairs (s, wF) with s ∈ T^{∗} and w ∈ W_{F} (s), taken modulo W -conjugation.

(iii) Distinct Deligne-Lusztig characters are pairwise orthogonal.

Note that R_{T^{wF}}^{H^{∘}^{ḃ^{−1}ȧF ḃ}}(s) where b ∈ A, becomes under conjugation by ḃ the character R_{T^{vaF}}^{H^{∘}^{ȧF}} (s′) where s′ = ^{b}s, so (i) is equivalent to its particular case when b = 1 which is R_{T^{wF}}^{H^{ȧF}} (s) = Ind_{H^{∘}^{ȧF}}^{H^{ȧF}} R_{T^{wF}}^{H^{∘}^{ȧF}} (s). Proof: As stated above, to prove (i) we may assume b = 1, i.e., wF = vaF with v ∈ W^{∘}. Let Y_{v̇,ȧ}^{∘} be the variety { y ∈ H^{∘} | y^{−1}^{ȧF}y ∈ v̇U }. We claim that the map (y, h) ↦ hy from the variety Y_{v̇,ȧ}^{∘} × H^{ȧF} to Y_{v̇,ȧ} is an epimorphism whose fibers are the orbits of H^{∘ȧF} if the action of h_{0} ∈ H^{∘ȧF} is given by (y, h) ↦ (h_{0}y, hh_{0}^{−1}). In fact, any element in v̇U can be written, by Lang’s theorem as y^{−1}^{ȧF}y with y ∈ H^{∘}, so any x ∈ Y_{v̇,ȧ} differs by an element of H^{ȧF} from an element of Y_{v̇,ȧ}^{∘}. The assertion about fibers is clear. On Y_{v̇,ȧ}^{∘} × H^{ȧF}, we have an action of H^{ȧF} × T^{wF} given by (y, k) ↦ (yt, hk). This action is compatible with the action of H^{∘}^{F}. As the above epimorphism is compatible with the actions of H^{ȧF} × T^{wF}, we get an isomorphism of varieties with H^{ȧF} × T^{wF} -actions (Y_{v̇,ȧ}^{∘} × H^{ȧF} ) /H^{∘}^{F} →^{∼}Y_{v̇,ȧ}. Using properties of l-adic cohomology, we get an isomorphism of H^{ȧF} × T^{wF} -modules

H_{c}^{∗}(Y_{v̇,ȧ})→^{∼}H_{c}^{∗}(Y_{v̇,ȧ}^{∘}) ⊗_{\overline{Q}_{l}[H^{∘ȧF}]} \overline{Q}_{l}[H^{ȧF}], whence (i).

To prove (ii) and (iii), we have to show that generalized Deligne-Lusztig characters R_{T^{wF}}^{H^{ȧF}} (s) and R_{T^{w′F}}^{H^{ȧF}} (s′) are equal if and only if (wF, s) and (w′F, s′) are conjugate under the action of W and are orthogonal otherwise. By conjugation by ḃ with b ∈ W, we may assume that the elements w and w′ are in the coset W^{∘}.a. By (i), and Mackey formula, as distinct Deligne-Lusztig characters of H^{∘ȧF} are orthogonal, we see that the scalar product ⟨ R_{T^{wF}}^{H^{ȧF}} (s), R_{T^{w′F}}^{H^{ȧF}} (s′) ⟩_{H^{ȧF}} is non zero if and only if ⟨ R_{T^{wF}}^{H^{∘ȧF}} (s), R_{T^{w′F}}^{H^{∘ȧF}} (s′) ∘ ad ẋ ⟩_{H^{∘ȧF}} is non zero for some representative ẋ ∈ H^{ȧF} of some x ∈ A^{F}. As R_{T^{w′F}}^{H^{∘ȧF}} (s′) ∘ ad ẋ is equal to R_{T^{xw′F x^{−1}}}^{H^{∘ȧF}} (xs′x^{−1}), the above condition is equivalent to the pairs (wF, s) and (xw′F x^{−1}, xs′x^{−1}) being W^{∘}-conjugate for some x ∈ A^{F}, which is equivalent to (wF, s) and (w′F, s′) being W -conjugate (note that as wF and w′F are in the same coset W^{∘}.aF, they are W -conjugate if and only if they are W^{∘} >< A^{F} -conjugate).

We then define (generalizing to non connected groups Lusztig’s definition) series of characters in Irr(H^{ȧF}) by

E(H^{ȧF} , s) = { χ ∈ Irr(H^{ȧF}) | ∃w ∈ W_{F} (s), ⟨ χ, R_{T^{wF}}^{H^{ȧF}} (s) ⟩_{H^{ȧF}} ≠ 0 }.

Note that in the above definition wF has to be conjugate to an element of W^{∘}aF.

2. Rational series.

From now on we will consider a connected reductive algebraic group G and will keep the notations of the previous section with H = H^{∘} = G and H^{∘}^{∗} = G^{∗}.

Given x ∈ G, we denote by A(x) the group C_{G}(x)/C_{G}(x)^{∘}. In the sequel we fix an element s ∈ T^{∗} such that W_{F} (s) is not empty, and we write simply A for A(s). We will write W (s) for the Weyl group of C_{G^{∗}}(s); we have W (s) = {w ∈ W | ^{w^{∗}}s = s}, so W_{F} (s) is a single coset in W (s)\W. We will denote by W^{∘}(s) the Weyl group of C_{G^{∗}}(s)^{∘}, and Φ_{s} its root system. According to 1.1, (i), if we identify A with {w ∈ W (s) | ^{w}Φ_{s}^{+} = Φ_{s}^{+}}, then we have W (s) = W^{∘}(s) >< A; and if we choose w_{1} ∈ W_{F} (s) such that ^{(w_{1}F)^{∗}}Φ_{s}^{+} = Φ_{s}^{+}, then by 1.1, (ii) w_{1}F induces an automorphism of W (s) which stabilizes W^{∘}(s) and A.

As we want to use results from the theory of groups with a connected center, we will embedd G in such a group: Let i : G ↪ G̃ be an embedding defined over IF_{q} of G in a group G̃ such that the center Z̃ of G̃ is connected, and G and G̃ have the same derived group (we may identify G̃ with G ×_{Z} Z̃, where Z is the center of G). We denote again by F the Frobenius endomorphism on G̃, so we have F ∘ i = i ∘ F.

We denote by T̃ the maximal torus of G̃ containing T (T̃ is T×_{Z} Z̃ in the above identification). Let G̃^{∗} be a group dual to G̃ and T̃^{∗} ⊂ G̃^{∗} be a torus dual to T̃. We fix i^{∗} : G̃^{∗} → G^{∗} dual to i. It is an epimorphism mapping T̃^{∗} onto T^{∗} and Ker i^{∗} is a central torus of G̃^{∗}. The embedding i gives a natural identification of W = N_{G}(T)/T with NG̃(T̃)/T̃.

2.1 Definition.

(i) For a ∈ A we denote by [[a, s]] ∈ Ker i^{∗} the commutator of any two preimages in G̃^{∗} of a and s (it clearly does not depend on the chosen preimages).

(ii) We denote by [[A, s]] the set { [[a, s]] | a ∈ A }.

With these notations, a straightforward computation shows that:

2.2 Proposition. [[A, s]] is a group and the map a ↦ [[a, s]] is a group isomorphism A → [[A, s]] which maps the action of w_{1}F on A to the action of F on [[A, s]] ↪ Ker i^{∗}.

We want to study how Lusztig series restrict from G̃^{F} to G^{F}. First we choose a suitable preimage of s.

2.3 Lemma. There exists a preimage s̃ of s such that W_{F} (s̃) = W^{∘}(s)w_{1}.

Proof: It is a straightforward computation, using Lang’s theorem in the connected group Ker i^{∗}.

In the sequel we fix s̃ as in 2.3. It is well known that for any t ∈ T̃^{∗}, the restriction of i^{∗} : CG̃^{∗}(t) → C_{G^{∗}}(i^{∗}(t))^{∘} is an isogeny of the derived groups; so the Weyl group W (t) is identified to W^{∘}(i^{∗}(t)). Next proposition relates series in Irr(G̃^{F}) corresponding to various preimages of s. Before stating this proposition we need the following definition:

2.4 Definition. We will call Lang’s map in an algebraic group defined over IF_{q}, with Frobenius endomorphism F, the map L : g ↦ g^{−1}^{F}g.

In this section, we will use only Lang’s map from Ker i^{∗} into itself, and denote it again by L. Recall that, as Ker i^{∗} is connected, Lang’s map is onto.

2.5 Proposition.

(i) For any pair (t, w) with t ∈ T̃^{∗} and w ∈ W_{F} (t), we have

ResG̃^{F} ; G^{F} RG̃^{F} ; T̃^{wF} (t) = R_{T^{wF}}^{G^{F}} (i^{∗}(t)).

(ii) Let s̃z be a preimage of s (where z ∈ Ker i^{∗}); then W_{F} (s̃z) is not empty if and only if z ∈ L^{−1}([[A, s]]), and then W_{F} (s̃z) = W^{∘}(s)aw_{1}, where L(z) = [[a, s]].

(iii) Two series E(G̃^{F} , s̃z) and E(G̃^{F} , s̃z′) are equal if and only if z and z′ differ by an element of [[A, s]].

Proof: We will show in part III that (i) is actually true for any morphism i inducing an isogeny of derived groups and with connected kernel. Statement (ii) results from a straightforward computation. Statement (iii) just reflects the fact that W -action by conjugation on elements s̃z is the same as [[A, s]]-action by translation.

By (ii) and (iii) above we see that the series E(G̃^{F} , s̃z) are parametrized by

L^{−1}([[A, s]])/[[A, s]].

For any t ∈ T̃^{∗}, let \overline{E}(G^{F} , t) be the subset of Irr(G^{F}) whose elements occur in the restriction of some χ ∈ E(G̃^{F} , t). By (i) above, \overline{E}(G^{F} , s̃z) is a subset of E(G^{F} , s) and the series E(G̃^{F} , s̃z) are all the series E(G̃^{F} , t) such that \overline{E}(G^{F} , t) intersects E(G^{F} , s).

The following properties are well known.

2.6 Proposition.

(i) The characters of G̃^{F} /G^{F} correspond by duality to elements of (Ker i^{∗})^{F}.

(ii) The characters of G̃^{F} /G^{F} whose kernel contains Z̃^{F} .G^{F} correspond by duality to elements of (Ker i^{∗})^{F} which are in the derived group of G̃^{∗}.

Using the fact that, if ẑ ∈ Irr(G̃^{F} /G^{F}) corresponds to z ∈ (Ker i^{∗})^{F}, then we have RG̃^{F} ; T̃^{wF} (s̃z′z) = RG̃^{F} ; T̃^{wF} (s̃z′) ⊗ ẑ and proposition 0.1, we get:

2.7 Proposition.

(i) The action of ⊗ẑ, for z ∈ (Ker i^{∗})^{F}, maps E(G̃^{F} , s̃z′) to E(G̃^{F} , s̃z′z). (ii) Two sets \overline{E}(G^{F} , s̃z) and \overline{E}(G^{F} , s̃z′) are equal or disjoint, and are

equal if and only if z and z′ differ by an element of (Ker i^{∗})^{F}.

As a consequence of 2.5, (iii) and of 2.7, we see that the sets \overline{E}(G^{F} , s̃z) are parametrized by the set L^{−1}([[A, s]])/[[A, s]].(Ker i^{∗})^{F} ≃ H^{1}(w_{1}F, A). So, for a representative a ∈ A of an element of H^{1}(w_{1}F, A), we will denote by E(G^{F} , s, a) the set \overline{E}(G^{F} , s̃z) when z is such that L(z) = [[a, s]]. We will call such sets rational series. We have

E(G^{F} , s) = ∐_{a∈H^{1}(w_{1}F,A)} E(G^{F} , s, a)

(note that

E(G^{F} , s, a) = {χ ∈ Irr(G^{F}) | ∃v ∈ W^{∘}(s), ⟨ χ, R_{T^{avw_{1}F}}^{G^{F}} (s) ⟩_{G^{F}} ≠ 0},

and that H^{1}(w_{1}F, A) also parametrizes the rational classes in G^{∗}^{F} which are geometrically conjugate to s).

3. Parametrization.

Lusztig has shown that, if Z is connected, there is a bijection π_{s} : E(G^{F} , s) → E(C_{G^{∗}}(s)^{(ẇ_{1}F)^{∗}}, 1) (where ẇ_{1} is any representative of w_{1} in N_{G^{∗}}(T^{∗})) such that (extending π_{s} by linearity to the ZZ-span of E(G^{F} , s))

3.1  π_{s}(R_{T^{wF}}^{G^{F}} (s)) = (−1)^{l(w_{1})}R_{T^{∗}^{(wF)^{∗}}}^{C_{G^{∗}} (s)^{(ẇ_{1}F)^{∗}}} (1)

for any w ∈ W_{F} (s) (it is an immediate consequence of [11, 4.23] applied once in the group G^{F} and once in the group C_{G^{∗}}(s)^{(ẇ_{1}F)^{∗}}).

We want to show that a similar result holds in general. In fact, as we will show in part II, condition 3.1 above determines uniquely π_{s} for classical groups (with connected centre), but there is some ambiguity for other groups; the main result of part II will be to give additional conditions which define uniquely π_{s} in all cases. In what follows, we will assume that for the group G̃ (cf. notations of section 2), the map π_{s} is defined as in part II. Using results from part II we prove the following:

3.2 Theorem. Let b ∈ A^{w_{1}F}; we denote by b̂ ∈ Irr(G̃^{F} /G^{F}) the character corresponding to [[b, s]] ∈ (Ker i^{∗})^{F} by 2.6. Then, if the representative ḃ^{∗} is chosen (ȧẇ_{1}F)^{∗}-fixed, the action of ⊗b̂ on E(G̃^{F} , s̃z) corresponds by π_{s} to that of ad ḃ^{∗} on E(CG̃^{∗}(s̃z)^{(ȧẇ_{1}F)^{∗}}, 1).

Proof: By 7.1 (iii), if ζ = [[b, s]], we have

π_{s̃zζ}(χ ⊗ b̂) = π_{s̃z}(χ) ∈ E(CG̃^{∗}(s̃z)^{(ȧẇ_{1}F)^{∗}}, 1).

By 7.3, we have the equality ad(ḃ^{∗}) ∘ π_{s̃zζ} = π_{s̃z}, whence the result. We need the following assumption:

3.3 Hypothesis. Restriction of representations from G̃^{F} to G^{F} are multiplicity-free, as well as restrictions of unipotent representations from C_{G^{∗}}(s)^{(ȧẇ_{1}F)^{∗}} to C_{G^{∗}}(s)^{∘}^{(ȧẇ_{1}F)^{∗}}, for any a ∈ A.

By 0.1, (iii), both parts of this hypothesis are true if G has no components of type D_{2n}, since both G̃^{F} /G^{F} and A^{w_{1}F} identify to subquotients of the fundamental group of G, which is then cyclic; Lusztig has proved the first part for type D_{2n} (cf. [13]). In the sequel we will assume that 3.3 is true.

By theorem 3.2, we see that, for ρ ∈ E(G̃^{F} , s̃z), the map a ↦ â is a group isomorphism from A(ρ) (cf. notations of 0.1) to Stab_{A^{w_{1}F}} (π_{s̃z}(ρ)); so, using 0.1 and assumption 3.3, we see that the number of irreducible components of ResG̃^{F} ; G^{F} (ρ) is | Stab_{A^{w_{1}F}} (π_{s̃z}(ρ))|.

Since i^{∗} defines an isogeny of the derived groups of CG̃^{∗}(s̃z) and of C_{G^{∗}}(s)^{∘} and since unipotent characters factorize through isogenies, we may consider π_{s̃z}(ρ) as an element of Irr(C_{G^{∗}}(s)^{∘}^{(ȧẇ_{1}F)^{∗}}) (where L(z) = [[a, s]]). By 0.1 and 3.3, the number of irreducible components

of Ind_{C_{G^{∗}} (s)^{∘}^{(ȧẇ_{1}F)^{∗}}}^{C_{G^{∗}} (s)^{(ȧẇ_{1}F)^{∗}}} π_{s̃z}(ρ) is | Stab_{A^{w_{1}F}} (π_{s̃z}(ρ))|. So we get:

3.4 Proposition. The sets E(G^{F} , s, a) and E(C_{G^{∗}}(s)^{(ȧẇ_{1}F)^{∗}}, 1) are in bijection.

We define π_{s} as being the collection over a ∈ H^{1}(w_{1}F, A) of these bijections. Note that the parametrization of individual components of a restriction is not completely defined, but, once a component μ of ResG̃^{F} ; G^{F} ρ and a component π_{s}(μ) of

Ind_{C_{G^{∗}} (s)^{∘}^{(ȧẇ_{1}F)^{∗}}}^{C_{G^{∗}} (s)^{(ȧẇ_{1}F)^{∗}}} π_{s̃z}(ρ) are chosen, we can define π_{s} uniquely for other

components of ResG̃^{F} ; G^{F} ρ: first note that by 0.1 Ker A(ρ) = G̃^{F} (ρ) under assumption 3.3; so A(ρ)→^{∼} Irr(G̃^{F} /G̃^{F} (ρ)), and we get a canonical isomorphism

G̃^{F} /G̃^{F} (ρ)→^{∼} Irr(Stab_{A^{w_{1}F}} (π_{s̃z}(ρ)));

if we demand that the action of G̃^{F} /G̃^{F} (ρ) on components of ResG̃^{F} ; G^{F} ρ be mapped by π_{s} on that of tensorization by linear characters of the group Stab_{A^{w_{1}F}} (π_{s̃z}(ρ)), we define canonically π_{s}. We now clearly have the analogue of 3.1 for groups with a non connected center, that is, for a ∈ A and v ∈ W^{∘}(s),

(3.5)  π_{s}(R_{T^{avF}}^{G^{F}} (s)) = (−1)^{l(aw_{1})}R_{T^{∗}^{(aw_{1}F)^{∗}}}^{C_{G^{∗}} (s)^{(ȧẇ_{1}F)^{∗}}} .

4. Families.

From section 3 it follows that, in order to know the multiplicities of the irreducible characters in the Deligne-Lusztig characters, it is enough to solve the problem for unipotent characters. We will write R_{wF} for R_{T^{wF}}^{G^{F}} (1) to abbreviate the notations. Lusztig [11] has shown in the connected-centre case how the multiplicities are given by a “Fourier transform” over “families” in the Weyl group. He defines a partition Ξ(W) of Irr(W) in “families”, and to each F ∈ Ξ(W) associates a finite group Γ_{F}, the set M(Γ_{F}) = {(x, χ) | x ∈ Γ_{F}, χ ∈ Irr(C_{Γ_{F}} (x))}/Γ_{F} (the action of Γ_{F} being by conjugation), and an embedding F ↪ M(Γ_{F}).

These data are functorial, i.e.,

4.1 Proposition. Given an isomorphism W ⟶^{π}W′ of Weyl groups (i.e., an isomorphism of groups coming from an isomorphism of root systems), we have π(Ξ(W)) = Ξ(W′), and for any F ∈ Ξ(W), there is a well-defined isomorphism π_{F} : Γ_{F} ⟶^{∼}Γ_{π(F)} (and π ↦ π_{F} is functorial); furthermore, if we again denote by π_{F} the induced isomorphism from M(Γ_{F}) to M(Γ_{π(F)}), the following diagram commutes:

M(Γ_{F}) ⟶^{π_{F}} M(Γ_{π(F)})
↑  ↑
F  ⟶^{π}  π(F)

Note that in the case of classical groups we get π_{F} using the facts that the groups Γ_{F} are commutative and that the image of F in M(Γ_{F}) contains the elements of the form (x, Id).

Proposition 4.1 implies that, given F ∈ Ξ^{F}, a well-defined automorphism of Γ_{F} is associated to F; we will again denote it by F. Lusztig shows that E(G^{F} , 1) is in bijection with the union over F ∈ Ξ^{F} of the sets

\overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >) = {(xF, χ) | x ∈ Γ_{F}, χ ∈ Irr(C_{Γ_{F}} (xF))}/Γ_{F} ><<F >

We will denote by ρ_{(xF,χ)} the irreducible unipotent character parametrized by (xF, χ). To describe the multiplicities ⟨ ρ_{(xF,χ)}, R_{wF} ⟩_{G^{F}} we first define another basis of the space spanned by the {R_{wF} }. For any F -class function f on W, we define R_{f} = ∑_{w∈W} f(w)R_{wF}. We can take as a basis of the space of F -class functions on W those obtained by choosing one extension to W ><<F > of each element of Irr(W)^{F} (the corresponding R_{f} are called the “almost-characters”).

4.2 Notations.

(i) For F ∈ Ξ, we will denote by E(G^{F} , 1, F) the set of unipotent characters parametrized by \overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >).

(ii) For F ∈ Ξ^{F} we will denote by F̃ the subspace of F -class functions spanned by extensions of elements of F^{F}.

With this notation, Lusztig defines an embedding of F̃ in the vector space

\underline{M}(Γ_{F} ⊂ Γ_{F} ><<F >) = ⊕_{y∈Γ_{F}} V_{y},

where V_{y} is the set of functions on C_{Γ_{F} .F} (y) (the set of elements in Γ_{F}.F which centralize y) invariant under conjugation by C_{Γ_{F}} (y) (note that only elements y such that C_{Γ_{F} .F} (y) is not empty are relevant).

4.3 Then, Lusztig shows that there is a pairing between \underline{M} and \overline{Q}_{l}\overline{M} given by

{(y, τ), (xF, χ)} = |C_{Γ_{F}} (y)|^{−1}|C_{Γ_{F}} (xF)|^{−1}Δ_{(xF,χ)}

∑_{\{g∈Γ_{F} |[^{g}y,xF]=1\}} \overline{χ(^{g}y)}τ(^{g^{−1}}(xF)),

(where Δ_{(xF,χ)} is a sign which is constant over each family except for socalled exceptional families i.e., families containing a character such that the corresponding representation of the Hecke algebra is not defined over Q[q, q^{−1}], which occurs only in types E_{7}, E_{8}) such that, if we consider all the above class functions as taking their values in \overline{Q}_{l}, the scalar product ⟨ ρ_{(xF,χ)}, R_{f} ⟩_{G^{F}} is given by {(y, τ), (xF, χ)} where f ∈ F̃ corresponds to (y, τ).

Let us remark that the pairing formula depends only on the action of F on Γ_{F} (not on the order of F). We also remark that this formula depends only on the restriction of τ to the centralizer of y in the coset Γ_{F}.F. We now introduce new bases in order to get simpler formulas for the pairing.

4.4 For x, y ∈ Γ_{F} such that [xF, y] = 1 we define “Mellin transforms”

(xF, y) = ∑_{χ∈Irr(C_{Γ_{F}} (xF))} χ(y)Δ_{(xF,χ)}(xF, χ) ∈ \overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >)

and

(y, xF) = ∑_{τ} τ(xF)(y, τ) ∈ \underline{M}(Γ_{F} ⊂ Γ_{F} ><<F >),

where the sum runs over a basis of the C_{Γ_{F}} (y)-class functions on C_{Γ_{F} .F} (y) which consists of the restrictions to the coset C_{Γ_{F} .F} (y) of one extension to C_{Γ_{F} ><<F >}(y) of each xF -invariant irreducible character of C_{Γ_{F}} (y). The pairs (xF, y) and (y, xF) are taken up to Γ_{F}-conjugacy. The following proposition gives a formula for the pairing using Mellin transforms.

4.5 Proposition. The pairing between \overline{Q}_{l}\overline{M} and \underline{M} is given by

{(xF, y), (y′, x′F)} = δ_{(xF,y),(x′F,y′)}|C_{Γ_{F}} ((xF, y))|.

Proof: By definition, we have

\{(xF, y), (y′, x′F)\} = ∑_{χ,τ} χ(y)Δ_{(xF,χ)}\overline{τ(x′F)}\{(xF, χ), (y′, τ)\}.

If we replace {(xF, χ), (y′, τ)} by its value given above, and if we use the fact that Δ_{(xF,χ)}^{2} = 1, we get

∑_{{g∈Γ_{F} |[^{g}y′,xF]=1}} |C_{Γ_{F}} (y′)|^{−1}|C_{Γ_{F}} (xF)|^{−1}

∑_{χ} \overline{χ(^{g}y′)}χ(y) ∑_{τ} \overline{τ(x′F)}τ(^{g^{−1}}(xF)).

By orthogonality formulas for characters and for the chosen basis of C_{Γ_{F}} (y)-class functions on C_{Γ_{F} .F} (y), we get the result.

5. Fourier coefficients.

In this section we show that the results stated in the preceding section hold in the non-connected centre case. For this we have to give the appropriate definition for families of a group of the form W >< A, and to generalize Lusztig’s parametrization of unipotent characters and of almost characters to non connected groups (the non connected groups we have to consider are the centralizers of semi-simple elements in G^{∗}, so we must keep in mind that the results of this section are to be applied when H, A, and F stand respectively for C_{G^{∗}}(s), A(s), and (w_{1}F)^{∗}). First we give the appropriate definition of families (cf. also [12]).

5.1 Definition. In the situation of 1.1, we define Ξ(W) to be the set of A-orbits in Ξ(W^{∘}), and, for F ∈ Ξ(W), we put Γ_{F} = Γ_{F_{0}} >< Stab_{A}(F_{0}), where F_{0} is in the A-orbit F. If F is F -stable, we define the action of F on Γ_{F} to be that induced by aF on Γ_{F_{0}} >< Stab_{A}(F_{0}) where a is such that ^{aF}F_{0} = F_{0}.

Note that the action of F is defined only up to an inner automorphism, but that this does not matter since all the constructions we made are invariant by Γ_{F}-conjugacy.

We can then consider objects such as \overline{M}, \underline{M}, etc… for non connected groups. We extend the definition of Δ (cf. 4.3) to our situation: given (xF, χ) ∈ \overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >) we define Δ_{(xF,χ)} = Δ_{(xaF,ψ)} for

any a such that ^{aF}F_{0} = F_{0} and any component ψ of Res_{C_{Γ_{F_{0}}} (xaF)}^{C_{Γ_{F}} (xF)} χ. This makes sense since Δ is invariant by any automorphism of W^{∘}. Next proposition will show that, with these definitions, ∐_{a∈H^{1}(F,A)} E(H^{ȧF} , 1) is again parametrized by ∐_{F∈Ξ(W)} \overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >).

5.2 Proposition. Let H be a non connected group such that A = H/H^{∘} is abelian. We keep the notation of 1.1 and of 1.3. We assume that restrictions of irreducible representations from H^{ȧF} to H^{∘ȧF} are multiplicity free for any a ∈ A. Then there is an isomorphism

⨁_{a∈H^{1}(F,A)} \overline{Q}_{l}E(H^{ȧF} , 1)→^{∼} ⨁_{F∈Ξ(W)^{F}} \overline{Q}_{l}\overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >)

such that for any family F_{0} ∈ Ξ(W^{∘}), the map

Ind_{H^{∘ȧF}}^{H^{ȧF}} : E(H^{∘ȧF} , 1, F_{0}) → E(H^{ȧF} , 1, F)

(with the notations of definition 5.1) corresponds to the linear map

\overline{Q}_{l}\overline{M}(Γ_{F_{0}} ⊂ Γ_{F_{0}} ><<aF >) → \overline{Q}_{l}\overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >)

defined on the basis of Mellin transforms by ((xaF, y) mod Γ_{F_{0}}) ↦ ((xF, y) mod Γ_{F}).

Proof: Irreducible characters in the left hand side are by definition extensions to H^{ȧF} of the sums of A^{F} -orbits of characters in E(H^{∘ȧF} , 1). By part II (cf. 6.4) we have a well defined bijection

E(H^{∘ȧF} , 1)→^{∼} ∐_{F∈Ξ(W^{∘})^{aF}} \overline{M}(Γ_{F} ⊂ Γ_{F} ><<aF >).

We denote by F_{0}, …, F_{r} the A-orbit of F_{0} ∈ Ξ(W^{∘}); by definition it is the element F of Ξ(W). It is clear from proposition 4.1 that there is a bijection from the set of A^{F} -orbits in ∐_{\{ i|^{aF}F_{i}=F_{i} \}} \overline{M}(Γ_{F_{i}} ⊂

Γ_{F_{i}} ><<aF >) onto { (ya′F, ψ) | y ∈ Γ_{F_{0}}, a′F ∼_{A} aF, ^{a′F}F_{0} = F_{0}, ψ ∈

Irr(C_{Γ_{F_{0}}} (ya′F)) }/Γ_{F}, given by

(xaF, χ) mod Γ_{F_{i}} ↦ (^{τ_{i}}xa_{i}a^{F}a_{i}^{−1}F,^{τ_{i}}χ) mod Γ_{F},

where Γ_{F} is identified to Γ_{F_{0}} >< Stab_{A}(F_{0}), where a_{i} is an element of A such that ^{a_{i}}F_{i} = F_{0}, and where τ_{i} : Γ_{F_{i}} →^{∼}Γ_{F_{0}} is the isomorphism associated to a_{i} as in 4.1.

We now claim that the number of extensions to H^{ȧF} of the sum of an A^{F} -orbit in E(H^{∘ȧF} , 1) corresponding via the parametrization of proposition II, 6.4 and the above bijection to the class of (ya′F, ψ) modulo Γ_{F_{0}} >< Stab_{A}(F_{0}) is equal to the number of irreducible components of the induced representation of ψ to C_{Γ_{F_{0}} >< Stab_{C}(F_{0})}(ya′F). By assumption, restrictions of irreducible characters from H^{ȧ′F} to H^{∘}^{ȧ′F} are multiplicity free. We need the same property for restrictions of irreducible characters from C_{Γ_{F_{0}} >< Stab_{A}(F_{0})}(ya′F) to C_{Γ_{F_{0}}} (ya′F). This can be proved in the following way: by 0.3 we can reduce ourselves to the case where W^{∘} is irreducible; then both a′F and Stab_{A}(F_{0}) act trivially on Γ_{F_{0}}; so C_{Γ_{F_{0}} >< Stab_{A}(F_{0})}(ya′F) is the direct product of C_{Γ_{F_{0}}} (ya′F) = C_{Γ_{F_{0}}} (y) by the abelian group Stab_{A}(F_{0}, F) and the result is clear.

Now, proving our claim is equivalent to show that the stabilizer in H^{ȧF} /H^{∘ȧF} of the character ρ of H^{∘ȧF} parametrized by (xaF, χ) ∈ \overline{M}(Γ_{F_{i}} ⊂ Γ_{F_{i}} ><<aF >) (a preimage of (ya′F, ψ) by the above bijection) has the same cardinality as Stab_{Γ_{F}} (ya′F, ψ)/C_{Γ_{F_{0}}} (ya′F). But this quotient group is isomorphic to the stabilizer in Stab_{A}(F_{0}) of ((ya′F, ψ) mod Γ_{F_{0}})), the isomorphism being induced by the projection of Γ_{F} onto Stab_{A}(F_{0}). On the other hand, the stabilizer of ρ in A^{F} is equal by II, 6.5 and II, 6.6 to the stabilizer of (xaF, χ) in Stab_{A^{F}} (F_{i}). As this latter group is isomorphic to Stab_{A^{F}} (F_{0}, (ya′F, ψ)) = Stab_{A}(F_{0}, (ya′F, ψ)), we get our claim. So we have a bijection

E(H^{ȧF} , 1, F)→^{∼}

{(ya′F, χ) | y ∈ Γ_{F_{0}}, a′F ∼_{A} aF, ^{a′F}F_{0} = F_{0}, χ ∈ Irr(C_{Γ_{F}} (ya′F))}/Γ_{F}

such that if ρ ∈ E(H^{∘ȧF} , 1) corresponds to (xaF, ψ) ∈ \overline{M}(Γ_{F_{0}} ⊂ Γ_{F_{0}} ><<aF >), the irreducible components of Ind_{H^{∘ȧF}}^{H^{ȧF}} ρ correspond to the elements (xaF, χ) in the right hand side set, where χ runs over the irreducible components of Ind_{C_{Γ_{F_{0}}} (xaF)}^{C_{Γ_{F}} (xaF)} ψ. If we take the union over

H^{1}(F, A), we then get

∐_{a∈H^{1}(F,A)} E(H^{ȧF} , 1, F)→^{∼}

{(ya′F, χ) | y ∈ Γ_{F_{0}}, a′ ∈ A, ^{a′F}F_{0} = F_{0}, χ ∈ Irr(C_{Γ_{F}} (ya′F))}/Γ_{F}.

The set in the above right hand side is isomorphic to

{ (xF, χ) | x ∈ Γ_{F}, χ ∈ Irr(C_{Γ_{F}} (xF)) }

by definition of Γ_{F}. So, taking the union over all F -invariant families F, we get the isomorphism of the proposition.

We now show the statement on Mellin transforms. It will be an immediate consequence of what we have shown above and of the following lemma.

5.3 Lemma. The map

Ind\overline{M} : \overline{Q}_{l}\overline{M}(Γ_{F_{0}} ⊂ Γ_{F_{0}} ><<aF >) → \overline{Q}_{l}\overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >)

defined by (xaF, ψ) ↦ (xF, Ind_{C_{Γ_{F_{0}}} (xaF)}^{C_{Γ_{F}} (xF)} ψ) maps ((xaF, y) mod Γ_{F_{0}})

on ((xF, y) mod Γ_{F}).

Proof: By definition we have

(xaF, y) = ∑_{ψ∈Irr(C_{Γ_{F_{0}}} (xaF))} ψ(y)Δ_{(xaF,ψ)}(xaF, ψ).

So its image is

∑_{ψ∈Irr(C_{Γ_{F_{0}}} (xaF))} ψ(y)Δ_{(xaF,ψ)}(xaF, Ind ψ) =

∑_{ψ∈Irr(C_{Γ_{F_{0}}} (xaF))} ψ(y)Δ_{(xaF,ψ)} ∑_{χ∈Irr(C_{Γ_{F}} (xF))} ⟨ Ind ψ, χ ⟩_{C_{Γ_{F}} (xF)}(xF, χ).

By definition we have Δ_{(xaF,ψ)} = Δ_{(xF,χ)} for any χ appearing in Ind ψ. So we can exchange the summations to get

∑_{χ∈Irr(C_{Γ_{F}} (xF))} Δ_{(xF,χ)}(xF, χ) ∑_{ψ∈Irr(C_{Γ_{F_{0}}} (xaF))} ⟨ Ind ψ, χ ⟩_{C_{Γ_{F}} (xF)}ψ(y).

As the second sum is equal to χ(y), we get

∑_{χ∈Irr(C_{Γ_{F}} (xF))} Δ_{(xF,χ)}(xF, χ)χ(y),

which is by definition the element (xF, y) of \overline{Q}_{l}\overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >), whence the lemma, and the proposition.

We will now prove an analogous statement for F -class functions on W and spaces \underline{M}. For this we generalize to W Lusztig’s embedding of F -class functions on W^{∘} mapping F_{0} into \underline{M}(Γ_{F_{0}} ⊂ Γ_{F_{0}} ><<aF >.

5.4 Proposition. Let F ∈ Ξ(W)^{F} be the orbit {F_{0}, …, F_{r}}. There is an embedding

F̃ ⊂ \underline{M}(Γ_{F} ⊂ Γ_{F} ><<F >)

such that the map Ind_{W^{∘}.ȧF}^{W.F} : F̃_{0} → F̃ is the restriction of the linear map \underline{M}(Γ_{F_{0}} ⊂ Γ_{F_{0}} ><<aF >) → \underline{M}(Γ_{F} ⊂ Γ_{F} ><<F >) defined on the basis of Mellin transforms by ((y, xaF) mod Γ_{F_{0}}) ↦ ((y, xF) mod Γ_{F}).

Proof: The elements of F are the components of Ind_{W^{∘}}^{W} φ for φ ∈ F_{0}. Let A_{0} = Stab_{A}(Γ_{F_{0}}) so that Γ_{F} = Γ_{F_{0}} >< A_{0}; if φ corresponds to (x, σ) ∈ M(Γ_{F_{0}}), by 4.1 we have Stab_{A}(φ) = Stab_{A_{0}}(x, σ), so the components of Ind_{W^{∘}}^{W}(φ) are in bijection with those of Ind_{C_{Γ_{F_{0}}} (x)}^{C_{Γ_{F}} (x)} σ,

whence an embedding F ⊂ M(Γ_{F}). Let us prove that the action of F on F corresponds via this embedding to the action of F on M(Γ_{F}): since components of Ind_{W^{∘}}^{W} φ (resp. components of Ind_{C_{Γ_{F_{0}}} (x)}^{C_{Γ_{F}} (x)} σ) differ from each other by characters of A, if we prove that there exists an F -stable component in each of these sets, then the action of F on these components will be in either case given by that of F on Irr(A). The following lemma proves that there exists an F -stable component in each of these sets.

5.5 Lemma.

(i) Let φ ∈ Irr(W^{∘}) have an F -invariant A-orbit. Then Ind_{W^{∘}}^{W} φ has an F -invariant irreducible component.

(ii) Let a ∈ A^{F} and let F ∈ Ξ(W)^{F} be the F -stable A-orbit {F_{0}, …, F_{r}} in Ξ(W^{∘})^{a}. For any (xa, χ) ∈ \overline{M}(Γ_{F_{0}} ⊂ Γ_{F_{0}} ><<a>), there exists a component χ̃ of Ind_{C_{Γ_{F_{0}}} (x)}^{C_{Γ_{F}} (x)} χ, such that (x, χ̃) is an F -stable element of M(Γ_{F}).

Proof: We first prove (i). By 0.2 φ extends to some χ ∈ Irr(Ĩ) where Ĩ = Stab_{W ><<F >} φ. Let I = Stab_{W} (φ); by hypothesis there exists a ∈ A such that aF ∈ Ĩ, so ResĨ ; I χ is an (irreducible) aF -stable character, and so Ind_{I}^{W} ResĨ ; I χ is F -stable and is an irreducible component of Ind_{W^{∘}}^{W} φ, whence the result.

The proof of (ii) is similar: the preceding argument shows that it suffices to show that χ extends to Stab_{Γ_{F′} ><<F >}(xa, χ). By using 0.3 we can reduce ourselves to the case where W is irreducible, and then the result is trivial since in this case Stab_{A^{F}} (F_{0}) acts trivially on Γ_{F_{0}}.

We now define the embedding of the proposition in the following way: for each F -invariant component τ of Ind_{W^{∘}}^{W} φ mapped onto (x, χ) (where χ is a component of Ind_{C_{Γ_{F_{0}}} (x)}^{C_{Γ_{F}} (x)} ψ) we choose a mapping τ̃ ↦

(x, χ̃) compatible with tensorization by characters of A. We then get x_{Ind}_{W^{∘} ><<aF >}^{W ><<F >} τ̃ = (x, Ind_{C_{Γ_{F_{0}} ><<aF >}(x)}^{C_{Γ_{F} ><<F >}(x)} ψ̃). The statement about Mellin transforms is then an immediate consequence of the following lemma:

5.6 Lemma. The map Ind\underline{M} : \underline{M}(Γ_{F_{0}} ⊂ Γ_{F_{0}} ><<aF >) → \underline{M}(Γ_{F} ⊂

Γ_{F} ><<F >) defined by (x, ψ̃) ↦ (x, Ind_{C_{Γ_{F_{0}} ><<aF >}(x)}^{C_{Γ_{F} ><<F >}(x)} ψ̃) maps the ele-

ment (y, xaF) mod Γ_{F_{0}} on (y, xF) mod Γ_{F}.

Proof: By definition we have (y, xaF) = ∑ψ̃ ψ̃(xaF)(y, ψ̃), where ψ̃ runs over a set consisting of one extension of each xaF -invariant irreducible character of C_{Γ_{F_{0}}} (y). So its image is

∑_{ψ̃} ψ̃(xaF)(y, Ind ψ) = ∑_{ψ̃} ψ̃(xaF) ∑_{χ} ⟨ Ind ψ̃, χ̃ ⟩(x, χ̃),

where χ̃ runs over a set consisting of one extension of each xF -invariant irreducible character of C_{Γ_{F}} (y). If we exchange the summations, we get

∑_{χ̃}(y, χ̃) ∑_{ψ̃} ⟨ Ind ψ̃, χ̃ ⟩ψ̃(xaF).

As the second sum is equal to χ̃(xF), we get

∑_{χ̃}(y, χ̃)χ̃(xF),

which is by definition the element (xF, y) of \underline{M}(Γ_{F} ⊂ Γ_{F} ><<F >), whence the lemma, and the proposition.

We will denote by f ↦ x_{f} the embedding defined by the previous proposition.

We want now to show that the scalar products ⟨ ρ_{(xF,χ)}, R_{f} ⟩_{H^{F}} (where R_{f} will be defined in an analogous way to the connected case, using generalized Deligne-Lusztig characters) are still given by 4.5. We first generalize the notation R_{f} for non connected groups.

5.7 Definition. For any F -class function f on W, we define

R_{f} = |W |^{−1} ∑_{w∈W} f(w)R_{T^{wF}}^{H^{ȧF}} (1)

(an element of the direct sum of the spaces of class functions on H^{ȧF} when ȧ runs over representatives of H^{1}(F, A)).

Note that the linear span considered above has a natural scalar product, which is the orthogonal sum of the scalar products of class functions on each group H^{ȧF}, and that similarly we can extend the pairing 4.3 to a pairing between ⊕_{F∈Ξ(W)^{F}} \overline{Q}_{l}\overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >) and ⊕_{F∈Ξ(W)^{F}} \underline{M}(Γ_{F} ⊂ Γ_{F} ><<F >). With these notations, we can now state:

5.8 Theorem. Let ρ ∈ E(H^{ȧF} , 1) be parametrized by

x_{ρ} ∈ ∐_{F∈Ξ(W)^{F}} \overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >)

and let f be an F -class function on W; then the scalar product ⟨ ρ, R_{f} ⟩ is given by {x_{ρ}, x_{f}}.

Proof: It is sufficient to prove the result for a basis of the space of F - class functions on W. We get such a basis, if we consider the functions γ_{wF}, where, for w ∈ W, we denote by γ_{wF} the function whose value is the cardinality of the centralizer C_{W} (wF) on the class of wF under W and zero outside. We have R_{γ_{wF}} = R_{T^{wF}}^{H^{ḃF}} (1), if w is of the form vbF, with b in the chosen set of representatives of H^{1}(F, A) and v ∈ W^{∘}. If b ≠ a, both the scalar product that we want to compute and the pairing {x_{ρ}, x_{γ_{wF}} } are zero, so we may assume b = a. By 1.3, (i), we have R_{γ_{wF}} = Ind_{H^{∘ȧF}}^{H^{ȧF}} (R_{γ_{wF}^{∘}}), where γ_{wF}^{∘} is the function defined similarly to γ_{wF} for the class of wF under the group W^{∘}. We have clearly γ_{wF} = Ind_{W^{∘}}^{W}(γ_{wF}^{∘}).

We want to compute ⟨ ρ, R_{γ_{wF}} ⟩. By Frobenius reciprocity and the preceding remarks, this is equal to ⟨ Res_{H^{∘ȧF}}^{H^{ȧF}} ρ, R_{γ_{wF}^{∘}} ⟩_{H^{∘ȧF}}. By 4.3, this scalar product is equal to \{x_{Res ρ}, x_{γ_{wF}^{∘}} \}, if x_{Res ρ} ∈ \overline{Q}_{l}\overline{M}(Γ_{F_{0}} ⊂

Γ_{F_{0}} ><<aF >) parametrizes Res_{H^{∘ȧF}}^{H^{ȧF}} ρ. So we are reduced to proving {x_{Res ρ}, x_{γ_{wF}^{∘}} } = {x_{ρ}, x_{Ind}_{γ_{wF}^{∘}} }.

With the notations of 5.3 and 5.6, we will show that in general \{Res\overline{M} \overline{m}, \underline{m}\} = \{\overline{m}, Ind\underline{M} \underline{m}\} for any \overline{m} ∈ \overline{Q}_{l}\overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >) and any \underline{m} ∈ \underline{M}(Γ_{F} ⊂ Γ_{F} ><<F >), where Res\overline{M} is the adjoint of Ind\overline{M} for the scalar product ⟨ , ⟩ on \overline{Q}_{l}\overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >) deduced from the scalar product on class functions via the isomorphism of 5.2 (i.e., Res x_{ρ} = x_{Res ρ}). For this scalar product the elements (xF, χ) form an orthonormal basis, so Mellin transforms are pairwise orthogonal and we have ⟨ (xF, y), (xF, y) ⟩ = |C_{Γ_{F}} ((xF, y))|. So ⟨ (xF, y), (x′F, y′) ⟩ = {(xF, y), (y′, x′F)}. Whence

{Res(xF, y), (x_{∘}F, y_{∘})} = ⟨ Res(xF, y), (x_{∘}F, y_{∘}) ⟩ = ⟨ (xF, y), Ind(x_{∘}F, y_{∘}) ⟩ = {(xF, y), Ind(x_{∘}F, y_{∘})};

whence the result.

II

6. Unicity of the parametrization.

The goal of this part is to give conditions which unambiguously determine the map π_{s} in the case of a group G connected and with a connected center. Our strategy is as follows:

We first determine (cf. 6.3) under which conditions 4.3 determines the map

ρ ∈ E(G^{F} , 1) → (xF, χ) ∈ ∐_{F∈Ξ^{F}} \overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >)

We need to add two conditions to define completely the map (cf. 6.4).

We then consider the general situation of E(G^{F} , s), and we have to deal with all the cases where 4.3 is not sufficient and 6.4 does not apply.

6.1 Since the families we consider depend on F, we will consider couples (W, F) where W is a Coxeter group and F a diagram automorphism of W. We say that (W, F) is irreducible if W is a product of irreducible Coxeter groups, permuted transitively by F.

With the above definition we have (W, F) = ∏_{i}(W_{i}, F_{i}) where the (W_{i}, F_{i}) are irreducible; each F ∈ Ξ(W)^{F} identifies to a product ∏_{i} F_{i} ∈ ∏_{i} Ξ(W_{i})^{F_{i}}, and we have Γ_{F} = ∏_{i} Γ_{F_{i}},

\overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >) = ∏_{i} \overline{M}(Γ_{F_{i}} ⊂ Γ_{F_{i}} ><<F_{i}>),

\underline{M}(Γ_{F} ⊂ Γ_{F} ><<F >) = ⨂_{i} \underline{M}(Γ_{F_{i}} ⊂ Γ_{F_{i}} ><<F_{i}>)

and F̃ = ⨂_{i} F̃_{i}, and the pairing between the spaces

\overline{Q}_{l}\overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >) and \underline{M}(Γ_{F} ⊂ Γ_{F} ><<F >)

is the product of the pairings between

\overline{Q}_{l}\overline{M}(Γ_{F_{i}} ⊂ Γ_{F_{i}} ><<F_{i}>) and \underline{M}(Γ_{F_{i}} ⊂ Γ_{F_{i}} ><<F_{i}>).

When (W, F) is irreducible where W = ∏_{i=1}^{r} W_{i} with W_{i} irreducible, then (cf. [11, 4.20]) Ξ(W)^{F} identifies to Ξ(W_{1})^{F^{r}}, Γ_{F} identifies to a product Γ_{F_{1}} × … × Γ_{F_{r}} permuted transitively by F and \overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >) identifies to \overline{M}(Γ_{F_{1}} ⊂ Γ_{F_{1}} ><<F^{r}>): any \overline{x} ∈ \overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >) has a representative of the form ((a, 1, …, 1)F, χ) where χ is a character of C_{Γ_{F}} ((a, 1, …, 1)F) ≃ C_{Γ_{Fr}} (a). Similarly there exists isomorphisms \underline{M}(Γ_{F_{1}} ⊂ Γ_{F_{1}} ><<F^{r}>) ≃ \underline{M}(Γ_{F} ⊂ Γ_{F} ><<F >) and F̃ ≃ F̃_{1} (relative to F^{r}), and the pairing is compatible with these isomorphisms.

Finally, when W is irreducible, for any automorphism F of W, any F with more than one element is in Ξ^{F} and is fixed pointwise by F (cf. [11, 4.19] for A_{n}, E_{6} and D_{4} (this last case with an automorphism of order 3), and for D_{n} the only characters not fixed by the diagram automorphism are those corresponding to symbols with identical parts, which are in a one-element family). So for any element F ∈ Ξ^{F}, we have \underline{M}(Γ_{F} ⊂ Γ_{F} ><<F >) ≃ \overline{Q}_{l}\overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >) ≃ \overline{Q}_{l}M(Γ_{F}) and the pairing between \underline{M}(Γ_{F} ⊂ Γ_{F} ><<F >) and \overline{Q}_{l}\overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >) has the same value as that between M(Γ_{F}) and itself, and F̃ ≃ \overline{Q}_{l}F.

6.2 So, in general, for any F ∈ Ξ(W)^{F} there is a set {F_{i} ∈ Ξ(W_{i})^{F_{i}}\}_{i} where W_{i} is irreducible, such that \overline{Q}_{l}\overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >) and M(Γ_{F} ⊂ Γ_{F} ><<F >) both identify to a product ⨂_{i} M(Γ_{F_{i}}), compatibly with the pairing, and F̃ identifies to ⨂_{i} \overline{Q}_{l}F_{i}. This will allow us to reduce the problem to the case of an irreducible W.

Proposition 6.3. Suppose that no component F_{i} of F is exceptional (cf. 4.4), and that \overline{x} ∈ \overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >) has no component x_{j} of the following types (using Lusztig’s [11, chap. 4] notations):

x_{j} = (g_{4}, i) or (g_{4}, −i) (Γ_{F_{j}} isomorphic to S_{4}).

x_{j} = (g_{3}, θ) or (g_{3}, θ^{2}) (Γ_{F_{j}} isomorphic to S_{3} or S_{4}).

Then the set of coefficients \{\overline{x}, xẼ\}, when Ẽ runs over F̃, determine the element \overline{x} ∈ \overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >).

Proof: In the identification of 6.2, we have Ẽ = (E_{1}, …, E_{r}) where E_{i} ∈ F_{i}, \overline{x} = (x_{1}, …, x_{r}) with x_{i} ∈ M(Γ_{F_{i}}) and \{xẼ, \overline{x}\} = ∏_{i}\{x_{E_{i}}, x_{i}\}. Let V (x_{i}) be the vector {x_{E_{i}}, x_{i}}_{E_{i}∈F_{i}}; to show the proposition we have to prove that the vectors V (\overline{x}) = V (x_{1}) ⊗ … ⊗ V (x_{r}) are all distinct when \overline{x} runs over \overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >) (unless an x_{i} is one of the exceptions of the proposition). If this were not true, then there would exist x_{i} ≠ x′_{i} and λ ∈ C^{×} such that V (x_{i}) = λV (x′_{i}).

We can do a case-by-case check that this is impossible for an irreducible W: when Γ_{F} ≃ S_{3}, S_{4} or S_{5} it is an easy check on the table of coefficients. In the case where Γ_{F} ≃ (ZZ/2ZZ)^{d} we first note that {x_{E}, x} are equal for all x when E is the special character in the family. So any λ as above has to be 1, and it is enough to check that the vector {x_{E}, x}_{E∈F} determines x.

The types of W to consider are B_{n}, D_{n} and ^{2}D_{n}.

• Case B_{n}. With Lusztig’s [11] notations, x is parametrized by a subset X of even cardinality of a set Z_{1} with 2d + 1 elements in which a fixed subset M_{0} of cardinal d has been distinguished. The elements x_{E} correspond to subsets Y ⊂ Z_{1} such that |Y ΔM_{0}| = d, and with these parametrizations, we have {x, x_{E}} = (−1)^{|X∩Y |}. Given a ∈ Z_{1} we can find two subsets S_{1} and S_{2} of cardinal d such that S_{1} ∪ S_{2} = Z_{1} − {a}. If we put Y_{i} = S_{i}ΔM_{0} (for i = 1, 2), then Y_{i} corresponds to some x_{E_{i}} and:

(−1)^{|X∩Y_{1}|+|X∩Y_{2}|} = 1 ⟺ a ∉ X

so X is clearly determined by the {x, x_{E_{i}}}.

• Case D_{n} or ^{2}D_{n}. This time x corresponds to a subset X of odd cardinality of a set Z_{1} with 2d elements, taken modulo the equivalence relation X ∼ Z_{1} − X, and elements x_{E} correspond to subsets Y of even cardinality of Z_{1} taken modulo the same equivalence relation and such that |Y ΔM_{0}| = d where M_{0} is a fixed subset of cardinality d; and we have:

{x, x_{E}} = (−1)^{|X∩Y |}.

Given two elements a, b ∈ Z_{1} there exists two subsets S_{1} and S_{2} of cardinal d such that S_{1}ΔS_{2} = {a, b}; if we set Y_{i} = S_{i}ΔM_{0} (for i = 1, 2) then:

(−1)^{|X∩Y_{1}|}(−1)^{|X∩Y_{2}|} = (−1)^{|X∩{a,b}|}

Modulo the equivalence X ∼ Z_{1} − X we may assume that a ∉ X, and then the above equality clearly determines X. This concludes the proof of proposition 6.3. We now give sufficient conditions for Lusztig’s parametrization ρ ↦

(xF, χ) to be uniquely determined in the case of unipotent characters.

For \overline{x} ∈ \overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >), we define a root of unity λ(\overline{x}) as follows: if \overline{x} identifies to (x_{1}, …, x_{r}) (cf. 6.2) then λ(\overline{x}) = λ(x_{1}) … λ(x_{r}); and if x_{1} = (a, χ) where a ∈ Γ_{F_{1}} and χ is a character of C_{Γ_{F_{1}}} (a), we put λ(x_{1}) = χ(a)/χ(1) unless F_{1} is exceptional and (x_{1}, χ) is not in the image of F_{1} by the embedding F_{1} ↪ M(Γ_{F}), when we put λ(x_{1}) = iχ(a)/χ(1).

Remark. In the next proposition we do not know a reference for the proof of (ii) when (W, F) is of type ^{2}D_{n} though we believe it is true. The same restriction will consequently apply in (ii) of theorem 7.1 about the unicity of Lusztig’s parametrization, but we will not use that part of the hypotheses in the proof of the unicity.

Proposition 6.4. Suppose (W, F) irreducible. Then, for F ∈ Ξ(W)^{F}, there is a unique bijection

ρ ↦ \overline{x}_{ρ} : E(G^{F} , 1, F) → \overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >)

such that:

(i) For any Ẽ ∈ F̃ and ρ ∈ E(G^{F} , 1, F) we have

⟨ρ, RẼ⟩ = Δ(\overline{x}_{ρ})\{\overline{x}_{ρ}, xẼ\}.

(ii) If F^{δ} is the smallest split power of F then the eigenvalues of F^{δ} associated to ρ ∈ E(G^{F} , 1, F) are equal, up to a power of q^{δ/2} to λ(\overline{x}_{ρ}).

(iii) The character ρ of the principal series associated (via Lusztig’s explicit isomorphism of W with the Hecke H(G, B)) to the special character of F is such that \overline{x}_{ρ} = (F, Id) ∈ \overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >).

Proof: Lusztig has shown the existence of a bijection compatible with (i)–(iii): indeed (i) is [11, 4.23]; to check that we may have a bijection satisfying (ii), if W = ∏_{i=1}^{r} W_{i} where the W_{i} are irreducible and cyclically permuted by F, the elements w = (w_{1}, 1, .., 1) span the F -classes of W and we have an isomorphism of the Deligne-Lusztig varieties: X_{w,G,F} ≃ X_{w_{1},G_{1},F^{r}}; moreover, if F^{δ} is split the eigenvalues of F^{δ} on H_{c}^{∗}(X_{w,G,F}) are equal to the eigenvalues of F^{rδ} on H_{c}^{∗}(X_{w_{1},G_{1},F^{r}}). So we may assume that W is irreducible. For irreducible groups, (ii) for untwisted groups is [11, 11.2], and for groups of type ^{2}E_{6} and ^{3}D_{4} can be deduced from [9, 7.3] (we can restrict ourselves to the case where ρ is cuspidal, and in these cases all cuspidal unipotent representations occur in H^{∗}(X_{w}, \overline{Q}_{l}) where w is the Coxeter element of W).

We note that at this stage we have already resolved all ambiguities left by proposition 6.3 (whence the unicity in the proposition) except for exceptional families, since the λ(x_{i}) differ in the listed cases.

Checking compatibility with (iii) can also be reduced to the case W irreducible, where it is a property of the embedding F ⊂ M(Γ_{F}) which can be easily checked case by case, except that in the case of exceptional families there is some ambiguity on the embedding that (iii) is precisely designed to resolve.

Proposition 6.5. Assume that (W, F) is irreducible and that we are given an automorphism φ of W ><<F > which stabilizes W, fixes F and induces a diagram automorphism of W. Then for any F ∈ Ξ^{F} we have either |F| = 1 or φ fixes F̃ pointwise.

Proof: Assume first that W is irreducible. If F has more than one element, then it is pointwise fixed by F and φ and the condition φ(F) = F ensures that an extension in F̃ of an element of F will be φ-invariant.

Suppose now that W = W_{1} × … × W_{r} permuted cyclically by F. Since φ and F commute, there exists i such that φF^{−i} stabilizes W_{j} for any j. Recall that we have an isomorphism: (F̃_{1}, W_{1}, F^{r}) ≃ (F̃, W, F). Write φ = φF^{−i}.F^{i}; φF^{−i} stabilizes all W_{j} and is clearly compatible with the above isomorphism, and from the first part it fixes pointwise F̃_{1} and so its image. We conclude by observing that F^{i} acts trivially on elements of F̃ since they are class functions on W ><<F >.

Proposition 6.6. Assume that (W, F) is irreducible and let φ be an isogeny of G commuting with F. Then for any F ∈ Ξ^{F} the set E(G^{F} , 1, F) is pointwise fixed by φ if it has more than one element.

Proof: Let ρ ∈ E(G^{F} , 1, F). For any Ẽ ∈ F̃ we have:

⟨ ρ, RẼ ⟩ = ⟨ ρ ∘ φ, RẼ ∘ φ ⟩ = ⟨ ρ ∘ φ, R_{Ẽ∘φ} ⟩

(the last equality by corollary 9.2). This means that:

\{\overline{x}_{ρ}, xẼ\} = \{\overline{x}_{ρ∘φ}, x_{Ẽ∘φ}\} = \{\overline{x}_{ρ∘φ}, xẼ\}

(the last equality since Ẽ = Ẽ ∘ φ by proposition 6.5). By proposition 6.3 this gives ρ = ρ ∘ φ unless F is one of the exceptions of the proposition. To deal with the remaining cases, note that φ induces an isomorphism: H_{c}^{∗}(X_{w}, \overline{Q}_{l})⟶^{∼}H_{c}^{∗}(X_{φ(w)}, \overline{Q}_{l}) which commutes with F, so the eigenvalues of F associated to ρ are equal to those associated to ρ ∘ φ; this gives the result by proposition 6.4 (ii) except if F is exceptional. In this last case we use the fact that a diagram isomorphism is compatible with Lusztig’s isomorphism from W to H(G, B).

7. Unicity of Lusztig’s parametrization.

We now study Lusztig’s parametrization of non-unipotent characters. If we fix s ∈ T^{∗} such that Z_{F} (s) is non empty, Lusztig’s parametrization of characters in E(G^{F} , s) may be precised as follows.

Theorem 7.1. Given s, there exists a unique bijection:

π_{s} : E(G^{F} , s)⟶^{∼}E(C_{G^{∗}}(s)^{(ẇ_{1}F)^{∗}}, 1)

satisfying the following conditions:

(i) For any χ ∈ E(G^{F} , s) and any w ∈ Z_{F} (s) we have:

⟨χ, R_{T^{wF}}^{G^{F}} (ŝ)⟩ = ⟨π_{s}(χ), (−1)^{l(w_{1})}R_{T^{∗(wF)^{∗}}}^{C_{G^{∗}} (s)^{(ẇ_{1}F)^{∗}}} (1)⟩.

(ii) If s = 1 then:

(a) The eigenvalues of F^{δ} associated to χ are equal, up to a power of q^{δ/2} to the eigenvalues of F^{∗δ} associated to π_{1}(χ).

(b) If χ is in the principal series then π_{1}(χ) and χ correspond to

the same character of the Hecke algebra.

(iii) If ζ ∈ G^{∗F^{∗}} is central and χ ∈ E(G^{F} , s) then π_{sζ}(χ ⊗ ζ̂) = π_{s}(χ), where ζ̂ is the character of G^{F} corresponding to ζ (see for instance [6, 6.7 and 6.8]).

(iv) If L is a standard Levi subgroup of G such that L^{∗} contains C_{G^{∗}}(s) and such that L is ẇF -stable (where ẇ is a representative of the element w reduced with respect to W_{L} in the class of w_{1}); then the following diagram is commutative:

E(G^{F} , s) ⟶^{π_{s}} E(C_{G^{∗}}(s)^{(ẇ_{1}F)^{∗}}, 1)
↑^{R_{L^{ẇF}}^{G^{F}}}  ∥
E(L^{ẇF} , s) ⟶^{π_{s}} E(C_{L^{∗}}(s)^{(v̇_{1}ẇF)^{∗}}, 1)

(where v̇_{1} is defined by v̇_{1}ẇ = ẇ_{1}).

(v) Assume that (W, F) is irreducible, that (G, F) is of type E_{8} and that (C_{G^{∗}}(s), (ẇ_{1}F)^{∗}) is of type E_{7}×A_{1} (resp. E_{6}×A_{2}, resp. ^{2}E_{6}×^{2}A_{2}) then we have the following commutative diagram:

C(G^{F} , s) ⟶^{π_{s}} C(C_{G^{∗}}(s)^{(ẇ_{1}F)^{∗}}, 1)
↑^{R_{L^{ẇ_{2}F}}^{G^{F}}}  ↑_{R_{L^{∗(ẇ_{2}F)^{∗}}}^{C_{G^{∗}} (s)^{(ẇ_{1}F)^{∗}}}}
E(L^{ẇ_{2}F} , s)^{•} ⟶^{π_{s}} E(L^{∗(ẇ_{2}F)^{∗}}, 1)^{•}

with L a Levi subgroup of G of type E_{7} (resp. E_{6}, resp. E_{6}) containing the corresponding component of C_{G^{∗}}(s), where the superscript • means the cuspidal part of the series, and where w_{2} = 1 (resp. 1, resp. the W_{L}-reduced element of Z_{F} (s) which is in a parabolic subgroup of type E_{7} of W).

(vi) Given an epimorphism with kernel a central torus:

(G, F)⟶^{φ}(G_{1}, F_{1})

and elements s_{1} ∈ G_{1}^{∗} and s = φ^{∗}(s_{1}) ∈ G^{∗}, then the following diagram is commutative:

E(G^{F} , s) ⟶^{π_{s}} E(C_{G^{∗}}(s)^{(φ(ẇ_{1})F)^{∗}}, 1)
↑^{^{t}φ}  ↓^{^{t}φ}^{∗}
E(G_{1}^{F_{1}}, s_{1}) ⟶^{π_{s_{1}}} E(C_{G_{1}^{∗}}(s_{1})^{(ẇ_{1}F_{1})^{∗}}, 1)

(vii) If G is a direct product ∏ G_{i}, then π_{∏ s_{i}} = ∏ π_{s_{i}}.

Proof: We have to show that conditions (i)–(vi) completely specify the map π_{s}, and that they are compatible with each other. Our strategy will be to consider each condition in turn, and each time to show that (together with the previous conditions) it specifies π_{s} for a larger set of situations, and the previous conditions hold for the new situations (as well as the new condition for the previous situations).

We first suppose that (W, F) is irreducible and that s = 1, and remark that by applying 6.4 in G and G^{∗}, we obtain that (i) and (ii) completely specify π_{s} in this case (and are compatible).

If now s is central, (iii) clearly specifies π_{s} from its value when s = 1. (ii) is not relevant unless s = 1, and (i) holds, since R_{T^{wF}}^{G^{F}} (ŝ⊗ζ̂) = R_{T^{wF}}^{G^{F}} (ŝ) ⊗ ζ̂ (cf. [3, 1.67]).

Suppose now that C_{G^{∗}}(s) is contained in a proper Levi subgroup L^{∗} of G^{∗}; then (iv) specifies π_{s} completely in G if π_{s} is uniquely defined in L, which we may assume by induction on the rank of G (we assume that the whole of 4.1 holds for groups of smaller rank); indeed in this case by [11, 6.21], R_{L^{ẇF}}^{G^{F}} is an isometry between E(L^{ẇF} , s) and E(G^{F} , s) which is such that (i) is compatible with (iv); (ii) is not relevant here, and (ii) holds since R_{L^{ẇF}}^{G^{F}} (χ ⊗ ζ̂) = R_{L^{ẇF}}^{G^{F}} (χ) ⊗ ζ̂ (cf. par exemple [6, 3.8]).

When (W, F) is irreducible, it only remains the case when s is exceptional (i.e., , C_{G^{∗}}(s) is not included in any proper Levi subgroup, but s is not central). Suppose first that W has no family which contains one of the exceptions of proposition 6.3; then (i) completely specifies π_{s}; (ii) and (iv) are not relevant here, and (iii) holds again because of R_{T^{wF}}^{G^{F}} (ŝ ⊗ ζ̂) = R_{T^{wF}}^{G^{F}} (ŝ) ⊗ ζ̂.

From the explicit knowledge of root subsystems of maximal rank in root systems (which parameterize centralizers of semi-simple elements in G^{∗}), we see that W (s) has a family which contains one of the exceptions in proposition 6.3 only if G is of type E_{8} and (C_{G^{∗}}(s), (ẇ_{1}F)^{∗}) is of type E_{7} × A_{1}, E_{6} × A_{2} or ^{2}E_{6} × ^{2}A_{2} (we assume that W is irreducible; it is easy to extend the arguments below to the case (W, F) irreducible by “descent of scalars”). Furthermore, in the above cases, the exceptions to proposition 6.3 are each time two characters occuring in a family where Γ_{F} ≃ S_{3} and coincide with the list of cuspidal unipotent characters of L^{ẇ_{2}F} (excepted that in the case ^{2}E_{6} the character parametrized by (1, 1) ∈ M(S_{3}) is also cuspidal). We will show that in this case, (v) can be assumed to hold, is compatible with (i)-(iv), and together with them completely specifies π_{s}.

We first prove:

Lemma 7.2. Suppose that s is central in L^{∗}, a Levi subgroup of G^{∗}, and that W (s) has a w_{1}F -stable direct product decomposition W (s) = W_{L}×W′. Let w_{2} ∈ W_{F} (s), let ρ ∈ E(L^{ẇ_{2}F} , 1), and let Ẽ be a w_{2}F -class function on W; write Ẽ = Ẽ_{L} ⊗ Ẽ′ where Ẽ_{L} is a w_{2}F -class function on W_{L} and Ẽ′ a w_{2}F -class function on W′. Then:

⟨ RẼ(s), R_{L^{ẇ_{2}F}}^{G^{F}} (ρ ⊗ ŝ) ⟩_{G^{F}} = Ẽ′(w_{2}F)⟨ RẼ_{L}, ρ ⟩_{L^{ẇ_{2}F}}

Proof: We first compute the scalar product ⟨R_{T^{ww_{2}F}}^{G^{F}} (s), R_{L^{ẇ_{2}F}}^{G^{F}} (ρ⊗ŝ)⟩ for w ∈ W (s). By Mackey’s formula [6, 2.1(b)] it is equal to:

∑_{L^{ẇ_{2}F} \{v∈G^{F} |^{v}T⊂L}/T^{ww_{2}F}} ⟨ R_{^{v}(T^{ww_{2}F})}^{L^{ẇ_{2}F}}(^{v}s), ρ ⊗ ŝ⟩_{L^{ẇ_{2}F}} =

|W_{L}|^{−1} ∑_{{v∈W_{G} \atop vw^{w_{2}F}v^{−1}∈W_{L}}} ⟨ R_{T^{vww_{2}F v^{−1}}}^{L^{ẇ_{2}F}} (^{v}s), ρ ⊗ ŝ⟩_{L^{ẇ_{2}F}}

Since ρ ⊗ ŝ ∈ E(L^{ẇ_{2}F} , s), the only non-zero summands are those where vs and s are conjugate in L^{∗}, i.e. equal since s is central in L^{∗}, and the above sum reduces to:

|W_{L}|^{−1} ∑_{{v∈W (s) \atop vw^{w_{2}F}v^{−1}∈W_{L}}} ⟨ R_{T^{vww_{2}F v^{−1}}}^{L^{ẇ_{2}F}} (1), ρ ⟩_{L^{ẇ_{2}F}} .

Write w = w′w_{L} where w′ ∈ W′ and w_{L} ∈ W_{L}. Then vw^{w_{2}F}v^{−1} is w_{2}F -conjugate to w_{L} in W_{L}, so R_{T^{vww_{2}F v^{−1}}}^{L^{ẇ_{2}F}} (1) = R_{T^{w_{L}w_{2}F}}^{L^{ẇ_{2}F}} (1) and the condition under the sum is that w′ is w_{2}F -conjugate to 1 in W′. So we get for the sum:

|W^{′w_{2}F} |⟨ R_{T^{w_{L}w_{2}F}}^{L^{ẇ_{2}F}} (1), ρ ⟩_{L^{ẇ_{2}F}} .

From that we get:

⟨ RẼ(s), R_{L^{ẇ_{2}F}}^{G^{F}} (ρ ⊗ ŝ) ⟩_{G^{F}}

= |W (s)|^{−1} ∑_{w∈W (s)} Ẽ(ww_{2}F)⟨ R_{T^{ww_{2}F}}^{G^{F}} (ŝ), R_{L^{ẇ_{2}F}}^{G^{F}} (ρ ⊗ ŝ) ⟩_{G^{F}}

= |W (s)|^{−1}|W_{L}|^{−1} ∑_{{w∈W (s),w_{L}∈W_{L}|w ∼^{w_{2}F}w_{L}∈W_{L}}} Ẽ(w_{L}w_{2}F)

|W^{′w_{2}F} |⟨ R_{T^{w_{L}w_{2}F}}^{L^{ẇ_{2}F}} (1), ρ ⟩_{L^{ẇ_{2}F}}

= |W_{L}|^{−1}Ẽ′(w_{2}F) ∑_{w_{L}∈W_{L}} Ẽ_{L}(w_{L}w_{2}F)⟨ R_{T^{w_{L}w_{2}F}}^{L^{ẇ_{2}F}} (1), ρ ⟩_{L^{ẇ_{2}F}}

= Ẽ′(w_{2}F)⟨ RẼ_{L}, ρ ⟩_{L^{ẇ_{2}F}}

By the general structure theorem for reductive groups, in the cases we consider we may write a (w_{1}F)^{∗}-stable decomposition of C_{G^{∗}}(s) as L^{∗} ×_{T} H where H is a reductive group of type A_{n} with Weyl group W′ and T is a maximal torus of H central in L^{∗}. Let ρ be a unipotent representation of L^{∗(ẇ_{2}F)^{∗}}. Since ρ is trivial on the center of L^{∗}, it factors through a representation of the form ρ′⊗Id of L_{ad}^{∗(ẇ_{2}F)^{∗}} ×T_{ad}^{(ẇ_{2}F)^{∗}} and we have:

R_{L_{ad}^{∗(ẇ_{2}F)^{∗}} ×T_{ad}^{(ẇ_{2}F)^{∗}}}^{C_{G^{∗}} (s)_{ad}^{(ẇ_{1}F)^{∗}}} ρ′ ⊗ Id = ρ′ ⊗ R_{T_{ad}^{(ẇ_{2}F)^{∗}}}^{H_{ad}^{(ẇ_{2}F)^{∗}}} Id =

∑_{E′∈Irr(W′)^{w_{2}F}} Ẽ′(w_{2}F)ρ′ ⊗ ρ_{E′}

where ρ_{E′} is the irreducible unipotent character of H parameterized by E′ (when H is split we take Ẽ′(w_{1}F) = 1 and when H is a unitary group the extension Ẽ′ is determined by the above formula).

Since H is of type A_{n}, the families of W′ are reduced to single characters, and the families in Ξ(W (s))^{w_{1}F} are of the form F ⊗ E′ with F ∈ Ξ(W_{L})^{w_{1}F} and E′ ∈ Irr(W′)^{w_{1}F}, and there is a bijection that we will denote x ↦ x ⊗ E′ : \overline{M}(Γ_{F} ⊂ Γ_{F} ><<w_{1}F >) → \overline{M}(Γ_{F⊗E′} ⊂ Γ_{F⊗E′} ><<w_{1}F >). If we denote by ρ′_{x} the unipotent character in the set E(L_{ad}^{∗(ẇ_{2}F)^{∗}}, 1, F) corresponding to x ∈ \overline{M}(Γ_{F} ⊂ Γ_{F} ><<w_{1}F >) we get:

R_{L_{ad}^{∗(ẇ_{2}F)^{∗}} ×T_{ad}^{(ẇ_{2}F)^{∗}}}^{C_{G^{∗}} (s)_{ad}^{(ẇ_{1}F)^{∗}}} ρ′_{x} ⊗ Id = ∑_{E′∈Irr(W′)^{w_{2}F}} Ẽ′(w_{2}F)ρ′_{x⊗E′}

And using (vi) of the theorem (which holds in C_{G^{∗}}(s) by induction) we get the analogous result in C_{G^{∗}}(s)^{(ẇ_{1}F)^{∗}}:

R_{L^{∗(ẇ_{2}F)^{∗}}}^{C_{G^{∗}} (s)^{(ẇ_{1}F)^{∗}}} (ρ_{x}) = ∑_{E′∈Irr(W′)^{w_{1}F}} Ẽ′(w_{2}F)ρ_{x⊗E′}

To prove (v) we will show that when ρ_{x} is cuspidal, R_{L^{ẇ_{2}F}}^{G^{F}} (ρ_{x}⊗ŝ) is of the form ∑_{E′∈Irr(W′)^{w_{1}F}} Ẽ′(w_{2}F)ρ_{x⊗E′}^{s} where ρ_{x⊗E′}^{s} ∈ E(G^{F} , s, F ⊗ E′) and

⟨ RẼ(s), ρ_{x⊗E′}^{s} ⟩_{G^{F}} = (−1)^{l(w_{1})}⟨ RẼ, ρ_{x⊗E′} ⟩_{C_{G^{∗}} (s)}_{(ẇ_{1}F)^{∗}}

for any E ∈ (F ⊗ E′)˜ (then if we define π_{s} by π_{s}(ρ_{x⊗E′}^{s}) = ρ_{x⊗E′} (i) and (v) are satisfied; (iii) is clearly satisfied also and (ii) and (iv) do not apply here, so we will have finished with the case (W, F) irreducible).

In the cases where (C_{G^{∗}}(s), (ẇ_{1}F)^{∗}) is of type E_{6} × A_{2} or E_{7} × A_{1}, we have w_{1} = 1 and L is a Levi subgroup of a rational parabolic subgroup of G so, by Howlett-Lehrer’s theory, we know that the decomposition of R_{L^{F}}^{G^{F}} (ρ⊗ŝ) for a cuspidal ρ ∈ E(L^{∗F^{∗}}, 1) (which corresponds to a cuspidal ρ ⊗ ŝ ∈ E(L^{F} , s)) is of the form ∑_{χ∈Irr(W (ρ))} χ(1)ρ_{χ}. When C_{G^{∗}}(s) is of type E_{6} × A_{2}, W (ρ) is equal to W′ since N_{W} (W_{L}) = W (s) ><<w> where w is an element of order 2 acting on L by the nontrivial diagram automorphism of E_{6} and w cannot fix ρ since it maps s to s^{−1}. Similarly W (ρ) is W′ if C_{G^{∗}}(s) is of type E_{7} × A_{1} since in that case N_{W} (W_{L}) = W (s). So we know that R_{L^{F}}^{G^{F}} (ρ ⊗ ŝ) has a decomposition of the form ∑_{E′∈Irr(W′)} E′(1)ρ_{x⊗E′}^{s}; it remains to show that ⟨ RẼ(s), ρ_{x⊗E′}^{s} ⟩_{G^{F}} = ⟨ RẼ, ρ_{x⊗E′} ⟩_{C_{G^{∗}} (s)}_{F^{∗}} (= ⟨ RẼ_{L}, ρ_{x} ⟩_{L^{∗F^{∗}}}).

Consider the projection ρ^{E′} of R_{L^{F}}^{G^{F}} (ρ_{x} ⊗ ŝ) on \overline{Q}_{l}E(G^{F} , s, F ⊗ E′); by lemma 7.2, we have ⟨ ρ^{E′}, RẼ(s) ⟩_{G^{F}} = E′(1)⟨ ρ_{x}, RẼ_{L} ⟩_{L^{F}} so ρ^{E′} is not 0; furthermore, since R_{L^{F}}^{G^{F}} (ρ_{x} ⊗ ŝ) has | Irr(W′)| isotypic components, ρ^{E′} must be one of them. Looking at the table of Fourier coefficients for M(S_{3}) which gives us ⟨ ρ_{x}, RẼ_{L} ⟩_{L^{F}} we see that |ρ^{E′}| ≥ 3/4E′(1) which implies |ρ^{E′}| ≥ E′(1) since both are integers and E′(1) ≤ 2; but ∑_{E′} |ρ^{E′}|^{2} = |R_{L^{F}}^{G^{F}} (ρ_{x} ⊗ ŝ)|^{2} = ∑_{E′} E′(1)^{2} so ρ^{E′} must be of the form E′(1)ρ_{x⊗E′}^{s} with ρ_{x⊗E′}^{s} irreducible.

Similarly, in the case where (C_{G^{∗}}(s), (ẇ_{1}F)^{∗}) is of type ^{2}E_{6} ×^{2}A_{2}, it is enough to show that

⟨ R_{L^{ẇ_{2}F}}^{G^{F}} (ρ_{x} ⊗ ŝ), R_{L^{ẇ_{2}F}}^{G^{F}} (ρ_{x} ⊗ ŝ) ⟩_{G^{F}} = 2

(then for each of the two elements E′ ∈ Irr(W′)^{w_{2}F} we can similarly deduce by using 7.2 and the table of fourier coefficients for M(S_{3}) that the projection ρ^{E′} of R_{L^{ẇ_{2}F}}^{G^{F}} (ρ_{x} ⊗ ŝ) on \overline{Q}_{l}E(G^{F} , s, F ⊗ E′) is an irreducible character with sign Ẽ′(w_{2}F)).

We have w_{2} = w_{0}^{L}w_{0}^{M} where w_{0}^{L} is the longest element of W_{L} and w_{0}^{M} is the longest element of the standard Levi subgroup M of G of type E_{7} which contains L. We will consider R_{L^{ẇ_{2}F}}^{M^{F}} (ρ_{x} ⊗ ŝ), which is an irreducible representation since C_{M^{∗}}(s) = C_{L^{∗}}(s) (in that case R_{L^{ẇ_{2}F}}^{M^{F}} is an isometry, cf. [10, proof of 7.9]) and we will conclude by proving that for any χ ∈ E(M^{F} , s), we have ⟨ R_{M^{F}}^{G^{F}} χ, R_{M^{F}}^{G^{F}} χ ⟩_{G^{F}} = 2. Indeed, ⟨ R_{M^{F}}^{G^{F}} χ, R_{M^{F}}^{G^{F}} χ ⟩_{G^{F}} = ⟨ χ,^{∗}R_{M^{F}}^{G^{F}} R_{M^{F}}^{G^{F}} χ ⟩_{M^{F}} and by Mackey’s formula (which we may apply since M is included in a rational parabolic subgroup):

∗R_{M^{F}}^{G^{F}} R_{M^{F}}^{G^{F}} χ = ∑_{w∈W_{M}\W/W_{M}} R_{(^{w^{−1}}M∩M)^{F}}^{M^{F}} ∘ ad w^{−1} ∘ ^{∗}R_{(M∩^{w}M)^{F}}^{M^{F}} χ

Here is a list of reduced representatives w ∈ W_{M}\W/W_{M}, and, below each, the value of M ∩ ^{w}M:

w = 1 w_{0}^{M}w_{0}^{G} s_{8} w_{0}^{G}w_{0}^{M}s_{8}w_{0}^{M}w_{0}^{L} w_{0}^{G}w_{0}^{M}s_{8}w_{0}^{M}w_{0}^{D}
M ∩ ^{w}M = M  M  L  L  D

where s_{8} is the generating reflection in W −W_{M} and where D is the standard Levi subgroup of G of type D_{6}. Since w_{0}^{M}w_{0}^{G} acts like the identity on M, the first two values of w give a contribution of 2χ to ^{∗}R_{M^{F}}^{G^{F}} R_{M^{F}}^{G^{F}} χ. We have to prove that the other terms are 0; for ^{∗}R_{(M∩^{w}M)^{F}}^{M^{F}} χ to be non-zero, s must have a W_{M}-conjugate s′ ∈ (M∩^{w}M) such that Z_{F} (s′) meets W_{M} ∩ ^{w}W_{M}. There is no conjugate of s in D, so ^{∗}R_{D^{F}}^{M^{F}} χ = 0. Now, Z_{F} (s) ∩ W_{M} = W_{L}w_{0}^{M}; when M ∩^{w}M = L we must find v ∈ W_{M} such that W_{L} ∩ ^{v}(W_{L}w_{0}^{M}) ≠ ∅ ⟺ W_{L}vW_{L} = W_{L}vW_{L}w_{0}^{M} but it can be checked that there is no element of W_{L}\W_{M}/W_{L} invariant by multiplication by w_{0}^{M}.

We now consider the general case of G connected reductive with connected center and use (vi) and (vii).

First we check that (vi) and (vii) hold in the previous situations: (vii) is irrelevant in the case (W, F) irreducible; (vi) holds when (i) specifies π_{s} by 6.3; (vi) holds when (ii) specifies π_{s} since φ induces an isomorphism of the Deligne-Lusztig varietes X_{w} for G et G_{1}, and of the Hecke algebras; when (iii) specifies π_{s}, (vi) clearly holds since ζ̂ factors through G_{1}; and when (iv) or (v) specify π_{s}, (vi) holds since φ commutes with a Lusztig functor R_{L}^{G} (cf. 9.2).

Now, in general, we can find an epimorphism with kernel a central torus ∏ G_{i} → G, where each G_{i} is a connected reductive algebraic group with a connected center, with a Weyl group W_{i} such that (W_{i}, F) is irreducible; so applying (vi) and (vii) π_{s} is completely determined from its value in the case (W_{i}, F) irreducible. We next show that the π_{s} thus constructed does not depend on the particular choice of ∏ G_{i} → G. If we have another such morphism ∏ H_{i} → G, then H_{i} and G_{i} have the same derived group and we can embed both in some K_{i} with still the same derived group such that the maps ∏ G_{i} → G and ∏ H_{i} → G factor through ∏ K_{i}. It is enough to show that the parametrization given through ∏ G_{i} factors through ∏ K_{i}, and that comes from the unicity in the case (W, F) irreducible and the fact that (vi) holds in that case so the embedding G_{i} → K_{i} is compatible with π_{s_{i}}.

The last thing we have to check is that the parametrization π_{s} defined above satisfies (i) to (v). It is clear for the parametrization in ∏ G_{i}, and the remarks above proving the compatibility of (vi) with (i)- (v) in the case (W, F) ireducible can be applied here too to prove that (i)-(v) are preserved via ∏ G_{i} → G.

Corollary 7.3. Let w ∈ W be an element such that ^{w}Φ_{s}^{+} ⊂ Φ^{+}.

Then the following diagram is commutative:

E(G^{F} , s) ⟶^{π_{s}}  E(C_{G^{∗}}(s)^{(ẇ_{1}F)^{∗}}, 1)
∥  ↓_{ad} _{}_{˙w^{∗}}
E(G^{F} ,^{w}s) ⟶^{π_{^{w}s}} E(C_{G^{∗}}(^{w}s)^{(ẇẇ_{1}F ˙w^{−1})^{∗}}, 1)

Proof: Note first that this corollary is a special case of 7.1(vi) when w ∈ W^{F} (taking then φ = ad ẇ^{∗}; the proof will be similar to that of 7.1(vi). We have to check that ad ẇ ∘ π_{s} satisfies (i)–(vii) of 7.1 applied replacing s by ^{w}s.

(i) results from 9.2 applied in G^{∗} with φ = ad ẇ^{∗} which gives

R_{T^{(vw_{1}F)^{∗}}}^{C_{G^{∗}} (s)^{(ẇ_{1}F)^{∗}}} ∘ ad ẇ(1) = R_{T^{(wvw_{1}F w^{−1})^{∗}}}^{C_{G^{∗}} (^{w}s)^{(ẇẇ_{1}F ˙w^{−1})^{∗}}} (1)

for any v ∈ W (s) and from R_{T^{wvw_{1}F w^{−1}}}^{G^{F}} (^{w}s) = R_{T^{vw_{1}F}}^{G^{F}} (ŝ).

(ii) and (iii) are trivial here (if s is central then w = 1).

To prove that (iv) holds we have to prove that the following diagram is commutative:

E(G^{F} ,^{w}s)  ⟶^{ad ˙w^{∗}∘π_{s}} E(C_{G^{∗}}(^{w}s)^{(ẇẇ_{1}F ˙w^{−1})^{∗}}, 1)
↑^{R_{^{w}L^{wvF w^{−1}}}^{G^{F}}}  ∥
E(ẇL^{wvF w^{−1}},^{w}s) ⟶^{π_{^{w^{∗}}s}} E(C_{ẇL}^{∗}(^{w}s)^{(ẇẇ_{1}F ˙w^{−1})^{∗}}, 1)

where L is a standard, vF -stable Levi subgroup of G such that L^{∗} ⊃ C_{G^{∗}}(s) and where v is the W_{L}-reduced element in W_{L}w_{1}. But since R_{L^{v̇F}}^{G^{F}} ∘ ad ẇ^{−1} = R_{^{w}L^{wvF w^{−1}}}^{G^{F}} the diagram above is the composed of the diagram given by 7.1(iv) (with v for w) and of the following diagram given by 7.1(vi) with φ = ad ẇ:

E(L^{v̇F} , s)  ⟶^{π_{s}}  E(C_{L^{∗}}(s)^{(ẇ_{1}F)^{∗}}, 1)
↑_{ad} _{}_{˙w^{−1}}  ↓_{ad} _{}_{˙w^{−1}}
E(^{w}L^{ẇv̇F ˙w^{−1}},^{w}s) ⟶^{π_{^{w}s}} E(C_{^{w}L}^{∗}(^{w}s)^{(ẇẇ_{1}F ˙w^{−1})^{∗}}, 1)

The proof for (v) is entirely similar; and (vi) and (vii) are quite obvious.

III

The aim of this part is to give a general conjecture about the action of Shintani descent on the parametrizations of parts I and II and to prove some results in the direction of that conjecture. We begin by recalling some facts about Shintani descent and the definition of “F - twisted induction” (cf. [4], [5] and [7]).

8. Shintani descent, F′-twisted induction. Let G be a connected reductive algebraic group defined over an algebraically closed field \overline{IF}_{q}. We consider two rational structures on G defined over some finite subfields of \overline{IF}_{q} and given by two Frobenius endomorphisms F and F′ which commute. In this background, we recall the definition of Shintani descent.

8.1 Definition.

(i) We denote by N_{F′/F} the map from F -classes on G^{F′} to F′-classes on G^{F} which maps the F -class of g ∈ G^{F′} to the F′-class in G^{F} of h^{−1}^{F′}h ∈ G^{F} where h ∈ G is such that h^{−1}^{F}h = g.

(ii) We call Shintani descent the map Sh_{F′/F} from F -class functions on G^{F′} to F′-class functions on G^{F} defined by Sh_{F′/F} = ^{t}N_{F′/F}^{−1} .

It is easy to show that Shintani descent is an isometry for the usual scalar products on C(G^{F} /F′) and C(G^{F′}/F).

Let us now assume that we are given a pair T ⊂ B of a maximal torus and a Borel subgroup both F and F′-stable. Let L be a standard Levi subgroup stable under F′ and let w be an element of W^{F′} such that L is stabilized by ẇF where ẇ is a fixed representative of w in the normalizer N_{G}(T)^{F′} (note that ẇF and F′ commute). With this background we want to define the F′-twisted induction which will be denoted by R_{L^{ẇF} /F′}^{G^{F} /F′} from the space C(L^{ẇF} /F′) to the space C(G^{F} /F′). We first define it for the elements of a basis of C(L^{ẇF} /F′): let π be an F′-invariant irreducible character of L^{ẇF} and π̃ be an extension of π to the semi-direct product L^{ẇF} ><<F′>; let [tildeπ] be a complex vector space affording a representation of L^{ẇF} with character π̃; let S be the algebraic variety

S = {x ∈ G | x^{−1}.^{F}x ∈ U.ẇ.^{F}U}

(the endomorphism F′ acts naturally on S). We define the F′-twisted induction of π̃ (actually of the restriction of π̃ to L^{ẇF} .F′; we shall not mention the restriction when the context is unambiguous) as being the restriction to G^{F} .F′ of the character of the representation of G^{F} ><<F′> on the space (H_{c}^{∗}(S) ⊗ [π̃])^{L^{ẇF}}, where F′ acts naturally on H_{c}^{∗}(S) and acts as defined on [π̃]. We immediately get:

8.2 Proposition. With the preceding notations we have:

Trace(gF′ | R_{L^{ẇF} /F′}^{G^{F} /F′}(π̃)) = |L^{ẇF} |^{−1} ∑_{l∈L^{ẇF}} π̃(lF′) Trace((g, l)F′ | H_{c}^{∗}(S)).

This allows us to extend the definition to the whole space of F′-class functions:

8.3 Definition. We call F′-twisted induction the map

R_{L^{ẇF} /F′}^{G^{F} /F′} : C(L^{ẇF} /F′) → C(G^{F} /F′)

which extends by linearity the formula of proposition 8.2.

9. Isogenies.

In this section we show that induction and F′-twisted induction from a Levi subroup behave well with respect to isogenies. We will consider a morphism φ : G → G_{1} of connected reductive algebraic groups such that φ induces an isogeny from the derived group of G to that of G_{1}. In this situation, if T is a given maximal torus of G, there is a unique maximal torus T_{1} of G_{1} such that φ(T) ⊂ T_{1}. In fact, we have T_{1} = φ(T).ZG_{1} since G_{1} = φ(G).ZG_{1}; we have also T_{1} ∩ φ(G) = φ(T). In addition, we have: Ker φ ⊂ ZG and N_{G_{1}}(T_{1}) = φ(N_{G}(T)).ZG_{1} so φ defines an isomorphism of Weyl groups: W → W_{1}.

Let F and F′ (resp. F_{1} and F′_{1}) be two commuting Frobenius endomorphisms on G (resp. on G_{1}) with F′ (resp. F′_{1}) split such that:

φ ∘ F′ = F′_{1} ∘ φ and φ ∘ F = F_{1} ∘ φ.

Let B be a Borel subgroup of G such that the pair T ⊂ B is F and F′-stable; then it is clear that the pair T_{1} ⊂ B_{1} is also F′ and F′_{1}-stable, where B_{1} = φ(B).ZG_{1}.

9.1 Theorem. In the above situation, assume in addition that Ker φ is connected. Let ẇ ∈ N_{G}(T)^{F′}, let ẇ_{1} = φ(ẇ), and let L_{1} be a standard Levi subgroup of G_{1} which is w_{1}F_{1} and F′_{1}-stable. Then for any F′_{1}-class function π̃_{1} on L_{1}^{ẇ_{1}F_{1}} we have:

R_{L_{1}^{ẇ_{1}F_{1}} /F′_{1}}^{G_{1}^{F_{1}} /F′_{1}}(π̃_{1}) ∘ φ = R_{L^{ẇF} /F′}^{G^{F} /F′}(π̃_{1} ∘ φ)

where L = φ^{−1}(L_{1}).

Proof: In view of 8.2 and the Lefschetz formula we have to show the equality of:

|L^{ẇF} |^{−1} ∑_{l∈L^{ẇF}} π̃(l) |{x ∈ G | x^{−1}.^{F}x ∈ Uẇ^{F}U and g^{F′}x = xl}| (1)

(where π̃ = π̃_{1} ∘φ, g ∈ G^{F} and U is the unipotent radical of a parabolic subgroup containing L) with the analogous expression for G_{1}:

|L_{1}^{ẇ_{1}F_{1}}|^{−1} ∑_{l_{1}∈L_{1}^{ẇ_{1}F_{1}}} π̃_{1}(l_{1})

|{x ∈ G_{1} | x^{−1}.^{F_{1}}x ∈ U_{1}ẇ_{1}^{F_{1}}U_{1} and φ(g)^{F′_{1}}x = xl_{1}}| (2)

The equalities G_{1} = L_{1}φ(G) (since ZG_{1} ⊂ L_{1}) and L_{1} ∩ φ(G) = φ(L) show that G_{1}/φ(G) ≃ L_{1}/(L_{1}∩φ(G)) ≃ L_{1}/φ(L) and this isomorphism is compatible with w_{1}F_{1} since φ(L) is w_{1}F_{1}-stable (because L is wF - stable).

So we get (G_{1}/φ(G))^{w_{1}F_{1}} ≃ (L_{1}/φ(L))^{w_{1}F_{1}} and the first group is also isomorphic to (G_{1}/φ(G))^{F_{1}} because G_{1} is central modulo φ(G). As φ(L) is connected we have: (L_{1}/φ(L))^{w_{1}F_{1}} ≃ L_{1}^{ẇ_{1}F_{1}}/φ(L)^{ẇ_{1}F_{1}}. Now U_{1} = φ(U) so that U_{1}ẇ_{1}^{F_{1}}U_{1} ⊂ φ(G); so any x_{1} ∈ G_{1} such that x_{1}^{−1}.^{F_{1}}x_{1} ∈ U_{1}ẇ_{1}^{F_{1}}U_{1} is F_{1}-stable modulo φ(G) and since we have (G_{1}/φ(G))^{F_{1}} = L_{1}^{ẇ_{1}F_{1}}/φ(L)^{ẇ_{1}F_{1}} we can find an element λ ∈ L_{1}^{ẇ_{1}F_{1}} such that y_{1} = x_{1}λ is in φ(G). Then y_{1}^{−1}.^{F_{1}}y_{1} ∈ U_{1}ẇ_{1}^{F_{1}}U_{1} and there are |φ(L)^{ẇ_{1}F_{1}}| elements y_{1} corresponding to |L_{1}^{ẇ_{1}F_{1}}| elements x_{1}. Moreover x_{1} satisfies x_{1}^{−1}φ(g)^{F′_{1}}x_{1} = l_{1} ∈ L_{1}^{ẇ_{1}F_{1}} if and only if y_{1} satisfies y_{1}^{−1}φ(g)^{F′_{1}}y_{1} = λ^{−1}l_{1}^{F′_{1}}λ ∈ L_{1}^{ẇ_{1}F_{1}} and, by assumption, π̃_{1}(l_{1}) = π̃_{1}(λ^{−1}l_{1}^{F′_{1}}λ). So expression (2) is equal to:

|φ(L)^{ẇ_{1}F_{1}}|^{−1} ∑_{l_{1}∈L_{1}^{ẇ_{1}F_{1}}} π̃_{1}(l_{1})

|{y_{1} ∈ φ(G) | y_{1}^{−1}.^{F_{1}}y_{1} ∈ U_{1}ẇ_{1}^{F_{1}}U_{1} and φ(g)^{F_{1}}y_{1} = y_{1}l_{1}}|. (3)

We show now that the map (x, l) ↦ (φ(x), φ(l)):

{(x, l) ∈ G × L^{ẇF} | x^{−1}.^{F}x ∈ Uẇ^{F}U and g^{F′}x = xl} →

{(y_{1}, l_{1}) ∈ φ(G) × L_{1}^{ẇ_{1}F_{1}} | y_{1}^{−1}.^{F_{1}}y_{1} ∈ U_{1}ẇ_{1}^{F_{1}}U_{1} and φ(g)^{F′_{1}}y_{1} = y_{1}l_{1}}

is surjective. If (y_{1}, l_{1}) is in the right-hand side set then y_{1} = φ(y) with y^{−1}.^{F}y ∈ Uẇ^{F}Uz for some z ∈ Ker φ. As Ker φ is connected (so that Lang’s theorem holds in Ker φ) we can modify y to get another element such that φ(y) = y_{1} and y^{−1}.^{F}y ∈ Uẇ^{F}U. Then we have φ(y^{−1}g^{F′}y) = l_{1} ∈ φ(L)^{ẇ_{1}F_{1}} which implies y^{−1}g^{F′}y ∈ L^{ẇF} Ker φ since φ(L)^{w_{1}F_{1}} ≃ (L/ Ker φ)^{wF} ≃ L^{ẇF} /(Ker φ)^{wF} as Ker φ is connected. Let us write y^{−1}g^{F′}y = lz with l ∈ L^{ẇF} and z ∈ Ker φ; then:

Fz = ^{F}l^{−1}^{F}y^{−1}g^{F′F}y = ^{F}l^{−1}^{F}y^{−1}y(y^{−1}g^{F′}y)^{F′}(y^{−1}^{F}y)

= ^{F}l^{−1}(^{F}y^{−1}.y)lz^{F′}(y^{−1}^{F}y).

Setting a = y^{−1}^{F}y ∈ Uẇ^{F}U we get

z^{−1}^{F}z = ^{F}l^{−1}a^{−1}l^{F′}a = b^{−1}^{wF}l^{−1}l^{F′}a

with b ∈ Uẇ^{F}U. But ^{wF}l = l, so z^{−1}^{F}zb = ^{F′}a ∈ Uẇ^{F}U, which implies z^{−1}^{F}z = 1 (using the Bruhat decomposition), so the element (y, lz) is a preimage of (y_{1}, l_{1}). The fibers of the above map are clearly all of cardinality |(Ker φ)^{F} |. So expression (3) becomes:

|φ(L)^{ẇ_{1}F_{1}}|^{−1}|(Ker φ)^{F} |^{−1}

∑_{l∈L^{ẇF}} π̃(l)|{y ∈ G | y^{−1}.^{F}y ∈ Uw^{F}U and g^{F′}y = yl}

which is the desired result since |L^{ẇF} | = |(Ker φ)^{F} ||φ(L)^{ẇ_{1}F_{1}}|.

9.2 Corollary. Under the assumptions of theorem 9.1 (except the existence of F′ and F′_{1}), for any π_{1} ∈ Irr(L_{1}^{ẇ_{1}F_{1}}) we have:

R_{L_{1}^{ẇ_{1}F_{1}}}^{G_{1}^{F_{1}}} (π_{1}) ∘ φ = R_{L^{ẇF}}^{G^{F}} (π_{1} ∘ φ).

Proof: Let d be such that F_{1}^{d} and F^{d} are split and such that L_{1} and π_{1} are fixed by F_{1}^{d}. Then, for any m ∈ IN, we can apply the theorem with F′ = F^{dm} and F′_{1} = F_{1}^{dm}. Applying the standard process of “taking the limit for m = 0” (cf. [3, 3.3 and 4.1.2] and [5, 3.2]) gives the result.

If G^{∗} (resp. G_{1}^{∗}) is a group dual to G (resp. G_{1}), a morphism φ as above induces a morphism φ^{∗} : G_{1}^{∗} → G^{∗} with a central kernel, which also induces an isogeny for the derived groups, and an isomorphism of the Weyl groups.

9.3 Corollary. Under the above hypotheses, assume in addition that ZG and ZG_{1} are connected. Then for any s_{1} ∈ T_{1}^{∗} such that W_{F_{1}}(s_{1}) is non empty we have:

(i) Let s = φ^{∗}(s_{1}); then φ^{∗} induces a morphism C_{G_{1}^{∗}}(s_{1}) → C_{G^{∗}}(s) which is an isogeny on the derived groups. We denote F ↦ φ^{∗}(F) the induced bijection from Ξ(W (s_{1})) to

Ξ(W (s)). Then (ii) χ ↦ χ∘φ defines a bijection: χ ∈ E(G_{1}^{F_{1}}, s_{1}, F) → E(G^{F} , s, φ^{∗}(F)).

Proof: Since ZG_{1} (resp. ZG) is connected, the group W (s_{1}) (resp. W (s)) is generated by reflections corresponding to the roots in Φ_{s_{1}} (resp. Φ_{s}), so to prove (i) we have to prove that α_{1} ∈ Φ_{s_{1}} (i.e. α_{1}(s_{1}) = 1) is equivalent to α(s) = 1 where α corresponds to α_{1} by φ^{∗}. This is true since (by definition of isogenies) α ∘ φ^{∗} = q(α)α_{1} for some power q(α) of the characteristic.

We now identify W (s) to W (s_{1}) by φ^{∗}. To prove (ii), since by 9.2

we have R_{T_{1}^{wF_{1}}}^{G_{1}^{F_{1}}} (ŝ_{1}) ∘ φ = R_{T^{wF}}^{G^{F}} (ŝ) for any w ∈ Z_{F} (s), it is enough to

prove that χ ∘ φ is irreducible for any χ ∈ E(G_{1}^{F_{1}}, s_{1}).

If χ and ψ are characters in Irr(G_{1}^{F_{1}}), by Clifford theory χ ∘ φ and ψ ∘φ are either disjoint or equal, and the latter occurs only if they differ by a (linear) character of G_{1}^{F_{1}}/φ(G)^{F_{1}}, and ⟨ χ ∘ φ, χ ∘ φ ⟩_{G^{F}} = |{θ ∈ Irr(G_{1}^{F_{1}}/φ(G)^{F_{1}}) | θ ⊗ χ = χ}|. Such a θ corresponds to an element z in Ker φ^{∗}. If z ≠ 1, the elements zs_{1} and s_{1} cannot be conjugate under G_{1}^{∗}: they would then be conjugate by some w ∈ W, but we would have ^{w}s = s so w ∈ W (s) = W (s_{1}) so ^{w}s_{1} = s_{1}, a contradiction. So if χ ∈ E(G_{1}^{F_{1}}, s_{1}) then θ ⊗ χ ∈ E(G_{1}^{F_{1}}, s_{1}z) which is disjoint from E(G_{1}^{F_{1}}, s_{1}), so we cannot have θ ∘ χ = χ, so χ ∘ φ is irreducible.

10. Shintani descent.

In this section, we want to express the action of Shintani descent in a connected reductive algebraic group G in terms of the parametrization of section 5. As in section 8, we assume that G has two rationnal structures given by Frobenius endomorphisms F and F′, both stabilizing a pair T ⊂ B. Furthermore, we assume that F′ is a split Frobenius endomorphism. We fix a semi-simple element s ∈ T^{∗} assumed to be F′-fixed, and such that W_{F} (s) is not empty and we choose w_{1} ∈ W_{F} (s) as in section 2. We keep all notations of part I. With these notations, we will give some evidence for the existence and commutativity of the following diagram (which properties we conjecture to hold in general): 10.1

C(G^{F ′} /F,s)  ⟶^{Φ∘Sh_{F ′/F}}  C(G^{F} /F′,s)
↓^{a}  ↓^{b}
∐_{a∈Aw1F} C(C_{G∗} (s)^{(ȧF ′)∗} /(ẇ_{1}F)^{∗},1)  ∐_{a∈H1(w1F,A)} C(C_{G∗} (s)^{(ȧẇ1F)∗} /(F′)^{∗},1)
↓^{c}  ↓^{d}
∐_{F∈Ξ(W (s))w1F} \underline{M}(Γ_{F} ⊂Γ_{F} ><<w_{1}F >) ⟶^{FT} ∐_{F∈Ξ(W (s))w1F} \overline{Q}_{l}\overline{M}(Γ_{F} ⊂Γ_{F} ><<w_{1}F >)

where map a (resp. b) is a bijection generalizing to F -class functions (resp F′-class function) the map π_{s} of section 3, and maps c and d are generalizations of the parametrization of section 5. We have denoted by F T the Fourier transform associated to the pairing between spaces \overline{Q}_{l}\overline{M} and \underline{M} and by Φ the endomorphism of C(G^{F} /F′, s) defined on the basis of extensions of F′-invariant irreducible characters by multiplication by the associated eigenvalue of F′. We will construct vertical maps under suitable assumptions: we keep the assumption 3.3 that restrictions of irreducible characters are multiplicity free and we will assume some degree of compatibility of π_{s} with the action of the Frobenius endomorphisms (cf. 10.3, 10.4 below). Then we will show that Shintani descent maps the subspace of C(G^{F′}/F, s) corresponding to principal series characters of C_{G^{∗}}(s)^{(ȧF′)^{∗}} into C(G^{F} /F′, s) and that when F′ = F_{0}^{m}, where F_{0} a split Frobenius and m sufficiently divisible, the restriction of the above diagram to the corresponding subspaces is commutative. We do not know in general if Shintani descent maps C(G^{F′}/F, s) on C(G^{F} /F′, s). Shoji and Asai ([14], [15] and [2]) proved this result for groups with connected center and for the special linear group. The proof of the commutativity of the restricted diagram relies heavily on the results of [7]. The idea is to show that the “limit for m=0” of this diagram ([7] shows the existence of the limit of Shintani descent) is known to be commutative by theorem 5.8, and to apply results of [7] which relate the value of Shintani descent to the value of its limit.

We need also the following result of [7] which is an easy generalization of [11, 2.20]:

10.2 Proposition. Let F′ be a split Frobenius endomorphism over the field IF_{q_{F′}} such that ^{F′}s = s; then

(i) Any ρ ∈ E(G^{F} , s) is fixed by F′.

(ii) For any extension ρ̃ of ρ to G^{F} ><<F′>, such that <F′> acts on ρ̃ via a finite quotient, there exists an algebraic number λρ̃ ∈ \overline{Q}_{l} all of which complex conjugates have absolute value 1 such that if ρ is a component of H_{c}^{∗}(Yẇ, \overline{Q}_{l}) (where Yẇ denotes a Deligne-Lusztig variety) then the isotypic component of type ρ in this module has a (G^{F} , F′)-stable filtration whose quotients are isomorphic to ρ̃, if we multiply the action of F′ in each quotient by λ−1 ; ρ̃q_{F′}^{−i} for some integer i.

So the right hand side of diagram 10.1 is given by 3.4 and 5.2 provided that F′ acts trivially on C(C_{G^{∗}}(s)^{(ȧẇ_{1}F)^{∗}}, 1). By 10.2 above, it acts trivially on C(C_{G^{∗}}^{∘}(s)^{ȧẇ_{1}F^{∗}}, 1). We assume that it acts again trivially on unipotent characters of the full centralizer (it may be possible to check case by case that this assumption holds):

10.3 Assumption. The action of F′ on E(C_{G^{∗}}(s)^{(ẇ_{1}F)^{∗}}, 1) is trivial.

We now look at the left part of diagram 10.1. Here the Frobenius endomorphism is F′. It is assumed to fix s, so W_{F′}(s) = W (s), i.e., the corresponding element w_{1} is equal to 1. As F′ is split, it acts trivially on the Weyl group and we have H^{1}(F′, A) = A. We will assume that all representatives of elements of W we consider are chosen in G^{F′}. To construct map a we have to show that the map π_{s} : E(G^{F′}, s) → ∐_{a∈A} E(C_{G^{∗}}(s)^{(ȧF′)^{∗}}, 1) restricts to a bijection from the set of F -fixed characters in the left hand side onto the set of (ẇ_{1}F)^{∗}-fixed characters in the right hand side. Recall that π_{s} was constructed using the embedding G ↪ G̃, and that it maps the elements of the G̃^{F′}/ StabG̃^{F′} (ρ)- orbit of ρ ∈ E(G^{F′}, s) onto the elements π_{s}(ρ) ⊗ ζ where ζ runs over Irr(C_{G^{∗}}(s)^{ȧF′}(π_{s}(ρ))/C_{G^{∗}}^{∘}(s)^{ȧF′}). Moreover, as stated after 3.4, there is a canonical isomorphism

G̃^{F′}/ StabG̃^{F′} (ρ)→^{∼} Irr(C_{G^{∗}}(s)^{ȧF′}(π_{s}(ρ))/C_{G^{∗}}^{∘}(s)^{ȧF′})

induced by the map a ↦ â of 3.2. So if both ρ is F -fixed and π_{s}(ρ) is ẇ_{1}F -fixed, the map π_{s} is compatible with the actions of F and ẇ_{1}F. To ensure this compatibility in the sequel, we will make the following assumption:

10.4 Assumption.

(i) If the A-orbit of ρ ∈ E(G^{F′}, s) is F -fixed, then it contains an F - fixed character.

(ii) For any a ∈ A and each (ẇ_{1}F)^{∗}-fixed A-orbit in E(C_{G^{∗}}^{∘}(s)^{ȧF′}, 1), there exists a (ẇ_{1}F)^{∗}-fixed extension to C_{G^{∗}}(s)^{ċF′} of the sum of the characters in that orbit.

This assumption gives us the desired existence of bijection a because a character ρ can be F -fixed only if its orbit is F -fixed, and a character in E(C_{G^{∗}}(s)^{F′}, 1) can be ẇ_{1}F -fixed only if its restriction to C_{G^{∗}}^{∘}(s)^{ȧF′} is ẇ_{1}F -fixed.

Let us remark that in the case where a = 1, and ρ is in the principal series of C_{G^{∗}}(s)^{F′}, property 10.4, (ii) holds by the lemma 5.5, (i) applied to W (s) with the automorphism w_{1}F. 10.5 We now construct map c. We simplify the notations by writing H for C_{G^{∗}}(s). Note that for any family F of W (s), as F′ acts trivially on Γ_{F}, we have

\overline{M}(Γ_{F} ⊂ Γ_{F} ><<F >) = M(Γ_{F}),

so the parametrization of 5.2 in this case is a bijection

⨁_{a∈A} \overline{Q}_{l}E(H^{aF′}, 1)→^{∼} ⨁_{F∈Ξ(W)} \overline{Q}_{l}M(Γ_{F}).

We now show that this parametrization maps ẇ_{1}F -invariant irreducible characters on w_{1}F -invariant elements in the right hand side, so we can restrict ourselves to w_{1}F -invariant families. Let F be such a family. It is defined as the A-orbit of a family F_{0} ∈ Ξ(W^{∘}(s))^{a} with a ∈ A^{w_{1}F}. The action of ẇ_{1}F on the orbit of E(H^{∘}^{aF′}, 1, F_{0}) corresponds to the action of w_{1}F on the A-orbit of \overline{M}(Γ_{F_{0}} ⊂ Γ_{F_{0}} ><<a>), which induces the action of w_{1}F on M(Γ_{F}). The parametrization maps the extensions to H^{aF′} of the sum of the A-orbit of ψ_{0} ∈ E(H^{∘}^{aF′}, s, F_{0}) onto elements (x, χ) ∈ M(Γ_{F}) where χ runs over irreducible components of Ind_{C_{Γ_{F_{0}}} (x)}^{C_{Γ_{F}} (x)}(χ_{0}) where (x, χ_{0}) ∈ \overline{M}(Γ_{F_{0}} ⊂ Γ_{F_{0}} ><<a>) parametrizes ψ_{0}. Extensions of the sum of the orbit of ψ_{0} differ by tensorization by linear characters of Stab_{A}(ψ_{0}). Irreducible components of Ind_{C_{Γ_{F_{0}}} (x)}^{C_{Γ_{F}} (x)}(χ_{0})

differ by tensorization by linear characters of Stab_{A}(F_{0}, (x, χ_{0})) which is equal to Stab_{A}(ψ_{0}). By assumption 10.4, (ii), the sum of any ẇ_{1}F - invariant orbit in E(H^{∘}^{aF′}, 1) has a ẇ_{1}F -invariant irreducible extension to H^{aF′}. So showing that ẇ_{1}F -invariant characters in E(H^{aF′}, 1, F) corresponds to w_{1}F -invariant elements of M(Γ_{F}) is equivalent to showing that for any (x, χ_{0}) ∈ \overline{M}(Γ_{F_{0}} ⊂ Γ_{F_{0}} ><<a>) there exists an irreducible component χ of Ind_{C_{Γ_{F_{0}}} (x)}^{C_{Γ_{F}} (x)}(χ_{0}) such that (x, χ) is invariant as an ele-

ment of M(Γ_{F}). But this is exactly the statement of 5.5, (ii) applied for the automorphism w_{1}F. So under assumptions 3.3 and 10.4 we have constructed map c.

Note that assumptions 10.3 and 10.4 are clearly true if the center of G is connected. So in this case, maps a, b, c and d are proved to exist.

Before giving the announced result about Shintani descent, we recall some of the main results of [7] (cf. [7] 4.2, 7.4, 8 and 9.7). We keep the same notations.

10.6 Theorem. We assume that F′ = F_{0}^{2m} where F_{0} is a split Frobenius endomorphism on G and m is such that ^{F_{0}^{m}}s = s. We denote by ψ_{G,F′} the linear operator from C(G^{F} /F′, s) to C(G^{F} , s) which maps any ρ̃ on λ−1 ; ρ̃ρ, where λρ̃ is the eigenvalue of F′ associated to the extension ρ̃ of ρ ∈ E(G^{F} , s) (cf. 10.2). Then, for any extension Ẽ to W (s) ><<w_{1}F > of E ∈ (Irr(W (s)))^{w_{1}F}, there exists an extension ρẼ of the character ρ_{E} (inverse image by π_{s^{−1}} of the principal series character corresponding to E) such that

ψ_{G,F′} ∘ Sh_{F′/F} ρẼ = π_{s^{−1}}(RẼ)

(and so is independent of m).

We can now state:

10.7 Proposition. As above, we assume that F′ = F_{0}^{m} where F_{0} is a split Frobenius endomorphism on G and m is such that ^{F_{0}^{m}}s = s; then for any ρ ∈ E(G^{F} , s^{−1}), any extension ρ̃ of ρ to G^{F} ><<F′> (where the action of F′ is of finite order) and any E ∈ (Irr(W (s)))^{w_{1}F}, if λρ̃ is the eigenvalue of F′ associated to ρ̃, we have

⟨ ρ̃, Sh_{F′/F} ρẼ ⟩_{G^{F} .F′} = λ−1 ; ρ̃(−1)^{l(w_{1})}{x_{π_{s^{−1}} (ρ)}, xẼ}.

Proof: It is an immediate consequence of 10.6, of 3.5 and of 5.8.

References.

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[2] T. Asai “Twisting operator on the space of class functions of finite special linear groups”, Proc. of Symp. in Pure Math. 47, 1987, 99– 148.

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[5] F. Digne and J. Michel “Fonctions L des variétés de Deligne-Lusztig; descente de Shintani” Mémoires de la SMF 20, Suppl. Bull. Soc. Math. France 113, 1985.

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[7] F. Digne and J. Michel “Lusztig functor and Shintani descent”, Rapports de Recherche du Laboratoire de Mathématiques de l’École Normale Supérieure 89-2, 1989.

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François DIGNE DMI, ENS, 45 rue d’Ulm, 75230 Paris Cedex 5; et Université de Picardie, 33 rue Saint-Leu, 80039 Amiens Cedex.

Jean MICHEL

DMI, ENS, 45 rue d’Ulm, 75230 Paris Cedex 5.