DMI École Normale Supérieure

Département de Mathématiques et d’informatique

Lusztig Functor and Shintani Descent

F.Digne J.Michel

Département de Mathématiques et d’Informatique École Normale Supérieure LMENS - 89 - 2

Janvier 1989

Rapport de Recherche du LMENS Laboratoire de Mathématiques de L’École Normale Supérieure

LMENS, Département de Mathématiques et d’Informatique, École Normale Supérieure, 45 Rue d’Ulm, 75230 PARIS Cédex 05, FRANCE.

Lusztig functor and Shintani descent

F.Digne and J.Michel

The results of this paper have been presented at the 1986 Arcata conference on representations of finite groups but have not yet been published anywhere. The reason was that the results here are mostly slight generalisations of those of Shoji [S] and [S2] (the generalisation is that we deal with groups with a not necessarily connected center, and that we consider all Shintani descents and not only their limit); we intended to publish them as part of a longer paper on Shintani descents of characters of reductive groups. But results and papers never appear out in the expected order, and now we need to be able to quote the results of this paper in our forthcoming paper on the parametrization of characters in the groups with non-connected center, which is why we publish them here.

This text is organized as follows: §1 introduces the notations. §2 contains the (now standard) computation relating the Lefschetz numbers of the Deligne-Lusztig varieties to Shintani descents (generalized here to the variety associated to a Lusztig functor). §3 expresses the Shintani liftings obtained in §2 in terms of the commuting algebra of an induced cuspidal representation from a Levi subgroup. The results of §2 and §3 have been found independently by B.Srinivasan, T.Shoji and us. They have been published first by Shoji [S] but we need them in a slightly more general form than found in [S].

To apply §3 we need some assumptions (cf. 4.1) that have been checked in the case of a group with connected center by Lusztig but that we don’t know how to prove in general (though we believe they always hold); however, we can prove them when the Levi subgroup is a torus, and that is the object of §4.

In §5 we check that some results of Lusztig’s book [L] about eigenvalues of Frobenius endomorphisms generalize to groups with non-connected center; and §6 contains results that we need about characters of endomorphism algebras of induced cuspidal representations (which are the semi-direct product of usual Hecke algebras by a finite group acting by diagram automorphisms).

§7 contains the main result: upon suitable assumptions (7.2) and (7.3) (mostly the periodicity of the Shintani descent of cuspidal representations) we get the periodicity of all Shintani descents (this type of result was first obtained in [DM1] for the principal series, and has since been generalized by Lusztig, Asai and Shoji).

Lusztig functor and Shintani descent F.Digne and J.Michel

In §8 we show that assumptions of §7 actually hold when the Levi subgroup is a torus. Finally §9 applies §7 by giving a commutation formula between Shintani descent and Lusztig functor.

1. General hypotheses, definitions and notations. Let G be a connected reductive algebraic group defined over an algebraically closed field IF. We consider two rational structures on G defined over some finite subfields of IF and given by two Frobenius endomorphisms F and F′ which commute. We call q_{F′} the power of the characteristic associated to F′.

In this background, we can define N_{F/F′} and the Shintani descent Sh_{F/F′} = ^{t} N_{F/F′}^{−1} (cf. [DM, 1.7.2]).

We consider a torus T contained in a Borel subgroup B such that T and B are stabilized by both F and F′. Given F, we may always modify F′ by an inner automorphism so that such a pair exists: suppose ^{F}(T, B) = (T, B) but ^{F′}(T, B) = ^{x}(T, B), then Fx(T, B) = ^{x}(T, B) so x^{−1}^{F}x ∈ T and by Lang’s theorem x^{−1}^{F}x = t^{−1}^{F}t with t ∈ T; then if we replace F′ by F′′ = tx^{−1}F′, then ^{F′′}(T, B) = (T, B), (G, F′) maps isomorphically to (G, F′′), F′′ still commutes with F and F′′ is still split if F′ was split.

We denote respectively by Σ, Π, and W the root system, its basis, and the Weyl group defined by the given pair T ⊂ B. On W and Σ, we have natural actions of F and F′. These actions stabilize Π. The group W^{F′} is a Coxeter group. We denote by l the length function in this group.

If J is a part of Π, we shall note L_{J} (resp. U_{J}) the standard Levi subgroup (resp. the unipotent radical of the standard parabolic subgroup P_{J}) associated with J.

If H is a finite group or a finite-dimensional semi-simple algebra over C, we note Irr(H) the set of irreducible characters of H. If <F > is a group generated by F and acting on H, we denote by C(H/F) the space of complex valued F -class functions on H, i.e. functions φ which verify φ(x.^{F}y) = φ(yx) for any x and y in H. We shall identify C(H/F) with the space of restrictions to the set H.F of class functions on the semi-direct product H ><<F >. This space admits as a basis the set of restrictions to H.F of an arbitrary extension to H ><<F > of each F-invariant irreducible character of H. If φ_{1} and φ_{2} are elements of C(H/F), we put

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

We recall that if φ_{1} (resp. φ_{2}) is the restriction to H.F of a character χ̃_{1} (resp. χ̃_{2}) of H ><<F > whose restriction χ_{1} (resp. χ_{2}) to H is irreducible, we have:

⟨ φ_{1}, φ_{2} ⟩_{H.F} = \{ \begin{matrix}0, & if χ_{1} ≠ χ_{2} \\ 1, & if χ̃_{1} = χ̃_{2}. \end{matrix}

Note that in the above the group <F > can be infinite.

If χ is a character (resp. a representation) of H, we denote by [χ] a vector space affording a representation of H of character χ (resp. a model of χ).

2. Generalized induction. If L is an F -stable Levi subgroup of G, we want to study Lusztig’s induction R_{L^{F}}^{G^{F}}. For this, we can, by a conjugation in G, bring L on a standard Levi subgroup L_{J}, the action of F becoming the action of ẇF for some representative ẇ of an element w of W. So, from now on, we shall consider a fixed part J of Π defining a standard Levi subgroup L_{J}, a standard parabolic subgroup P_{J} and the unipotent radical U_{J}. We shall write L (resp. U, P) for L_{J} (resp. U_{J}, P_{J}). Assumption 2.1. We assume that F′ stabilizes L(this is automatically true if F′ is split).

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. 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 [π̃] 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: Proposition 2.2. 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:

Lusztig functor and Shintani descent F.Digne and J.Michel Definition 2.3. 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 2.2.

With these notations we have: Proposition 2.4. Let μ̃ be any ẇF -class function on L^{F′}, considered as a class function on the coset L^{F′}.ẇF of L^{F′} ><<ẇF >. Let g be any element of G^{F′} then:

R_{L^{ẇF} /F′}^{G^{F} /F′}(Sh_{F′/ẇF} μ̃)(N_{F′/F} g.F′) = \frac{1}{|L^{F′}|} ∑_{l∈L^{F′}} μ̃(lẇF)#\{y ∈ G^{F′} | y^{−1}g.^{F}y∈lUẇ^{F}U\}.

Proof: By 2.3 and the Lefshetz formula, the left hand side is equal to

|L^{ẇF} |^{−1} ∑_{l∈L^{ẇF}} (Sh_{F′/ẇF} μ̃)(lF′)#{x ∈ S | (N_{F′/F} g, l).^{F′}x = x}  .

The cardinality of the set appearing in this expression is equal to:

#{x ∈ G | N_{F′/F} g.^{F′}x.l^{−1} = x and x^{−1}.^{F}x ∈ Uẇ^{F}U}.

Let h ∈ G be such that g = h.^{F}h^{−1}; then we have N_{F′/F} g = h^{−1}.^{F′}h. Let λ ∈ L be such that l = λ^{−1}.^{F′}λ; then the last expression is equal to

#{x ∈ G | ^{F′}(hxλ^{−1}) = hxλ^{−1} and (hxλ^{−1})^{−1}.g.^{F}(hxλ^{−1}) ∈ λUẇ^{F}U^{F}λ^{−1}}. But we have λUẇ^{F}U^{F}λ^{−1} = λ.^{wF}λ^{−1}Uẇ^{F}U. So, if we put y = hxλ^{−1} and l_{1} = λ.^{wF}λ^{−1} we get for the last expression :

|L^{F′}|^{−1}|L^{ẇF} |#{y ∈ G^{F′} | y^{−1}.g.^{F}y ∈ l_{1}Uẇ^{F}U}.

As we have (Sh_{F′/ẇF} μ̃)(lF′) = μ̃(l_{1}ẇF), we get the result. Corollary 2.5. With the same notations as in 2.4 we have:

R_{L^{ẇF} /F′}^{G^{F} /F′}(Sh_{F′/ẇF} μ̃)(N_{F′/F} g.F′)

= |L^{F′}|^{−1} ∑_{l∈L^{F′}} μ̃(lẇF)#{y ∈ G^{F′} | y^{−1}.g.^{F}y∈lU^{F′} ẇ(^{F}U)^{F′}}.

Proof: We have to show that if an element of the form uẇ.^{F}u_{1} with u and u_{1} in U is fixed by F′, then it is in fact in U^{F′} ẇ(^{F}U)^{F′}. But then we have u^{−1}.^{F′}u = ^{ẇF}(u_{1}^{F′}u_{1}^{−1}) ∈ U ∩ ẇFU. We can apply Lang’s theorem in the connected group U ∩^{ẇF}U to get v ∈ U ∩^{ẇF}U such that v.^{F′}v^{−1} = u^{−1}.^{F′}u, so that uv and ^{(ẇF)^{−1}}v^{−1}u_{1} are elements of U^{F′}. Then the element u.^{ẇF}u_{1} is equal to uv.^{ẇF}(^{(ẇF)^{−1}}v^{−1}u_{1}), and so lies in U^{F′}(^{ẇF}U)^{F′}.

Remark 2.6. Corollary 2.5 shows that the expression R_{L^{ẇF} /F′}^{G^{F} /F′}(Sh_{F′/ẇF} μ̃)(N_{F′/F} g.F′) does not change if we replace F by F_{1} such that F and F_{1} have the same action on G^{F′}. It also shows that this expression does not change if we multiply ẇ on the left by an element of N_{L}(T)^{F′}.

3. Right hand side of 2.5. We shall now, in the following paragraphs, transform both sides of 2.5, interpreting the right hand side as the trace of an operator on the representation R_{L^{F′}}^{G^{F′}} (μ), and using in the left hand side an analogue of Lusztig’s result [L, 2.20] for eigenvalues of F′. In this paragraph, we shall transform the right hand side in the case when μ̃ comes from extending a cuspidal representation. Let δ be a cuspidal representation of L^{F′}.

Assumption 3.1. There exists an extension δ̃ of δ to the group L̃ = <x∈N_{G}(L)^{F′} | ^{x}δ ≃ δ>.

Remark 3.2. Assumption 3.1 is proved to be true when the center of G is connected in [L, 8.6], and has been proved by Lusztig (private communication) when F′ is split, with no assumption on the center of G.

We consider now the group \overline{W} (δ) = L̃/L^{F′}. This group is the isomorphic image of W (δ) = {w∈W^{F′} | ^{w}J = J and ^{w}δ ≃ δ}. By a result of Howlett and Lehrer (cf. [HL, 2.7]) we know that W (δ) has a semi-direct decomposition of the form W (δ) = W^{∘}(δ) >< C(δ) where W^{∘}(δ) is a Coxeter group associated to a set of roots Γ ⊂ Σ whose image \overline{Γ} in <Σ> ⊗ IR/<J> ⊗ IR is a root system. We consider also the right W (δ)-coset Z(δ) defined by Z(δ) = \{w∈W^{F′} | ^{wF}J = J and ^{wF}δ ≃ δ\} and we put \overline{Z}(δ) = W_{L}^{F′}\Z(δ).

We note Δ ⊂ Γ the subset such that \overline{Δ} is the basis of \overline{Γ} (for the order on the quotient induced by the chosen order which makes Π a basis of Σ). In the following we fix an element w_{1} in the coset Z(δ) such that ^{w_{1}F}Δ = Δ, and a representative ẇ_{1} of w_{1} in N_{G}(T)^{F′}. This is possible because of the following lemma:

Lemma 3.3. We have ^{wF}Γ = Γ for any w ∈ Z(δ).

Proof: We use the notations of Howlett-Lehrer [HL, 2.3]: let Ω̂ = {a ∈ Σ | w(J ∪ {a}) ⊂ Π for some w ∈ W }. For a ∈ Ω̂, let v(a, J) = w_{∘}^{J∪{a}}w_{∘}^{J} where w_{∘}^{K} is the longest element in W_{K}. The set Ω̂ is normalized by wF, and for any element a of this set we have wFv(a, J) = v(^{wF}a, J). So the action of wF preserves Γ′ = {a ∈ Ω̂ | v(a, J) ∈ W (δ) and v(a, J)^{2} = 1} (cf. [HL, 3.21]). Then, by a proof similar to that of [HL, 3.22], we get the result.

Lusztig functor and Shintani descent F.Digne and J.Michel

With these notations we have Z(δ) = W (δ).w_{1}. We now define a vector space E(δ) which is a model for the induced representation R_{L^{F′}}^{G^{F′}} (δ). Definitions 3.4.

(i) Let E be the vector space:

E = {f : C[G^{F′}] → [δ]}

where [δ] is a vector space affording a model of the representation δ. (ii) Let E(δ) = {f ∈ E | f(lux) = δ(l)f(x) for any l ∈ L^{F′} and any u ∈ U^{F′}}

More generally, for any bijective endomorphism j of G commuting with F′, we define: (iii) E(^{j}δ) = {f ∈ E | f(^{j}l.^{j}u.x) = δ(l)f(x)} = E ∘ j. (iv) We note H(δ) the endomorphism algebra End_{G^{F′}} (E(δ)).

On these spaces we shall define some operators (in the following definitions, j is as in 3.4.) Definitions 3.5.

(i) We define an action of F on E by ^{F}f(g) = f(^{F^{−1}}g) for any f ∈ E and any g ∈ G^{F′}. (ii) For x ∈ G^{F′}, we define x : E → E by (xf)(g) = f(gx), for any f ∈ E and any

g ∈ G^{F′}. (iii) We assume that j normalizes T. For ẇ ∈ N_{G}(T)^{F′} such that wjJ ⊂ Π, we define

B_{^{j}δ,ẇ} : E(^{j}δ) → E(^{ẇj}δ) by:

(B_{^{j}δ,ẇ}f)(g) = |U_{wjJ}^{F′}|^{−1} ∑_{u∈U_{wjJ}^{F′}} f(ẇ^{−1}ug).

(iv) If Λ is any operator on [δ], we define an action of Λ on E by (Λf)(x) = Λ.f(x) for

any f ∈ E and any x ∈ G^{F′}. Remark 3.6. Definition (iv) applies in particular to the case Λ = δ̃(x) where x ∈ L̃.

As the representations δ and ^{ẇ_{1}F}δ are equivalent, there exists a linear operator Φ on [δ] which realizes this equivalence, and, if (ẇ_{1}F)^{i} acts trivially on G^{F′}, then Φ^{i} is a scalar. Dividing Φ by an i-th root of this scalar, we get the following proposition: Proposition 3.7. There exists an operator Φ of finite order on [δ] such that for any l ∈ L^{F′} we have Φ.δ(l).Φ^{−1} = δ(^{ẇ_{1}F}l).

With these notations we have the following easy to check properties:

Proposition 3.8.

(i) ^{F}E(^{j}δ) = E(^{F j}δ). (ii) For x ∈ G^{F′} we have xE(^{j}δ) = E(^{j}δ). (iii) ΦE(^{j}δ) = E(^{j(ẇ_{1}F)^{−1}}δ). (iv) For x ∈ L̃ we have δ̃(x)E(^{j}δ) = E(^{jx^{−1}}δ).

We give now some relations involving these operators.

Proposition 3.9.

(i) For any x ∈ L̃ ∩ N_{G}(T) and any ẇ such that wxJ ⊂ Π, we have

B_{δ,ẇ}δ̃(x) = δ̃(x)B_{^{x}δ,ẇ}

(ii) For any x in L̃, we have

F δ̃(x) = δ̃(x)F

(iii) F B_{δ,ẇ} = B_{^{F}δ,^{F}(ẇ)}F (iv) ΦB_{^{ẇ_{1}F}δ,ẇ}Φ^{−1} = B_{δ,ẇ}.

These relations are easily verified. To simplify formulas we introduce the following notation:

Definition 3.10. B′_{^{j}δ,ẇ} = q_{F′}^{l(w)/2}B_{^{j}δ,ẇ}.

We need now:

Proposition 3.11. Let ẇ be an element of N_{G}(T)^{F′} such that w^{−1}J ⊂ Π and such that for any positive root a ∈ Γ^{+} the root w^{−1}a is positive; let v be an element of W (δ); then we have

B′_{δ,v̇}B′_{^{˙w^{−1}}δ,ẇ} = B′_{δ,v̇ẇ}.

Proof: We have only to symmetrize the proofs in [HL, 4.1 and 4.2] (with the notations of [HL], the hypothesis on w is N(w^{−1}) ∩ Γ = ∅).

We recall now some known facts about the endomorphism algebra H(δ) of R_{L^{F′}}^{G^{F′}} δ.

Definition 3.12. We define a function l̃: W (δ) → Q by p_{w} = q_{F′}^{l̃(w)}, where p_{w} is as in [HL, 4.9].

Note that l̃ is zero on C(δ) (cf. [HL, 4.10]), and is additive, i.e. we have l̃(w_{1}w_{2}) = l̃(w_{1}) + l̃(w_{2}) if and only if the lengths in W^{∘}(δ) of the elements w_{1} and w_{2} of W^{∘}(δ) add (cf. proof of [HL, 4.11]). In all cases that we know, l̃ actually takes its values in IN.

Note that the restriction to E(δ) of the operator δ̃(ẇ)B′_{δ,ẇ} does not depend on the representative ẇ of w ∈ W (δ). This gives a well defined meaning to the notation T_{w} in the following proposition:

Proposition 3.13. Under assumption 3.1, there exists an extension δ̃ of δ to L̃ defined up to multiplication by a linear character of C(δ) such that the algebra H(δ) admits as a basis the set {T_{w} | w∈W (δ)}, where the operators T_{w} are the restriction to E(δ) of q_{F′}^{l̃(w)/2}δ̃(ẇ)B′_{δ,ẇ}, and satisfy the relations:

(i) For any w_{1} and w_{2} such that l̃(w_{1}w_{2}) = l̃(w_{1}) + l̃(w_{2}), we have:

T_{w_{1}}T_{w_{2}} = T_{w_{1}w_{2}}.

(ii) For any a ∈ Δ :

(T_{v(a,J)} + 1)(T_{v(a,J)} − q_{F′}^{l̃(v(a,J))}) = 0.

Proof: By [HL, 4.11], for any extension δ̂ of δ, there exists a linear character w ↦ ε_{w} of W (δ), which is concentrated on W^{∘}(δ), such that the operators T_{w} = ε_{w}q_{F′}^{l̃(w)/2}δ̂(ẇ)B′_{δ,ẇ} form a basis satisfying (i) and (ii) above (cf. [HL, 4.14]). The representation ẇ ↦ δ̃(ẇ) = ε_{w}δ̂(ẇ) is another extension of δ for which (i) and (ii) hold. For this extension the (same) operators T_{w} are defined as stated in the proposition. In addition, if ζδ̃ is any other extension of δ (where ζ is a linear character of W (δ)) then an easy computation shows that the proposition (specifically relation (ii)) holds for this extension if and only if ζ equals 1 on W^{∘}(δ) i.e., is a character of C(δ).

We now look at the obstruction to extend δ̃ to L̃ ><<ẇ_{1}F >.

Lemma 3.14. For any x∈W (δ), we have:

Φδ̃(ẋ)Φ^{−1} = λ_{x}δ̃(^{ẇ_{1}F} ẋ),

where the map x ↦ λ_{x} is a linear character of W (δ) (ẋ denotes a representative of x).

Proof: For any l ∈ L^{F′}, we have:

(Φδ̃(ẋ)Φ^{−1})δ(l)(Φδ̃(ẋ)Φ^{−1})^{−1} = δ(^{^{ẇ_{1}F} ẋ}l) = δ̃(^{ẇ_{1}F} ẋ)δ(l)δ̃(^{ẇ_{1}F} ẋ)^{−1}.

Therefore (as δ is irreducible) there exists a constant λẋ such that Φδ̃(ẋ)Φ^{−1} = λẋδ̃(^{ẇ_{1}F} ẋ). We check now that we have got a linear character; this comes at once from the identity

Φδ̃(ẋẏ)Φ^{−1} = Φδ̃(ẋ)Φ^{−1}Φδ̃(ẏ)Φ^{−1}.

Finally, from the definition of Φ (cf. 3.7), we see that if the element ẋ of 3.14 is in L^{F′}, we have λẋ = 1, so λẋ does not depend on the representative ẋ. We shall write λ_{x} for λẋ.

In addition to the preceding result we prove:

Lemma 3.15. The character x → λ_{x} of W (δ) depends neither on the choice of the representative ẇ_{1}, nor on the particular element w_{1} in the right coset W (δ)w_{1}.

Proof: Let v̇ be a representative of an element v ∈ W (δ), the operator δ̃(v̇)Φ satisfies 3.7 for v̇ẇ_{1}F, and we have:

δ(v̇)Φδ̃(ẋ)Φ^{−1}δ(v̇)^{−1} = λ_{x}δ̃(^{v̇ẇ_{1}F}x),

whence the result.

Lemma 3.16. The action of w_{1}F stabilizes the groups W^{∘}(δ) and C(δ); it induces on W^{∘}(δ) a diagram automorphism, and preserves the function l̃.

Proof: By definition of ẇ_{1}, we know that w_{1}F normalizes Γ^{+}, so, as C(δ) = {w∈W (δ) | wΓ^{+} = Γ^{+}} we see that w_{1}F normalizes C(δ). The other statements are consequences of lemma 3.3 and its proof, and of the definition of l̃ (cf. 3.12).

Definition 3.17. We call Ψ the operator on E(δ) defined by the action of ΦB′_{^{F}δ,ẇ_{1}}F.

Proposition 3.18 (cf. also [S, 1.12.2]).

(i) The conjugation by Ψ stabilizes H(δ) and the automorphism φ which it induces is given by: For any y ∈ W (δ),

φ(T_{y}) = λ_{y}T_{^{w_{1}F}y}.

(ii) The character x ↦ λ_{x} is equal to 1 on W^{∘}(δ).

Proof: Let us first prove that for any ẏ ∈ W (δ), we have

(ΦB′_{^{F}(ẏδ),ẇ_{1}}F)B′_{δ,ẏ}(ΦB′_{^{F}δ,ẇ_{1}}F)^{−1} = B′_{δ,^{w_{1}F} ẏ}  .

We know by 3.9,(iii) that F B′_{δ,ẏ}F^{−1} = B′_{^{F}δ,^{F} ẏ} (we use here the fact that l(^{F}y) = l(y), which is true because F stabilizes the F′-orbits in Π). We then apply 3.11 and [HL, 4.2] to get:

B′_{^{F}(ẏδ),ẇ_{1}}B′_{^{F}δ,^{F} ẏ} = B′_{^{w_{1}F}δ,^{w_{1}F} ẏ}B′_{^{F}δ,^{F} ẏ}

(We apply 3.11 to get the right hand side and Howlett-Lehrer’s result to get the left hand side). We then get the result by 3.8 (iii) and 3.9 (iv).

We see, from 3.9 (i), (ii), 3.14, 3.16 and the preceding assertion that the conjugate of T_{y} by Ψ is equal to λ_{y}T_{^{w_{1}F}y}, whence (i). Now, if we apply φ to the relation 3.13 (ii), we see that λ_{v(a,J)} must be 1 if l̃(v(a, J)) is not 0, in particular if a is in Δ. As the set {v(a, J) | a ∈ Δ} is a set of generators for W^{∘}(δ), we get (ii).

We give now a condition on δ to be extendible to L̃ ><<w_{1}F >.

Proposition 3.19. Let D be the abelianized of C(δ). The set of linear characters of C(δ) identifies to Irr(D) and:

(i) The restriction μ of λ to D^{w_{1}F} does not depend on the extension δ̃ satisfying 3.13. (ii) There exists an extension of δ to L̃ ><<w_{1}F > (and satisfying 3.13) if and only if the

character μ of D^{w_{1}F} defined in (i) is trivial.

Proof: Let δ̃ and ζδ̃ be two extensions of δ satisfying 3.13 where ζ is a linear character of C(δ). Then if the image of c in D is in D^{w_{1}F} we have ζ(c) = ζ(c^{w_{1}F}) so:

Φ.(ζδ̃)(ċ).Φ^{−1} = ζ(c)Φδ̃(ċ)Φ^{−1} = λ_{c}ζ(c)δ̃(^{ẇ_{1}F} ċ) = λ_{c}(ζδ̃)(^{ẇ_{1}F} ċ),

whence assertion (i). Let us prove (ii). The condition of the proposition is clearly necessary. On the other hand the sequence:

0 → Irr(D)^{w_{1}F} → Irr(D) →^{L} Irr(D/D^{w_{1}F}) → 0

is exact where L is the map ζ ↦ ζ.^{w_{1}F}ζ^{−1} (surjectivity of L can be seen by a counting argument). So, if the character λ associated to an extension δ̃ of δ is trivial on D^{w_{1}F}, it can be written ε.^{w_{1}F}ε^{−1} for some character ε of C(δ). Then the extension εδ̃ is clearly w_{1}F invariant, whence the result.

We want now to compute the order of Ψ. For this we need:

Proposition 3.20 (cf. also [S, 1.12.3]). For any integer i ≥ 1,

Ψ^{i} = Φ^{i}B′_{^{F}δ,ẇ_{1}.^{F} ẇ_{1}...^{F^{i−1}}ẇ_{1}}F^{i}

Proof: By 3.9 (iii) and (iv), the operator Ψ^{i} is the action of

Φ^{i}B′_{^{F (w_{1}F)^{i−1}}δ,ẇ_{1}} … B′_{^{F^{i−1}(w_{1}F)}δ,^{F^{i−2}}ẇ_{1}}.B′_{^{F^{i}}δ,^{F^{i−1}}ẇ_{1}}F^{i}  (1).

To compute this product, we use:

Lemma 3.21. Let w be an element of W such that ^{w}J ⊂ Π and ^{w}Δ(δ) ⊂ Δ(^{w}δ); let w′ be an element of W such that ^{w′w}J ⊂ Π, then:

B′_{ẇδ, ˙w′}.B′_{δ,ẇ} = B′_{δ, ˙w′ẇ}.

Proof: We write a standard decomposition of w′ (cf. [HL, 2.11]):

ẇ′ = v̇(a_{n}, K_{n}) … v̇(a_{1}, K_{1}),

where K_{1} = ^{w}J, and where the lengths in the Coxeter group W^{F′} add. Then we have:

B′_{^{w}δ, ˙w′} = ∏_{i=n}^{i=1} B′_{δ_{i},v̇(a_{i},K_{i})},

where δ_{i} stands for ^{v̇(a_{i},K_{i})}δ_{i+1}, and δ_{n} is ẇδ.

Now we prove by induction on m that

∏_{i=m}^{i=1} B′_{δ_{i},v̇(a_{i},K_{i})}B′_{δ,ẇ} = B_{δ,v(a_{m},K_{m})...v(a_{1},K_{1})ẇ}  .

In fact, if a is a positive root (i.e. in Σ^{+}), then the root w^{−1}(a) is either positive or not in Γ(δ), because w(Γ^{−}(δ)) = Γ^{−}(^{w}δ) ⊂ Σ^{−}. But, by [HL, 2.11], we have:

v(a_{1}, K_{1})^{−1}v(a_{2}, K_{2})^{−1} … v(a_{i}, K_{i})^{−1}(a_{i+1}) ∈ Σ^{+}.

Let us call this root a; then we have either w^{−1}(a) ∈ Σ^{+} or w^{−1}(a) ∉ Γ(δ). So, we may apply [HL, 4.1] to get:

B′_{δ_{j+1},v̇(a_{j+1},K_{j+1})}B′_{δ,v̇(a_{m},K_{m})...v̇(a_{1},K_{1})ẇ} = B′_{δ,v̇(a_{j+1},K_{j+1})...v̇(a_{1},K_{1})ẇ}.

This proves the result.

To get proposition 3.20, we have now to check the hypothesis of lemma 3.21 at each step of the product (1), i.e.

(^{F^{j}}w_{1}^{F^{j+1}}w_{1} …^{F^{i−1}}w_{1})^{F^{i}}J ⊂ Π  (2)

and

(^{F^{j}}w_{1}^{F^{j+1}}w_{1} …^{F^{i−1}}w_{1})Δ(^{F^{i}}δ) = Δ(^{F^{j}w_{1}...F_{i−1}w_{1}F^{i}}δ)  (3)

(we apply 3.21 with ^{j}δ and ^{j}J in place of δ and J respectively). First we remark that the element (^{F^{j}}w_{1}^{F^{j+1}}w_{1} …^{F^{i−1}}w_{1})F^{i} is equal to F^{j}(w_{1}F)^{i−j}. But, as w_{1}F normalizes J, and F normalizes Π, we get (2). The same remark shows that the left hand side and the right hand side of (3) are equal to ^{F^{j}}Δ(δ) = Δ(^{F^{j}}δ).

We now come to our goal which is to express the right hand side of 2.5.

Proposition 3.22. Let ẇ be an F′-fixed representative of an element w ∈ Z(δ), and let μ̃ be the ẇF -class function on L^{F′} defined by μ̃(lẇF) = Trace(lẇẇ_{1}^{−1}Φ | δ̃). Then:

R_{L^{ẇF} /F′}^{G^{F} /F′}(Sh_{F′/ẇF} μ̃)((N_{F′/F} g)F′) = \frac{|U^{F′}|^{2}}{|U^{F′} ∩ ^{ẇF}U^{F′}|} Trace(gδ̃(ẇẇ_{1}^{−1})ΦB_{^{F}δ,ẇ}F | E(δ)) .

Proof: We consider the projector p from E to E(δ) defined by:

p(f)(x) = |P^{F′}|^{−1} ∑_{l∈L^{F′}} ∑_{u∈U^{F′}} δ(l)f(l^{−1}ux),

where f∈E and x ∈ G^{F′}.

Then, we have:

Trace(gδ̃(ẇẇ_{1}^{−1})ΦB_{^{F}δ,ẇ}F | E(δ)) = Trace(gδ̃(ẇẇ_{1}^{−1})ΦB_{^{F}δ,ẇ}F p | E).

Let f be an element of E; for x ∈ G^{F′}, we have:

(gδ̃(ẇẇ_{1}^{−1})ΦB_{^{F}δ,ẇ}F pf)(x) = δ̃(ẇẇ_{1}^{−1})Φ|U_{F J}^{F′}|^{−1} ∑_{v∈U_{F J}^{F′}}(F pf)(ẇ^{−1}vxg)

= |P^{F′}|^{−1}|U_{F J}^{F′}|^{−1} ∑_{l∈L^{F′}} ∑_{u∈U^{F′}} ∑_{v∈U_{wF J}^{F′}} δ̃(ẇẇ_{1}^{−1})Φδ(l)f(l^{−1}.u.^{F^{−1}}(ẇ^{−1}vxg)).

We take for f the element f_{x,i} of the basis {f_{x,i}}_{x∈G^{F′} ,i∈I} (where {e_{i}}_{i∈I} is a basis of [δ]) defined at y ∈ G^{F′} by 0 if y ≠ x and e_{i} if y = x. Then the coefficient of

gδ̃(ẇẇ_{1}^{−1})ΦB_{^{F}δ,ẇ}F pf_{x,i}

on f_{x,i} is the coefficient on e_{i} of

|P^{F′}|^{−1}|U_{wF J}^{F′}|^{−1} ∑_{l∈L^{F′}} δ̃(ẇẇ_{1}^{−1})e_{i}Φδ(l)#{(u, v) | l^{−1}.u.^{F^{−1}}(ẇ^{−1}vxg) = x}  .

As we have wF J = J, the trace we want to compute is equal to:

|U^{F′}|^{−1}|P^{F′}|^{−1} ∑_{l∈L^{F′}} Trace(lẇẇ_{1}^{−1}Φ | [δ])
#{x ∈ G^{F′}, u ∈ U^{F′}, v ∈ U^{F′} | l^{−1}.u.^{F^{−1}}(ẇ^{−1}vxg) = x}.

But the cardinality of the set appearing in this last expression is equal to

#{x ∈ G^{F′}, v ∈ U^{F′} | x.g.^{F}x^{−1} ∈ v^{−1}.ẇ.^{F}U^{F′}.^{F}l}
= |U^{F′} ∩ ^{ẇF}U^{F′}|#{x ∈ G^{F′} | x.g.^{F}x^{−1} ∈ U^{F′}.ẇ.^{F}U^{F′}.^{F}l}

= |U^{F′} ∩ ^{ẇF}U^{F′}|#{x ∈ G^{F′} | x.g.^{F}x^{−1} ∈ ^{ẇF}l.U^{F′}.ẇ.^{F}U^{F′}}.

So, if we put l_{1} = ^{ẇF}l, we get for the trace we compute, exactly the right hand side of 2.5 multiplied by the factor |U^{F′} ∩ ^{ẇF}U^{F′}||U^{F′}|^{−2}, whence the result.

We shall now state the main theorem of this paragraph:

Notation 3.23. We consider the semi-direct product of the algebra H(δ) by an infinite cyclic group whose generator acts by φ. We call T_{w_{1}F} the generator of this cyclic group.

Theorem 3.24. Let μ̃ be as in 3.22 and let y be the element ww_{1}^{−1} of W (δ). For any extension χ̃ to H(δ) ><<T_{w_{1}F} > of a T_{w_{1}F} -invariant character χ of H(δ), there is an extension ρχ̃ to G^{F′} ><<F > of the character ρ_{χ} of G^{F′} associated to χ such that:

R_{L^{ẇF} /F′}^{G^{F} /F′}(Sh_{F′/ẇF} μ̃) = q_{F′}^{l(w_{∘}^{G})−l(w_{∘}^{L})+l(w)/2−l̃(y)/2} ∑_{χ̃∈Irr(H(δ))^{T_{w_{1}F}}} χ̃(T_{y}T_{w_{1}F}) Sh_{F′/F} ρχ̃

where the expression χ̃ ∈ Irr(H(δ))^{T_{w_{1}F}} means that we choose one extension χ̃ of each T_{w_{1}F} -invariant character χ of H(δ), and where w_{∘}^{G} (resp. w_{∘}^{L}) is the longest element in the Coxeter group W_{G}^{F′} (resp. W_{L}^{F′}). In addition, if <T_{w_{1}F} > acts via a finite quotient on [χ̃], then <F > acts via a finite quotient on [ρχ̃].

Proof: Let ẏ be an F′-fixed representative of y = ww_{1}^{−1}. By remark 2.6, the trace which appears in proposition 2.2 does not depend on the representative ẇ of w. So we can take ẇ = ẏẇ_{1}. By 3.11 the expression δ̃(ẏ)ΦB_{^{F}δ,ẏẇ_{1}} is equal to δ̃(ẏ)ΦB_{^{w_{1}F}δ,ẏ}B_{^{F}δ,ẇ_{1}}, and, by 3.9 (iv), this is equal to:

δ̃(ẏ)B_{δ,ẏ}ΦB_{^{F}δ,ẇ_{1}}.

So the trace which appears in proposition 3.22 is equal to:

q_{F′}^{1/2(−l̃(y)−l(y)−l(w_{1}))} Trace(gT_{y}Ψ | E(δ)).

Now, we have the following decomposition of E(δ):

E(δ) = ⨁_{χ∈Irr(H(δ))}[χ] ⊗ [ρ_{χ}],

where ρ_{χ} is the irreducible character of G^{F′} associated to the character χ of the endomorphism algebra H(δ). As the operator Ψ commutes with H(δ) in the same way as T_{w_{1}F} (cf. 3.18), the operator gT_{y}Ψ induces on the summands of E(δ) the same permutation as T_{w_{1}F} induces on Irr(H(δ)). So to compute its trace, we have only to consider summands [χ] ⊗ [ρ_{χ}] where χ is a T_{w_{1}F} -invariant character of H(δ). Let χ̃ be an arbitrarily chosen extension of such a χ to H(δ) ><<T_{w_{1}F} >. Then, as T_{w_{1}F} Ψ^{−1} commutes with H(δ), the action of Ψ on [χ]⊗[ρ_{χ}] is of the form Ψ′ ⊗Ψ′′, where Ψ′ is the action of T_{w_{1}F} in the chosen extension of χ. So we get a representation ρχ̃ of G^{F′} ><<F > extending the representation of character ρ_{χ} if we make F act by Ψ′′.

We then have:

Trace(gδ̃(ẇẇ_{1}^{−1})ΦB_{^{F}δ,ẇ}F | E(δ)) = qF′^{\frac{−l̃(y)−l(y)−l(w_{1})}{2}} ∑_{χ̃∈Irr(H(δ))^{T_{w_{1}F}}} χ̃(T_{y}T_{w_{1}F})ρχ̃(gF),

whence the formula 3.24, because |U^{F′} ∩ ẇU^{F′}| = q_{F′}^{l(w_{∘}^{G})−l(w)−l(w_{∘}^{L})} and l(yw_{1}) = l(y) + l(w_{1}). In addition, by 3.20, we know that Ψ is of finite order (by 3.7 and 3.20 it is equal to 1 for any i such that (w_{1}F)^{i} is equal to F^{i} and acts trivially on G^{F′}). So if Ψ′ is of finite order, so is Ψ′′.

4. Further properties of endomorphism algebras; case of a torus. We first state some assumptions which will be needed in the following paragraphs when applying §3.

Assumptions 4.1.

(i) The representation δ has an extension δ̃ to the group L̃ (cf. 3.1). (ii) The function l̃defined in 3.12 takes integral values, and this function depends only on

the action of F′ as a diagram automorphism of the Coxeter group W′(δ) = <v(a, J) ∈ W (δ) | v(a, J)^{2} = 1>. This has been proved by Lusztig when the center of G is connected, cf. [L, 8.6].

(iii) The group C(δ) is abelian.

Proposition 4.2. When L is a maximally split torus assumptions 4.1 hold. (We conjecture assumptions 4.1 actually always hold).

We shall denote T for L in this case. The proof is slightly involved because we do not assume F′ to be split. The character δ of §3 is here a character of T^{F′}. Proof of 4.1(iii): In the present case, C(δ) is an abelian p′-group by [HK, 2.25(iii)].

Proof of 4.1(ii): As in [BT, §6] we consider a maximal split subtorus S of T (so S = T if F′ is split). The group W^{F′} identifies with the relative Weyl group of G with respect to S. We denote by Σ_{F′} the set of relative roots, which correspond with orbits of F′ in Σ. The group W′(δ) is generated by reflections relative to the roots in Γ′ = {a ∈ Σ_{F′} | r_{a} ∈ W (δ)} (in our case v(a, ∅) is the reflection r_{a} relative to the root a). The parameters p_{a} (cf. [HL, 3.21 and 3.19] for the definition of p_{a}) can be computed in the group <T, U_{α}, U_{−α}>_{α∈a}; this group is a restriction of scalars from IF_{q} to IF_{q^{F′}} of a group of type A_{1} or ^{2}A_{2} (where q is some power of q_{F′}); it is easy to prove that p_{a} = q if this group is of type A_{1}(q) and p_{a} = q^{3} in the other case, so in any case p_{a} will be a power of q_{F′}. We then get 4.1 (ii) since l̃ is 0 on the canonical complement (see [HL, 2.2 and 4.10]) of W′(δ) in W (δ). Proof of 4.1(i): Note in the proof above that in any case p_{a} ≠ 1 so W^{∘}(δ) is generated by reflexions relative to the roots Γ = {a ∈ Σ_{F′} | ^{r_{a}}δ = δ}. For a ∈ Γ let T_{a} ⊂ S be the one-parameter subtorus corresponding to the root a. We will first show that the restriction of δ to ∏_{a∈Γ} T_{a} is trivial. Let us recall how the restriction of δ to S^{F′} corresponds to an element of the kernel of F′ − 1 : X(S) ⊗ Q/ZZ → X(S) ⊗ Q/ZZ. If \overline{L/n} ∈ X(S) ⊗ Q/ZZ is the image of L ⊗ 1/n ∈ X(S) ⊗ Q and if (F′ − 1)\overline{L/n} = 0 then (F′ − 1)L/n is an element of X(S); we can choose L and n such that the restriction of (F′ − 1)L/n to S^{F′} coincides with δ. To compute δ on T_{a}^{F′} we need to evaluate δ(ǎ(ζ)) where ζ is a generator of IF _{q_{F′}}^{×}; since ζ corresponds to the element \frac{1}{q_{F′} −1} ∈ Q/ZZ this corresponds to the scalar product ⟨ (F − 1)L/n, \frac{ǎ}{q_{F′} −1} ⟩ which is equal to ⟨ L/n, ǎ ⟩ since S is split. But this last scalar product is equal to 0 ∈ Q/ZZ since:

r_{a}δ = δ ⟺ ^{r_{a}}\overline{L/n} = \overline{L/n} ⟺ ^{r_{a}}L/n − L/n ∈ X(S) ⟺ ⟨ L/n, ǎ ⟩ ∈ ZZ.

If we choose Tits representatives for elements of W^{∘}(δ) (i.e. we choose a representative for r_{a} in the rank 1 semi-simple subgroup G_{a} corresponding to a), then <ẇ | w ∈ W^{∘}(δ)>∩ T ⊂ ∏_{a∈Γ} T_{a}, so we can extend δ to <T^{F′}, ẇ | w∈W^{∘}(δ)> by putting δ̃(ẇ) = 1. This extension is C(δ)-invariant; indeed, for c∈C(δ) we have ^{c}G_{a} = G_{^{c}a} so C(δ) normalizes the group <∏_{a∈Γ} T_{a}, ẇ | w ∈ W^{∘}(δ)> and so ^{c}ẇ(^{c}ẇ)^{−1} ∈ ∏_{a∈Γ} T_{a} so

δ(^{c}ẇ) = δ( ˙\overline{^{c}w}) = 1 = δ(ẇ).

Since C(δ) is an abelian group and δ̃ is 1-dimensional we can then extend δ̃ to T̃.

Remark 4.3. From the proof of 4.1(ii) above we see that when F′ is split l̃ is the usual length function on W^{∘}(δ), extended by 0 to C.

Proposition 4.4. Assume that there exists an extension δ̃ of δ to L̃ such that the character λ defined in 3.14 is trivial (or equivalently δ̃ is ẇ_{1}F -invariant). Then (i) there exists an extension of δ̃ to L̃ ><<w_{1}F > whose restriction to L̃ satisfies 3.13; (ii) let us again note δ̃ the extension to L̃ ><<w_{1}F > defined in (i); if we take Φ = δ̃(ẇ_{1}F)

in 3.22, then for any ẇ the function μ̃ of 3.22 coincides with the restriction of δ̃ to L^{F′}.ẇF.

Proof: We need to check that, if ε_{w} ≠ 1, where ε is as in the proof of 3.13, and if we change δ̃ as in that proof, then λ is still trivial. But then λ_{w} becomes ε_{^{w_{1}F}w}ε_{w}^{−1}, so is concentrated on W^{∘}(δ). As, by 3.18 (ii), λ is a character of C(δ), it is still equal to 1.

Remark 4.5. The assumption of 4.4 does not necessarily hold, even if L is a torus and F and F′ are split.

Counter-example to the hypothesis of 4.4: We show that the condition of proposition 3.19 (ii) does not necessarily hold. Let us take G = Sl_{2} and let F_{∘} be the usual Frobenius endomorphism such that G^{F} = Sl_{2}(IF_{q}) where q ≡ 3(mod 4). Let us take

F = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) F_{∘} and let T be the diagonal torus and δ be the character of T^{F′} corre-

sponding to s = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ∈ T^{∗} ⊂ PGl_{2}. Then δ is F -invariant and C(δ) = \{1, c\} where

we can take ċ = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ). The condition of 3.19 (ii) does not hold since ^{F}c = c but

λ_{c} = δ(ċ^{F} ċ^{−1}) = −1.

5. Left hand side of 2.5. We want now to transform the left hand side of 2.5 using results on the eigenvalues of F′ when F′ is split. First we have to generalize results of [L, chapters 1–3] to the non connected center case.

As in §4 and in [L] we consider L ∈ X(T) and \overline{L/n}, the image of L ⊗ 1/n in X(T) ⊗ (Q/ZZ)_{p′}; if T^{∗} is a torus dual to T, the element \overline{L/n} corresponds to some semi-simple element s ∈ T^{∗}. We put Z_{F} (s) = {z∈W (T) | ^{zF}s = s} (note that the action of zF on T^{∗} is well defined since any isogeny dual to zF has the same action on T^{∗}). Let F_{∘} be a split Frobenius on G defining a rational structure over IF_{p}. Given a F_{∘}-stable representative ẇ of w ∈ Z_{F} (s) Lusztig defines a sheaf F_{ẇ,s} on the Deligne-Lusztig variety X_{w}. Note that our notations correspond to Lusztig’s as follows: our F is Lusztig’s F′ and our F_{∘} is Lusztig’s F; Z_{F} (s) is the image by F of the set called Z_{L,n} in [L, 2.1.1], so that whenever we write wF Lusztig writes F′w; F_{ẇ,s} corresponds to Lusztig’s F_{ẇ,n}^{L} [L, 2.3].

Definition 5.0. We define:

E(G^{F} , s) = {ρ ∈ Irr(G^{F}) | ρ occurs in H_{c}^{∗}(X_{w}, F_{ẇ,s}) for some w∈Z_{F} (s)}

and we will denote by C(G^{F} , s) the \overline{Q}_{l}-span of E(G^{F} , s).

According to [DL, 1.19] this definition depends neither on the representative ẇ of w chosen nor on F_{∘}.

If F′ is split there exists a split Frobenius F_{∘} over IF_{p} such that F′ = F_{∘}^{m} for some m. If F′s = s then to the element s corresponds a character θ of T^{F′} (see [L, 2.9.5]; θ corresponds to Lusztig’s θ_{∘}). Lusztig defines W_{L,n} = {w ∈ W (T) | ^{w}L/n−L/n ∈ X(T)_{ad}}. This group is generated by {r_{a} | ^{r_{a}}L/n − L/n∈X(T)}, so according to §4 we have W_{L,n} = W^{∘}(θ).

It is known that the Weyl group of Z_{G^{∗}}(s) is isomorphic to W (θ) and the Weyl group of Z_{G^{∗}}(s)^{∘} to W_{L,n} = W^{∘}(θ). The set Z_{F} (s) is a coset modulo W (θ).

As in [L, 1.23] we choose a coherent lifting of each W^{∘}(θ)-coset in Z_{F} (s). We also fix an element w_{1}∈Z_{F} (s) of minimal length. Thus W^{∘}(θ)-cosets in Z_{F} (s) have the form W^{∘}(θ)cw_{1} where c ∈ C(θ). We define a lifting ċ of each c∈C(θ) by ċẇ_{1} = ˙ \overline{cw_{1}}.

We can apply §3 to End_{G^{F′}} (R_{T^{F′}}^{G^{F′}} (θ)); according to §4 assumptions 4.1 hold and if F′ is split our function l̃ becomes the usual length function on the Coxeter group W^{∘}(θ), extended to W (θ) by 0 on C(θ).

We have the following generalization of [L, 2.17]:

Lemma 5.1. Suppose F′ is split and ^{F′}s = s; then the trace of F′ on [IH^{∗}(\overline{X}_{w}, F_{ẇ,s}) ⊗ IH^{∗}(\overline{X}′_{w}, F_{˙w′,s^{−1}})]^{G^{F}} is equal to the trace on End_{G^{F′}} (R_{T^{F′}}^{G^{F′}} (θ)) of the operator defined by:

T_{y} ↦ q_{F′}^{(l(w)+l(w′)−l̃(z)−l̃(z′))/2} ∑_{v′∈W^{∘}(θ)} P_{v′,z′}(q_{F′})T_{(v′c′)^{−1}}T_{(w_{1}F)^{−1}(y)} ∑_{v∈W^{∘}(θ)} P_{v,z}(q_{F′})T_{vc}

where w = zcw_{1} and w′ = z′c′w_{1} with z, z′ ∈ W^{∘}(θ) and c, c′ ∈ C(θ)

(here y ∈ W (θ) and P_{v,z} and P_{v′,z′} denote the Kazhdan-Lusztig polynomials defined with respect to the Coxeter group W^{∘}(θ)). Proof: We follow the proof of [L]. We use [L, 2.8 and 2.11] which don’t assume the center to be connected. Then we need an analogue of [L, 2.16], i.e. we have to consider the operator on H(θ) given by h ↦ F^{−1}τ_{}_{˙w^{′−1}}h τẇF where the operator τẇ corresponds to the operator defined in [L, 2.9] and w, w′ ∈ Z_{F} (s). An easy computation shows that: (τẇ)|_{E(^{F}θ)} = q_{F′}^{l(w)/2} B′_{^{F}θ,ẇ}; also note that, as θ is 1-dimensional, we can take Φ = 1. Putting w = zw_{1} and w′ = z′w′_{1} with z, z′ ∈ W (θ), we have:

F^{−1}τ_{}_{˙w^{′−1}}T_{y}τẇF = q_{F′}^{(l(w)+l(w′))/2}F^{−1}B′_{θ, ˙w^{′−1}}T_{y}B′_{^{F}θ,w}F
= q_{F′}^{(l(w)+l(w′))/2}F^{−1}B′_{θ,w_{1}^{−1}}B′_{θ,z′^{−1}}T_{y}B′_{θ,z}B′_{^{F}θ,w_{1}}F

(by 3.11 and [HL, 4.2])

= q_{F′}^{(l(w)+l(w′))/2}F^{−1}B_{^{F}θ,w_{1}}^{′−1}B′_{θ,z′^{−1}}T_{y}B′_{θ,z}B′_{^{F}θ,w_{1}}F

(since 3.21 applied with w_{1} for w′ and w_{1}^{−1} for w gives B_{^{F}θ,w_{1}}^{′−1} = B′_{θ,w_{1}^{−1}})

= q_{F′}^{(l(w)+l(w′)−l̃(z)−l̃(z′))/2}T_{(w_{1}F)^{−1}(z′)^{−1}}T_{(w_{1}F)^{−1}(y)}T_{(w_{1}F)^{−1}(z)}

(by 3.18 and the definition of T_{w}).

The last expression is the analogue of [L, 2.16], from which as in [L] we can deduce 5.1 (note that P_{v,z} = P_{(w_{1}F)(v),(w_{1}F)(z)}).

So in our case [L, 2.18] is still true and we get the following analogue of 2.20:

Proposition 5.2. Suppose F′ is split and ^{F′}s = s;

(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 quo-

tient, there exists an algebraic number λF′ ; ρ̃ ∈ \overline{Q}_{l} all of whose complex conjugates have absolute value 1 such that if ρ is a component of IH^{∗}(\overline{X}_{w}, F_{ẇ,s}) then the isotypic component IH_{ρ}^{∗} 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′ by (λF′ ; ρ̃)^{−1}q_{F′}^{−i/2} on IH_{ρ}^{i}.

As in [S, 1.4] we deduce the analogous result for H_{c}^{∗}(X_{w}, F_{ẇ,s}) (the result is the same as 5.2 excepted that we do not know the exact power of q_{F′} involved in the eigenvalues of F′).

Definition 5.3.

(i) We note C(G^{F} /F′, s) the subspace of C(G^{F} /F′) generated by the restrictions to G^{F} .F′ of characters of G^{F} ><<F′> whose restriction to G^{F} is in E(G^{F} , s).

(ii) We note E(G^{F} /F′, s) a basis of C(G^{F} /F′) given by the choice of an extension of each irreducible F′-stable character in E(G^{F} , s). Note that when F′ is split and ^{F′}s = s, then by 5.2(i) all characters in E(G^{F} , s) are

F′-stable. (iii) If F′ is split and ^{F′}s = s, we define ψ_{G,F′} : C(G^{F} /F′, s) → C(G^{F}) by linearity,

its value on a basis being given by the formula: ψ_{G,F′}(ρ̃|_{G^{F} .F′}) = (λF′ ; ρ̃)^{−1}ρ for ρ̃ ∈ E(G^{F} /F′, s), where λF′ ; ρ̃ is as in 5.2.

We can now apply the preceding results to the left hand side of 2.5. We use the notations of §2, but we assume that the element w of §2 is in Z_{F} (s) (so s ∈ L^{∗} and wFs = s).

Then we have:

Proposition 5.4. Let f ∈ C(L^{ẇF} /F′, s), then R_{L^{ẇF} /F′}^{G^{F} /F′}f ∈ C(G^{F} /F′, s) and if F′ is split and fixes s:

R_{L^{ẇF} /F′}^{G^{F} /F′}f = ∑_{ρ̃∈E(G^{F} /F′,s)} λF′ ; ρ̃ ρ̃ ∑_{h∈IN} ⟨ (H_{c}^{∗}(S) ⊗ ψ_{L,F′}f)^{L^{ẇF} ,(h)}, ρ ⟩q_{F′}^{h/2}

Proof: Let π be an irreducible representation in E(L^{ẇF} , s) and let w′ ∈ W_{L} be such that π is a component of H_{c}^{∗}(X_{w′,ẇF}^{L} , F_{˙w′,s}^{L}). Let V be an irreducible L^{ẇF} -submodule of type π stabilized by F′ in H_{c}^{∗}(X_{w′,ẇF}^{L} , F_{˙w′,s}^{L}). Then we have the following inclusions of G^{F} ><<F′>-modules:

(H_{c}^{∗}(S) ⊗ V)^{L^{ẇF}} ↪ (H_{c}^{∗}(S) ⊗ H_{c}^{∗}(X_{w′,ẇF}^{L} , F_{˙w′,s}^{L}))^{L^{ẇF}} ˜→H_{c}^{∗}(X_{ww′}^{G}, F_{ẇ ˙w′,s}^{G})

the last isomorphism coming from the transitivity of R_{L}^{G}. This proves the first part of the proposition. Now, if F′ is split and fixes s, by 5.2 the last module has a (G^{F} , F′)-stable filtration whose quotients have the form ρ̃⊗ E where F′ acts by ρ̃(F′) ⊗ λF′ ; ρ̃q_{F′}^{h/2} for some h ∈ IN. So the same must be true for (H_{c}^{∗}(S) ⊗ V)^{L^{ẇF}}. Therefore:

Trace(gF′ | (H_{c}^{∗}(S) ⊗ V)^{L^{ẇF}}) = ∑_{ρ̃∈E(G^{F} /F′,s)} λF′ ; ρ̃ ρ̃(gF′) ∑_{h∈IN} q_{F′}^{h/2}⟨ (H_{c}^{∗}(S) ⊗ V)^{L^{ẇF} ,(h)}, ρ ⟩

As, by 5.2, F′ acts on V by π̃(F′)λF′ ; π̃q_{F′}^{h′/2}, we get:

Trace(gF′ | (H_{c}^{∗}(S) ⊗ π̃)^{L^{ẇF}}) = λF′ ; π̃q_{F′}^{−h′/2} Trace(gF′ | (H_{c}^{∗}(S) ⊗ V)^{L^{ẇF}})

so we get the result for π̃, whence for any f by linearity.

6. Characters of endomorphism algebras. We now study the irreducible characters of algebras which appear in §3 as endomorphism algebras of R_{L^{F′}}^{G^{F′}} (δ). First we consider the following situation:

Let W be a finite Coxeter group, S its set of simple reflections, and D a finite group which acts on W by diagram automorphisms (i.e., the action of an element of D is given by a permutation of S). Here W represents W^{∘}(δ) of §3 and D will represent either C(δ) or C(δ) ><<γ> where <γ> is a finite quotient of <w_{1}F >.

We denote by H the generic algebra H(W, \underline{x}) of type W over an algebraically closed field k; it is a k[\underline{x}]-algebra, where \underline{x} is a set of indeterminates associated to S such that x_{s} = x_{t} if s and t are conjugate under W. We denote by {a_{w}}_{w∈W} the natural basis of H.

We suppose that x_{s} = x_{t} if s and t are conjugate under D; we can then consider the semi-direct product H >< D where D acts on H by ^{d}a_{w} = a_{^{d}w}. We note {a_{v}}_{v∈W >< D} the natural basis of H >< D.

Lemma 6.1. Let χ be an irreducible character of H and let I_{D}(χ) be the subgroup of D normalizing χ; there exists an extension ψ of χ to the algebra H >< I_{D}(χ) and an integer c_{χ} ∈ {1, 2} such that:

ψ(a_{v}) ∈ ZZ[\underline{x}^{1/c_{χ}}] for any v ∈ W >< I_{D}(χ).  (∗)

Proof:

1) If W is irreducible, except if W is of type D_{4} and I_{D}(χ) acts as S_{3}, I_{D}(χ) must act via a cyclic group and we can apply [DM],II,3.4. In the excepted case, as in the proof of [DM],II,3.3 we may identify H >< S_{3} with a generic algebra of type F_{4}, and conclude using the rationality of characters of an algebra of type F_{4} (cf. [DM], II, 3.1).

2) In general, W is of the form W_{1}×…×W_{l} with W_{i} irreducible; then H = H_{1}⊗…⊗H_{l}, χ = χ_{1} ⊗ … ⊗ χ_{l} and I_{D}(χ) permutes the components (W_{i}, χ_{i}). We may clearly assume that the action of I_{D}(χ) on the components has only one orbit. Let I_{i} be the subgroup of I_{D}(χ) normalizing W_{i} (and so χ_{i}). By applying case 1), we see that χ_{i} has an extension to H_{i} >< I_{i} satisfying property (*). We now use the following lemma:

Lemma 6.2. With the above notations, if for each i, the character χ_{i} has an extension to H_{i} >< I_{i} satisfying (*), then χ_{1} ⊗ … ⊗ χ_{l} has an extension to H >< I_{D}(χ) satisfying (*). Proof: Let us choose for each i an element γ_{i} ∈ I_{D}(χ) such that ^{γ_{i}}(W_{1}, χ_{1}) = (W_{i}, χ_{i}) that we shall use to identify (W_{1}, χ_{1}) to (W_{i}, χ_{i}) (we choose γ_{1} = 1). Let χ̃_{1} be an extension of χ_{1} to H_{1} >< I_{1} which satisfies (*) and let χ̃_{i} be the extension of χ_{i} to H_{i} >< I_{i} deduced from χ̃_{1} by “transport de structure” by means of γ_{i}.

We now define a representation of H >< I_{D}(χ) on [χ_{1}]⊗…⊗[χ_{l}]: let γ be any element of I_{D}(χ); then γ induces a permutation σ of the components (W_{1}, χ_{1}), …, (W_{l}, χ_{l}). The element γ_{i}γ_{σ(i)}^{−1}γ lies in I_{i}. We put τ_{i} = χ̃_{i}(γ_{i}γ_{σ(i)}^{−1}γ) and we make γ act by σ ∘(τ_{1} ⊗…⊗τ_{l}) i.e. :

γ(v_{1} ⊗ … ⊗ v_{l}) = τ_{σ^{−1}(1)}(v_{σ^{−1}(1)}) ⊗ … ⊗ τ_{σ^{−1}(l)}(v_{σ^{−1}(l)}) where v_{i} ∈ [χ_{i}]

To check that this formula defines a representation of H >< I_{D}(χ), we note the two following facts:

• Let γ′ be another element of I_{D}(χ) inducing the permutation σ′ and put τ′_{i} = χ̃_{i}(γ_{i}γ_{σ′(i)}^{−1}γ′) ; we have:

\begin{aligned}χ̃_{i}(γ_{i}γ_{σσ′(i)}^{−1}γγ′) & = χ̃_{i}(γ_{i}γ_{σ′(i)}^{−1}γ_{σ′(i)}γ_{σσ′(i)}^{−1}γγ_{σ′(i)}γ_{i}^{−1})χ̃_{i}(γ_{i}γ_{σ′(i)}^{−1}γ′) \\ & = χ̃_{σ′(i)}(γ_{σ′(i)}γ_{σσ′(i)}^{−1}γ)χ̃_{i}(γ_{i}γ_{σ′(i)}^{−1}γ′) \end{aligned}

so the element τ′′_{i} corresponding to γ′′ = γγ′ equals τ_{σ′(i)}τ′_{i} and so the formula for the action of γ′′ is: σσ′ ∘ (τ_{σ′(1)} ⊗ … ⊗ τ_{σ′(l)}) ∘ (τ′_{1} ⊗ … ⊗ τ′_{l}) = σ ∘ (τ_{1} ⊗ … ⊗ τ_{l}) ∘ σ′ ∘ (τ′_{1} ⊗ … ⊗ τ′_{l}) so the formula for the action is compatible with product.

• An easy computation shows that for γ ∈ I_{D}(χ) and w ∈ W, the element γ.a_{w} acts like a_{^{γ}w}.γ. Finally, the values of the character of the extension we have defined are products of

values of the χ̃_{i} so they satisfy (*) if all χ̃_{i} satisfy it.

We now deduce from lemma 6.1:

Theorem 6.3. Suppose that D is abelian; let Φ be an irreducible character of H >< D; then for any x ∈ W >< D we have Φ(a_{x}) ∈ θ(x)Q[\underline{x}^{1/c_{χ}}] where θ is some irreducible character of D, and χ is any irreducible component of the restriction of Φ to H.

Proof: We need two well-known lemmas which hold when D is abelian:

Lemma 6.4. let χ be an irreducible character of H; then

Ind_{H}^{H >< I_{D}(χ)} χ = ∑_{θ∈Irr(I_{D}(χ))} ψ ⊗ θ

where ψ is an irreducible character of H >< I_{D}(χ) such that Res_{H}^{H >< I_{D}(χ)} ψ = χ; in this decomposition ψ ⊗ θ runs over the different extensions of χ to H >< I_{D}(χ).

Lemma 6.5. let ψ be a component of Ind_{H}^{H >< I_{D}(χ)} χ; then

(i) Ind_{H >< I_{D}(χ)}^{H >< D} ψ = ψ̃ is irreducible. (ii) For x ∈ W >< D we have:

ψ̃(a_{x}) = \{ \begin{matrix}0 & if x ∉ W >< I_{D}(χ) \\ ∑_{i=1}^{k} ψ_{i}(a_{x}) & if x ∈ W >< I_{D}(χ) \end{matrix}

where ψ_{1}, …, ψ_{k} are the distinct conjugates of ψ under the action of D. (iii) For any irreducible character Φ of H >< D and any component χ of Res_{H}^{H >< D} Φ, there

is a component ψ of Ind_{H}^{H >< I_{D}(χ)} χ such that Φ = Ind_{H >< I_{D}(χ)}^{H >< D} ψ.

We only prove 6.5(iii) which is perhaps less well-known:

Let χ be an irreducible component of Res_{H}^{H >< D} Φ. By applying Frobenius reciprocity, we see that Res_{H >< I_{D}(χ)}^{H >< D} Φ and Ind_{H}^{H >< I_{D}(χ)} χ must have a common component ψ. But then Ind_{H >< I_{D}(χ)}^{H >< D} ψ = Φ since the left-hand side is an irreducible character by 6.5(i).

We now prove theorem 6.3:

Let χ be an irreducible component of Res_{H}^{H >< D} Φ. By 6.1 there exists an extension ψ_{1} of χ to H >< I_{D}(χ) satisfying (*). By 6.5 (iii) there is another extension ψ such that Φ = Ind_{H >< I_{D}(χ)}^{H >< D} ψ, and by 6.4 ψ = ψ_{1} ⊗ θ for some θ ∈ Irr(I_{D}(χ)), whence the result.

We now suppose that D is of the form C ><<γ> where C is abelian and prove a rationality result for characters of H >< D.

Theorem 6.6. Let Φ be an irreducible γ-invariant character of H >< C; there exists an extension Φ̃ of Φ to H >< D such that for any x ∈ W >< D:

Trace(a_{x} | Φ̃)∈K[\underline{x}^{1/c_{χ}}]

where K is the field generated by the values of the characters of C and the values of the characters of <γ> and where χ is any component of Res_{H}^{H >< C} Φ.

Proof: By 6.4 and 6.5(iii) we get Φ = Ind_{H >< I_{C}(χ)}^{H >< C} ψ ⊗ θ where ψ satisfies (*), where θ is a linear character of I_{C}(χ) and where χ = Res_{H}^{H >< I_{C}(χ)} ψ. Moreover we can choose ψ as follows: by 6.1 we get an extension ψ̃ of χ to H >< I_{D}(χ) satisfying (*), and we take for ψ the restriction of ψ̃ to H >< I_{C}(χ) (note that χ is determined by Φ up to C-conjugacy). We then use:

Lemma 6.7. Let c ∈ C such that cγ ∈ I_{D}(χ), then ^{cγ}θ = θ.

Proof: Since ψ̃ is a character of H >< I_{D}(χ) we have ^{cγ}ψ̃ = ψ̃; the element cγ normalizes I_{C}(χ) since χ is cγ-invariant, so we can take restrictions to H >< I_{C}(χ) and we get ^{cγ}ψ = ψ. But ^{cγ}Φ = Φ since Φ is γ-invariant, so by 6.5(ii) applied with C for D, Φ for ψ̃ and ψθ for ψ we must have ^{cγ}(ψθ) = ^{c′}(ψθ) for some c′ ∈ C. Taking restrictions to H we get cγχ = ^{c′}χ, but ^{cγ}χ = χ by hypothesis, so c′ ∈ I_{C}(χ) and ^{c′}ψ = ψ. So we get ^{cγ}(ψθ) = (ψθ) and comparing with ^{cγ}ψ = ψ we conclude ^{cγ}θ = θ (since ψθ ≠ ψθ′ for θ ≠ θ′ by 6.4).

An element c such as in 6.7 always exists since ^{γ}Φ = Φ so γ permutes the components of the restriction of Φ to H, which are conjugate under C because they are the restrictions of the ψ_{i} of lemma 6.5(ii), so ^{γ}χ = ^{c^{−1}}χ for some c. Thus we have I_{D}(χ) = <I_{C}(χ), cγ>; so from 6.7 we can extend θ to a character θ̃ of I_{D}(χ), and we now define Φ̃ = Ind_{H >< I_{D}(χ)}^{H >< D}(ψ̃θ̃). We claim that Φ̃ satisfies the requirements of 6.6. First we prove:

Proposition 6.8. Res_{H >< C}^{H >< D} Φ̃ = Φ.

Proof: We have

deg(Φ̃) = \frac{|W >< D|}{|W >< I_{D}(χ)|} deg(ψ̃θ̃) = \frac{|D|}{|I_{D}(χ)|} deg(ψ)

but

|I_{D}(χ)| = ∑_{i=1}^{o(γ)} #{a∈C|aγ^{i} ∈ I_{D}(χ)} = ∑_{i=1}^{o(γ)} #{b∈C | b(cγ)^{i} ∈ I_{D}(χ)} = o(γ)|I_{C}(χ)|

where o(γ) is the order of γ; so that

|D|/|I_{D}(χ)| = |C|/|I_{C}(χ)| and deg(Φ̃) = |C|/|I_{C}(χ)| deg(ψ) = deg(Φ).

It is now sufficient to prove that Φ occurs in Res_{H >< C}^{H >< D} Φ̃; but ψ̃θ̃ is a component of Ind_{H >< I_{C}(χ)}^{H >< I_{D}(χ)} ψθ, so Φ̃ = Ind_{H >< I_{D}(χ)}^{H >< D}(ψ̃θ̃) must occur in Ind_{H >< I_{C}(χ)}^{H >< D}(ψθ) = Ind_{H >< C}^{H >< D} Φ, and we conclude by using Frobenius reciprocity.

It remains to prove that for x ∈ W >< D we have Trace(a_{x} | Φ̃)∈K[\underline{x}^{1/c_{χ}}]. As Φ̃ is induced from ψ̃θ̃, we have a direct decomposition:

[Φ̃] = ⨁_{d∈D/I_{D}(χ)} ^{d}[ψ̃θ̃]

If x = w.d′ is an element of W >< D, then a_{x} permutes the spaces ^{d}[ψ̃θ̃], and the trace of a_{x} is the sum of the trace on those spaces which are stabilized by a_{x}. On such a space d[ψ̃θ̃] the trace is ψ̃θ̃(a_{(^{d^{−1}}x)}) (note that ^{d^{−1}}x is in W >< I_{D}(χ)). We conclude the proof of 6.6 since ψ̃ satisfies (*) and θ̃ has values in K.

We now give a formula for scalar products. For v = wc ∈ W >< C, let us put x_{v} = ∏_{j=1}^{r} x_{s_{i_{j}}} where w = s_{i_{1}} … s_{i_{r}} is a reduced decomposition of w. Then:

Theorem 6.9. Let ψ_{1} (resp. ψ_{2}) be an irreducible γ-invariant character of H >< C and ψ̃_{1} (resp ψ̃_{2}) be an extension of ψ_{1} (resp. ψ_{2}) to H >< D; then:

∑_{v∈W >< C} ψ̃_{1}(a_{v}a_{γ})ψ̃_{2}(a_{γ^{−1}}a_{v^{−1}})x_{v}^{−1} = \{ \begin{matrix}0 & if ψ_{1} ≠ ψ_{2} \\ |C| ∑_{w∈W} χ(a_{w})χ(a_{w^{−1}})x_{w}^{−1} & if ψ̃_{1} = ψ̃_{2} \end{matrix}

for any component χ of Res_{H}^{H >< C} ψ_{1}.

Proof: Let λ be the linear form on H >< C defined by:

λ(A) = coefficient of A on a_{1} (a_{1} is the identity element of H)

For this linear form, H >< C is a Frobenius algebra with dual bases {a_{v}}_{v∈W >< C} and {a_{v^{−1}}x_{v}^{−1}}_{v∈W >< C}.

So, applying [DM, II, 2.10] we get:

∑_{v∈W >< C} ψ̃_{1}(a_{v}a_{γ})ψ̃_{2}(a_{γ^{−1}}a_{v^{−1}})x_{v}^{−1} = \{ \begin{matrix}0 & if ψ_{1} ≠ ψ_{2} \\ ∑_{v∈W >< C} ψ_{1}(a_{v})ψ_{1}(a_{v^{−1}})x_{v}^{−1} & if ψ̃_{1} = ψ̃_{2} \end{matrix}

But, by 6.5(iii), ψ_{1} is of the form Ind_{H >< I_{C}(χ)}^{H >< C} ψ for some component ψ of Ind_{H}^{H >< I_{C}(χ)} χ, and by 6.5(ii):

ψ_{1}(a_{v}) = \{ \begin{matrix}0 & if v ∉ W >< I_{C}(χ) \\ ∑_{i=1}^{k} ψ^{(i)}(a_{v}) & if v ∈ W >< I_{C}(χ) \end{matrix}

where ψ^{(i)} are the distinct conjugates of ψ under C. (note that k = \frac{|C|}{|I_{C}(χ)|}). So we have:

\begin{aligned}∑_{v∈W >< C} ψ_{1}(a_{v})ψ_{1}(a_{v^{−1}})x_{v}^{−1} & = ∑_{i,j} ∑_{c∈I_{C}(χ)} ∑_{w∈W} ψ^{(i)}(a_{w}a_{c})ψ^{(j)}(a_{c^{−1}}a_{w^{−1}})x_{w}^{−1} \\ & = ∑_{i} ∑_{c,w} ψ^{(i)}(a_{w}a_{c})ψ^{(i)}(a_{c^{−1}}a_{w^{−1}})x_{w}^{−1} \end{aligned}

(the terms with i ≠ j are equal to 0 since H >< I_{C}(χ) is a Frobenius algebra with dual bases {a_{w}a_{c}} and {a_{c^{−1}}a_{w^{−1}}x_{w}^{−1}}). But, applying [DM, II, 2.10] to H ><<c> for one element c, we get:

∑_{w∈W} ψ^{(i)}(a_{w}a_{c})ψ^{(i)}(a_{c^{−1}}a_{w^{−1}})x_{w}^{−1} = ∑_{w∈W} χ^{(i)}(a_{w})χ^{(i)}(a_{w^{−1}})x_{w}^{−1}

where χ^{(i)} is the restriction of ψ^{(i)} to H (a conjugate of χ under C). We now get the theorem because the right-hand side of the last equation does not depend on i, since by assumption x_{w} and x_{v} are equal if v and w are conjugate under C.

Definition 6.10. We put P_{H >< C} = |C|P_{H} where P_{H} is the Poincaré polynomial of H, and for an irreducible character ψ of H >< C we put d_{ψ} = \frac{ψ(1)}{χ(1)}d_{χ} where χ is any component of Res_{H}^{H >< C} ψ and d_{χ} is the generic degree of χ.

(Note that P_{H >< C} ∈ K[\underline{x}] and d_{ψ} ∈ K(\underline{x}^{1/2})). We now get from the definition of the generic degree and from 6.9 :

Corollary 6.11. Under the assumptions of 6.9, we have:

∑_{v∈W >< C} ψ̃_{1}(a_{v}a_{γ})ψ̃_{1}(a_{γ^{−1}}a_{v^{−1}})x_{v}^{−1} = ψ(1)P_{H >< C}/d_{ψ}

Theorem 6.12 (Deformation theorem). Let \overline{k[\underline{x}]} be the integral closure of k[\underline{x}] in some algebraic closure of k(\underline{x}). Let f be a k-linear homomorphism k[\underline{x}] → k such that the k-algebra H ⊗_{k[\underline{x}]} k (where k is given a k[\underline{x}]-algebra structure by f) is semi-simple, and let \overline{f} : \overline{k[\underline{x}]} → k be an extension of f. Then (H >< D) ⊗_{k[\underline{x}]} k is a semi-simple algebra which has the same numerical invariants as H >< D, and for any irreducible character χ of H >< D the linear map (H >< D) ⊗_{k[\underline{x}]} k → k defined by a_{v}a_{γ} ⊗ 1 ↦ \overline{f}(χ(a_{v}a_{γ})) is an irreducible character of (H >< D) ⊗_{k[\underline{x}]} k. We get thus a bijection from irreducible characters of H >< D to those of (H >< D) ⊗_{k[\underline{x}]} k.

(note that it makes sense to apply \overline{f} to χ(a_{v}a_{γ}) since if χ(a_{v}a_{γ}) ≠ 0 then χ is an extension of a character of H >< C and by 6.6 we have χ(a_{v}a_{γ}) ∈ K[\underline{x}^{1/c_{χ}}] ⊂ \overline{k[\underline{x}]}).

This proposition is proved as the usual deformation theorem (see [DM, 1.3, 1.6 and 1.7]).

7. Main theorem

Proposition 7.1. Let π be an F -invariant irreducible representation of G^{F′} in a (geometric) series E(G^{F′}, s); then

(i) π is a component of R_{L^{F′}}^{G^{F′}} δ for some standard Levi L and some cuspidal representation δ of L^{F′}.

(ii) There exists w∈W^{F′} such that ^{wF}L = L and ^{wF}δ = δ. (iii) Any element of \overline{Z}(δ) has a representative w∈Z(δ).W_{L} such that ^{wF}s = s.

Proof: (i) is a consequence of Harish-Chandra theory, which further asserts that ^{F}L and L must be conjugate by G^{F′} because R_{L^{F′}}^{G^{F′}} δ = R_{^{F}L^{F′}}^{G^{F′}} ^{F}δ (since they have π as a common component); in addition, since T is F -stable, and T ⊂ B where B is F′-stable, the groups FL and L must be conjugate under W^{F′}, so there exists v ∈ W^{F′} such that ^{vF}L = L. Now, again by Harish-Chandra theory, ^{vF}δ and δ must be conjugate under N_{G}(L)^{F′} so there exists a v′ ∈ W^{F′} normalizing L such that ^{v′vF}δ = δ; we get (ii) by putting w = v′v. Let now v be a representative in W^{F′} of an element \overline{v} ∈ Z(δ); the L^{∗}-conjugacy class of s must be vF -stable, so there exists v′ ∈ W_{L} such that ^{v′vF}s = s. We get (iii) by taking v′v instead of v for the representative of \overline{v} (note that v′v is not necessarily F′-stable).

We suppose now that F′ is equal to F_{∘}^{m}, a power of a split Frobenius (in particular W^{F′} = W). We want to consider Shintani descents Sh_{F_{∘}^{m}/F} of non-cuspidal characters of G^{F_{∘}^{m}} and prove that they have a “limit” when m “tends to 0”, assuming this is true for cuspidal characters of proper Levi subgroups of G. We shall now state precisely the induction hypothesis, which will also clarify what we mean by “limit” of Shintani descents.

We fix a representative s ∈ T^{∗} of a given semi-simple geometric conjugacy class. Let d be the smallest positive integer such that ^{F_{∘}^{d}}s = s (d does not depend on the representative s ∈ T^{∗} since any two such representatives are conjugate under W = W^{F_{∘}}). Let L be a standard Levi subgroup of G; we now state the induction hypothesis for L:

Assumptions 7.2.

(i) The cuspidal representations in E(L^{F_{∘}^{d}}, s) are in one-to-one correspondence with the cuspidal representations of E(L^{F_{∘}^{m}}, s) for any m multiple of d. So we can parameterize cuspidal representations in E(L^{F_{∘}^{m}}, s) by “symbols” δ indepen-

dent of m multiple of d. We shall note δ_{m} the representation of E(L^{F_{∘}^{m}}, s) corresponding to a given δ.

(ii) Let F_{1} be a Frobenius on L which stabilizes T and B ∩ L and which commutes with

F_{∘}^{d}. Then δ_{m} is F_{1}-stable if and only if δ_{d} is F_{1}-stable. (iii) If δ_{d} is F_{1}-stable, there exists a multiple d′ of d (equal to d or 2d) such that for any m

multiple of d′, and any extension δ̃_{m} of δ_{m} to L^{F_{∘}^{m}} ><<F_{1}> we have Sh_{F_{∘}^{m}/F_{1}} δ̃_{m} ∈ C(L^{F_{1}}/F_{∘}^{m}, s^{−1}) and:

ψ_{L,F_{∘}^{m}} ∘ Sh_{F_{∘}^{m}/F_{1}} δ̃_{m} = η_{δ,m}^{F_{1}} ∑_{π∈E(L^{F_{1}} ,s^{−1})} c_{π,δ}π

where c_{π,δ} are algebraic numbers independent of m, and where η_{δ,m}^{F_{1}} is a root of unity.

We will denote by Sh_{F_{1}}^{0}(δ̃) the summation appearing in the right-hand side of the above equality. Note that Sh_{F_{1}}^{0}(δ̃) is only defined up to some root of unity (its value depends on how we split the product η_{δ,m}^{F_{1}}c_{π,δ} in η_{δ,m}^{F_{1}} and c_{π,δ}).

We need another series of hypotheses relative to the embedding of L as a standard Levi subgroup of G.

We now fix δ; first note that Z_{F} (δ_{m}) (and hence W (δ_{m})) does not depend on m multiple of d by assumption (ii) above. We shall call them respectively Z_{F} (δ) and W (δ). Note also that W′(δ_{m}) and its root system do not depend on m (this is clear from their definition and the fact that W (δ_{m}) does not depend on m).

Assumptions 7.3.

(i) We assume that assumptions 4.1 hold for any m multiple of d. Note that 4.1 (ii) implies that l̃, hence C(δ_{m}), does not depend on m; we shall note it

C(δ). (ii) We assume that the hypothesis of 4.4 holds for any m multiple of d′, i.e. that there

exists an extension δ̃_{m} of δ_{m} to <L^{F_{∘}^{m}}, W (δ)> ><<w_{1}F >. As in §3, we fix an element w_{1} in Z(δ) such that ^{w_{1}F}Δ^{+} = Δ^{+} (cf. definition above

3.3). We shall note \overline{w}_{1} the image of w_{1} in \overline{Z}(δ).

Let Ż_{F} (s) be a coherent lifting (cf. [L, 1.23]) of Z_{F} (s). By 7.1 (iii) any element of \overline{Z}(δ) has a representative in Z(δ)W_{L} ∩Z_{F} (s). So, for each element \overline{w} ∈ \overline{Z}(δ) we can choose a representative ẇ in Ż_{F} (s). (iii) We assume that with such a choice of ẇ for each \overline{w} ∈ \overline{Z}(δ), we may choose η_{δ,m}^{ẇF}

in 7.2(iii) above (applied with F_{1} = ẇF and with δ̃_{m} being the restriction of δ̃_{m} of (ii) above to L^{F_{∘}^{m}} ><<ẇF >) of the form λ_{δ,m}θẇ ; δ,m where the map \overline{w}F ↦ θẇ ; δ,m is the restriction to \overline{Z}(δ) of a linear character of \overline{W} (δ) ><<\overline{w_{1}F} >. Moreover, we assume that there is a choice of δ̃_{m} in (v) above such that θ_{δ,m} factors through \overline{C}(δ) ><<\overline{w_{1}F} >.

We consider now an F -invariant representation π of G^{F_{∘}^{m}} in a series E(G^{F_{∘}^{m}}, s); according to 7.1, π is a component of R_{L^{F^{m}_{∘}}}^{G^{F^{m}_{∘}}} δ_{m} for some standard Levi subgroup L and some cuspidal representation δ_{m} of L^{F_{∘}^{m}}. We assume that 7.2 and 7.3 hold for L. We will denote by H_{m} the endomorphism algebra of R_{L^{F^{m}_{∘}}}^{G^{F^{m}_{∘}}} (δ_{m}) and by H the corresponding generic algebra which is of the form form H(W^{∘}(δ), \underline{x}) >< C(δ). Since by 7.3(i) we assumed that 4.1 holds, C(δ) is abelian, λ of 3.18 is trivial, and H_{m} is the specialization of H for the function f(x_{s}) = q_{F_{∘}}^{ml̃(s)} (in what follows we note q for q_{F_{∘}}). We choose an extension \overline{f} of f as in 6.12

by setting \overline{f}(x_{s}^{1/2}) = q_{F_{∘}}^{ml̃(s)/2}; if H ><<γ> is a finite quotient of H ><<a_{w_{1}F} >, by 6.12 there is a bijection χ̃ ↦ χ̃_{m} between characters of H ><<γ> and those of H_{m} ><<γ>. This bijection does not depend on the finite quotient considered since \overline{f} is always chosen the same. So there is a bijection χ̃ ↦ χ̃_{m} between characters of H ><<a_{w_{1}F} > where a_{w_{1}F} acts through a finite quotient and those of H_{m} ><<T_{w_{1}F} >. Our representation π is the representation ρ_{χ_{m}} associated to some character χ_{m} ∈ Irr(H_{m})^{T_{w_{1}F}} (cf. proof of 3.24).

Theorem 7.4. Let χ̃ be a character of H ><<a_{w_{1}F} > on a model of which a_{w_{1}F} is of finite order and whose restriction to H is χ, and let ρχ̃_{m} be the extension to G^{F_{∘}^{m}} ><<F > of ρ_{χ_{m}} corresponding to χ̃_{m} (cf. 3.24) (for m multiple of d). Then if m is multiple of d′, we have Sh_{F_{∘}^{m}/F} ρχ̃_{m} ∈ C(G^{F} /F_{∘}^{m}, s^{−1}) and:

λ_{δ,m}ψ_{G,F_{∘}^{m}} ∘ Sh_{F_{∘}^{m}/F} ρ_{χ̃_{m}⊗θ_{δ,m}} = ∑_{ρ∈E(G^{F} ,s^{−1})} c_{ρ,χ̃}ρ

where c_{ρ,χ̃} are algebraic numbers independent of m (note that, as θ_{δ,m} factors through C ><<\overline{w_{1}F} >, it makes sense to tensorize χ̃_{m} with θ_{δ,m} so that χ̃_{m} ⊗θ_{δ,m} is still a character of H_{m} ><<T_{w_{1}F} > whose restriction to H_{m} is irreducible).

We will denote by Sh^{0}(ρχ̃) the right-hand side of the above equality.

Proof: Note first that if the theorem is true for an extension χ̃, it is true for any other extension χ̃′ of χ if a_{w_{1}F} is still of finite order on χ̃′; indeed χ̃′|_{H.a_{w_{1}F}} = λχ̃|_{H.a_{w_{1}F}} for some root of unity λ, and the theorem holds for χ̃′ if we set c_{ρ,χ̃′} = λ^{−1}c_{ρ,χ̃}. Using this remark, we may assume in the following proof that χ̃ is the extension described in theorem 6.6, where <γ> is the finite quotient through which <a_{w_{1}F} > acts. Let w′ be any element of Z(δ). We have w′ = yw_{1} with y ∈ W (δ). As above 7.3(iii) we choose a representative w in Z(δ)W_{L} ∩ Z_{F} (s) of the image \overline{w}′ of w′ in \overline{Z}(δ), and a representative ẇ ∈ Ż_{F} (s) of w. Let ẇ′ be a representative of w′; we can apply 3.24 with ẇ′; but by remark 2.6 the Lusztig functor and Shintani descent F.Digne and J.Michel left-hand side of 3.24 is the same for ẇ and ẇ′ so:

R_{L^{ẇF} /F_{∘}^{m}}^{G^{F} /F_{∘}^{m}} Sh_{F_{∘}^{m}/ẇF} δ̃_{m} =

q^{m(l(w_{∘}^{G})−l(w_{∘}^{L})+l(w′)/2−l̃(y)/2)} ∑_{χ̃∈Irr(H)^{T_{w_{1}F}}} χ̃_{m}(T_{y}T_{w_{1}F})Sh_{F_{∘}^{m}/F} ρχ̃_{m}

By summing over y ∈ W^{∘} >< C = W (δ), and applying 6.11 and 6.12, we get:

Sh_{F_{∘}^{m}/F} ρχ̃_{m} = A ∑_{y∈W (δ)} χ̃_{m}(T_{y^{−1}}T_{(w_{1}F)^{−1}})q^{−m(l(w′)+l̃(y))/2}R_{L^{ẇF} /F_{∘}^{m}}^{G^{F} /F_{∘}^{m}} Sh_{F_{∘}^{m}/ẇF} δ̃_{m} (1)

where A = \frac{d_{χ}(q^{m})}{χ_{∘}(1)P(q^{m})}q^{m(l(w_{∘}^{L})−l(w_{∘}^{G}))}. We now apply 5.4 with Sh_{F_{∘}^{m}/ẇF} δ̃_{m} for f (which is possible since ẇ ∈ Ż_{F} (s) and since, by 7.2(iii), Sh_{F_{∘}^{m}/ẇF} δ̃_{m} ∈ C(L^{ẇF} /F_{∘}^{m}, s)); let us write

λρ̃ for λF_{∘}^{d} ; ρ̃, so λF′ ; ρ̃ is λm/d ; ρ̃; we get:

R_{L^{ẇF} /F_{∘}^{m}}^{G^{F} /F_{∘}^{m}} Sh_{F_{∘}^{m}/ẇF} δ̃_{m} =

∑_{ρ̃∈E(G^{F} /F_{∘}^{m},s^{−1})} λm/d ; ρ̃ρ̃ ∑_{h∈IN} ⟨ (H_{c}^{∗}(S) ⊗ ψ_{L,F_{∘}^{m}} ∘ Sh_{F_{∘}^{m}/ẇF} δ̃_{m})^{L^{ẇF} ,(h)}, ρ ⟩q^{mh/2}

According to 7.2(iii), the last expression can be rewritten as:

∑_{ρ̃∈E(G^{F} /F_{∘}^{m},s^{−1})} λm/d ; ρ̃λ_{δ,m}^{−1}θ_{δ,m}(wF)ρ̃ ∑_{h∈IN} ∑_{π∈E(L^{ẇF} ,s^{−1})} c_{π,δ}⟨ (H_{c}^{∗}(S) ⊗ π)^{L^{ẇF} ,(h)}, ρ ⟩q^{mh/2}

This is of the form ∑_{ρ̃∈E(G^{F} /F_{∘}^{m},s^{−1})} λm/d ; ρ̃λ_{δ,m}^{−1}θ_{δ,m}(wF)ρ̃P_{ρ,δ}^{y}(q^{m/2}) where the P_{ρ,δ}^{y}’s are

polynomials of Q(c_{π,δ})[X]. Let us put Q_{ρ,δ}^{y} = X^{−(l(w′)+l̃(y))}P_{ρ,δ}^{y}.

Replacing in (1) we get:

Sh_{F_{∘}^{m}/F} ρχ̃_{m} = A ∑_{ρ̃∈E(G^{F} /F_{∘}^{m},s^{−1})} λm/d ; ρ̃ρ̃ ∑_{y∈W (δ)} θ_{δ,m}(wF)λ_{δ,m}^{−1}χ̃_{m}(T_{y^{−1}}T_{(w_{1}F)^{−1}})Q_{ρ,δ}^{y}(q^{m/2}).

This proves immediately that Sh_{F_{∘}^{m}/F} ρχ̃_{m} ∈ C(G^{F} /F_{∘}^{m}, s^{−1}). Using now the scalar product of F_{∘}^{m}-class functions (see §1), we get:

λ_{δ,m}\overline{θ}_{δ,m}(wF)⟨ Sh_{F_{∘}^{m}/F} ρχ̃_{m}, ρ̃⟩_{G^{F} .F_{∘}^{m}} = Aλm/d ; ρ̃ ∑_{y∈W (δ)} χ̃_{m}(T_{y^{−1}}T_{(w_{1}F)^{−1}})Q_{ρ,δ}^{y}(q^{m/2}).

We rewrite the last equality as:

λ_{δ,m}⟨ Sh_{F_{∘}^{m}/F} ρ_{χ̃_{m}⊗θ_{δ,m}}, ρ̃⟩_{G^{F} .F_{∘}^{m}} = Aλm/d ; ρ̃ ∑_{y∈W (δ)} χ̃_{m}(T_{y^{−1}}T_{(w_{1}F)^{−1}})Q_{ρ,δ}^{y}(q^{m/2}) (2)

Let K be the extension of Q containing c_{π,δ} and the values of the characters of C and of <w_{1}F >. By 6.6 the right hand side of the last equation is of the form λm/d ; ρ̃f_{ρ,χ̃}(q^{m/2}) where f_{ρ,χ̃} ∈ K(X).

Consider now c_{m} = |G^{F} |λ_{δ,m}⟨ Sh_{F_{∘}^{m}/F} ρ_{χ̃_{m}⊗θ_{δ,m}}, ρ̃⟩_{G^{F} .F_{∘}^{m}}: it is an algebraic integer as a sum of character values times λ_{δ,m}, we have |c_{m}| < |G^{F} | since Sh_{F_{∘}^{m}/F} is an isometry and |λ_{δ,m}| = 1, and we have c_{m} ∈ K′ = K(q^{1/2}, λρ̃) by (2). Furthermore all complex conjugates of c_{m} are of module less than |G^{F} | since for any σ ∈ Gal(K′/Q) we have:

σ(⟨ Sh_{F_{∘}^{m}/F} ρ_{χ̃_{m}⊗θ_{δ,m}}, ρ̃⟩_{G^{F} .F_{∘}^{m}}) = ⟨ Sh_{F_{∘}^{m}/F} ^{σ}ρ_{χ̃_{m}⊗θ_{δ,m}},^{σ}ρ̃⟩_{G^{F} .F_{∘}^{m}}

As there is a finite number of algebraic integers all of which conjugates are of bounded module in a given algebraic number field, c_{m} must be constant for all m in an infinite subset E of multiples of d′. Let m, m′ be two elements of E; we then have:

λ(m−m′)/d ; ρ̃ = \frac{f_{ρ,χ̃}(q^{m′/2})}{f_{ρ,χ̃}(q^{m/2})}

Fixing m′ and letting m vary, the right hand side of this equality can be written \frac{P (q^{m/2})}{R(q^{m/2})}

where P, R ∈ K[X^{1/2}]. For all m ∈ E we have: R(q^{m/2})λ(m−m′)/d ; ρ̃ = P (q^{m/2}). Let aX^{p}

(resp. bX^{r}) be the term of highest degree of P (resp. R); taking equivalents when m tends to infinity we get: aq^{pm/2} ∼ λ(m−m′)/d ; ρ̃bq^{rm/2}. Since |λρ̃| = 1 we must have p = r and λm/d ; ρ̃ constant for large enough m; so λρ̃ is a root of unity. Then we must have P = R, so f_{ρ,χ̃} is a constant that we shall note c_{ρ,χ̃}.

Putting things back in (2), we get for any m multiple of d′:

λ_{δ,m}⟨ Sh_{F_{∘}^{m}/F} ρ_{χ̃_{m}⊗θ_{δ,m}}, ρ̃⟩_{G^{F} .F_{∘}^{m}} = λm/d ; ρ̃c_{ρ,χ̃}

which proves the theorem.

8. Case of a torus We want to prove now that in the case where L is a maximally split torus T, assumptions 7.2 and 7.3 hold taking d′ = 2d (d when the center of G is connected).

For d|m we note δ_{m} the only element of E(T^{F_{∘}^{m}}, s); 7.2(i) and (ii) are trivially true since we have δ_{m} = δ_{d} ∘ N_{F_{∘}^{m}/F_{∘}^{d}}. We put T̃_{m} = <n∈N_{G}(T)^{F_{∘}^{m}} | ^{n}δ_{m} = δ_{m}>.

According to 4.1(i) there exists an extension δ̃_{d} of δ_{d} to T̃_{d}, which we can take such that δ̃_{d}(ẏ) = 1 for the Tits representatives ẏ (which we take in N_{G}(T)^{F_{∘}}) of elements of W^{∘}(δ) which appear in the proof of 4.1(i). So we can choose a coherent lifting Ż_{F} (s) of Z_{F} (s) and an extension δ̃_{d} of δ_{d} such that if ẇ, ẇ_{1} ∈ Ż_{F} (s) and ww_{1}^{−1} ∈ W^{∘}(δ) then δ̃_{d}(ẇẇ_{1}^{−1}) = 1. We can define an extension δ̃_{m} of δ_{m} to T̃_{m} as follows: we can choose a set N of elements of N_{G}(T)^{F_{∘}} which are representatives of T̃_{m}/T^{F_{∘}^{m}} for all m; for v̇ ∈ N we put δ̃_{m}(v̇) = δ̃_{d}(v̇)^{m/d}. This formula defines a character since if we put v̇v̇′ = t ˙\overline{vv′} then:

δ̃_{m}(v̇v̇′) = δ̃_{m}(t ˙\overline{vv′}) = δ_{m}(t)δ̃_{d}( ˙\overline{vv′})^{m/d} = δ_{d}(N_{F_{∘}^{m}/F_{∘}^{d}}(t))δ̃_{d}( ˙\overline{vv′})^{m/d}

= δ_{d}(t^{m/d})δ̃_{d}( ˙\overline{vv′})^{m/d} (since t ∈ T^{F_{∘}}) = δ_{d}(t ˙\overline{vv′})^{m/d} = δ_{d}(v̇)^{m/d}δ_{d}(v̇′)^{m/d} = δ̃_{m}(v̇)δ̃_{m}(v̇′).

We can take Φ = 1 (in 3.7) and extend δ̃_{m} to <ẇF > by δ̃_{m}(ẇF) = δ̃_{m}(ẇẇ_{1}^{−1}); this will define an extension of δ̃_{m} to T̃_{m} ><<w_{1}F > if the condition of lemma 3.19(ii) holds. Since we have chosen Tits representatives for elements of W (δ), if c ∈ C(δ) is w_{1}F - stable then ċ^{w_{1}F} ċ^{−1} is an element of T of order 2 and if 2d|m then δ_{m}(ċ^{w_{1}F} ċ^{−1}) = (δ_{d}(ċ^{w_{1}F} ċ^{−1})^{2})^{m/2d} = 1 so λ_{c} = 1 and the hypothesis of 3.19(ii) holds, so 7.3(ii) holds.

To check 7.2(iii) we need to compute ψ_{T,F_{∘}^{m}} ∘ Sh_{F_{∘}^{m}/ẇF} δ̃_{m}. Let π be the character of T^{ẇF} associated to s^{−1}; we have δ_{m} = \overline{π} ∘ N_{F_{∘}^{m}/ẇF} and (Sh_{F_{∘}^{m}/ẇF} δ̃_{m})(F_{∘}^{m}) = δ̃_{m}(ẇF) = δ̃_{m}(ẇẇ_{1}^{−1}). In our case the only X_{w} is X_{1} and H_{c}^{∗}(X_{1}, F_{1,s^{−1}}) is a one-dimensional space affording a model of π on which F_{∘}^{m} acts trivially, so the “eigenvalue” λπ̃ of F_{∘}^{m} associated to an extension π̃ of π to T^{wF} ><<F_{∘}^{m}> as in 5.2 is π̃(F_{∘}^{m})^{−1}.

So we get:

ψ_{T,F_{∘}^{m}} Sh_{F_{∘}^{m}/ẇF} δ̃_{m} = Sh_{F_{∘}^{m}/ẇF} δ̃_{m}(F_{∘}^{m})π = δ̃_{m}(ẇẇ_{1}^{−1})π

so we see that 7.2(iii) holds with c_{δ,π} = 1, λ_{δ,m} = 1 and θ_{δ,m}(wF) = δ̃_{m}(ẇẇ_{1}^{−1}) so Sh_{F}^{0} (δ̃) = π. Indeed, if 2d|m, then y ↦ θ_{δ,m}(yw_{1}F) = (δ̃_{d}( ˙\overline{yw_{1}}ẇ_{1}^{−1})^{2})^{m/2d} defines a character of C since the kernel of the projection <ẏ | y∈W (δ)> → W (δ) is a group of exponent 2. This character is w_{1}F -invariant by 4.4.

9. Shintani descent and Lusztig induction. In this section, our aim is to give a commutation formula between F′-twisted induction and “Shintani descent” (Theorem 9.2). Until 9.7, we shall not assume that F′ is split.

We consider the following situation: let L be a standard F′-stable Levi subgroup L_{K} of G. There exists a w ∈ W^{F′} such that ^{wF}L = L. Let μ be an irreducible wF - invariant representation of L^{F′}. By 7.1, there exists a standard F′-stable Levi subgroup L_{∘} = L_{J} included in L and a cuspidal representation δ of L_{∘}^{F′} such that μ is a component of the representation R_{L_{∘}^{F′}}^{L^{F′}} δ. We shall use the same notations as in the preceding sections, excepted that the group L_{∘} will play the part of L and either G or L will play the part of G. We shall specify by an index G or L which of these groups we are dealing with. By 7.1(ii) there exists an element v ∈ W_{L}^{F′} such that L_{∘} and δ are vwF -fixed. The representation μ is equal to some component ρ_{ψ} of the representation R_{L_{∘}^{F′}}^{L^{F′}} δ, where ψ is an irreducible, vwF -invariant character of the endomorphism algebra H_{L}. We may assume that wF normalizes K (resp. vwF normalizes J) (if not, multiply w (resp. v) by some element in W_{L} (resp. W_{L_{∘}})). As in §3, we define an element w_{1} such that:

vw = y_{G}w_{1} with y_{G} ∈ W_{G}(δ) and ^{w_{1}F}Δ_{G}(δ) = Δ_{G}(δ).

We define an analogous element w′_{1} in L, satisfying:

v = y_{L}w′_{1} with y_{L} ∈ W_{L}(δ) and ^{w′_{1}wF}Δ_{L}(δ) = Δ_{L}(δ).

Then we have w = w′_{1}^{−1}y_{L}^{−1}y_{G}w_{1}. As in §3, the element w_{1}F (resp. w′_{1}wF) defines an automorphism of W_{G}(δ) (resp. of W_{L}(δ)). Let y be the element y_{L}^{−1}y_{G}; then w′_{1}wF = yw_{1}F.

As in 3.23 we consider the semi-direct products W_{G}(δ) ><<w_{1}F >, W_{L}(δ) ><<yw_{1}F > and H_{G} ><<T_{w_{1}F} >, H_{L} ><<T_{yw_{1}F} >. We have a natural embedding of W_{L}(δ) into W_{G}(δ). Let us prove that via this embedding W_{L}^{∘}(δ) is a parabolic subgroup of W_{G}^{∘}(δ). We use the notations of [HL] (cf. proof of 4.1). It is clear from the definition of these sets of roots that we have Γ′_{L} ⊂ Γ′_{G}. But by 3.12, q_{F′}^{l̃(w)} is the product of the p_{a}’s over the positive roots a ∈ Γ′ which become negative under the action of w. Whence we see that the “length” l̃_{L} is the restriction to W_{L}(δ) of the “length” l̃_{G} (in the following, we shall denote both by l̃). As a consequence, we get Δ_{L}(δ) ⊂ Δ_{G}(δ), so W_{L}^{∘}(δ) is a parabolic subgroup of W_{G}^{∘}(δ). We also get C_{L}(δ) = C_{G}(δ)∩W_{L}(δ), as C(δ) is characterized by l̃= 0. By general properties of endomorphism algebras we have a natural embedding of H_{L} into H_{G}. Since in the definition of the the operators B_{δ,ẇ} in 3.5 we can restrict the summation over the root subgroups corresponding to roots which change sign under the action of w, and since l and l̃restrict well from W_{G}(δ) to W_{L}(δ), we see that the embedding of H_{L} into H_{G} maps T_{w} on T_{w}.

We assume that 4.1 holds for L_{∘} ⊂ G (hence also for L_{∘} ⊂ L).

The algebra H_{L} ><<T_{yw_{1}F} > can be embedded in H_{G} ><<T_{w_{1}F} > by identifying T_{yw_{1}F} to T_{y}T_{w_{1}F}, which is possible according to the following lemma:

lemma 9.1. The element T_{y}T_{w_{1}F} acts on H_{L} as the element T_{yw_{1}F}.

Proof: We have to show that:

T_{y}T_{w_{1}F} T_{v}T_{w_{1}F}^{−1}T_{y}^{−1} = T_{yw_{1}F} T_{v}T_{yw_{1}F}^{−1}

for any element v ∈ W_{L}(δ). From the definition of λ (cf. 3.14), it can be seen that the λ defined for L_{∘} ⊂ L is the restriction of the λ defined for L_{∘} ⊂ G, if we take the operator Φ_{G} of 3.7 equal to δ̃(ẏ^{−1})Φ_{L}. The equality to be proved is then equivalent to T_{y}T_{^{w_{1}F}v} = T_{^{yw_{1}F}v}T_{y}. This last equality is true because of the additivity of “lengths”: l̃(vy) = l̃(v) + l̃(y) and l̃(y^{w_{1}F}v) = l̃(y) + l̃(^{w_{1}F}v) for any v ∈ W_{L}(δ), which is true since y^{−1}Δ_{L} = ^{w_{1}F}Δ_{L} ⊂ Σ^{+} and ^{y}(^{w_{1}F}Δ_{L}) = Δ_{L} ⊂ Σ^{+}.

We choose representatives as follows: we choose arbitrarily ẇ_{1} and ẇ′_{1}. Then we choose representatives of W_{G}(δ) such that when l̃(xx′) = l̃(x) + l̃(x′) the representative of xx′ is ẋẋ′. We define ẏ by v̇ = ẏẇ_{1} and we put ẇ = v̇^{−1}ẏ_{G}ẇ_{1}. We then have ẏ_{G} = ẏ_{L}ẏ, since by the proof of 9.1 we have l̃(y_{G}) = l̃(y_{L}) + l̃(y). As the operators Φ_{G} and Φ_{L} have been chosen so that Φ_{G} = δ̃(ẏ^{−1})Φ_{L}, then with the notations of 3.22, the function μ̃_{L} is the restriction of the function μ̃_{G}, since we have v̇ẇẇ_{1}^{−1}ẏ^{−1} = v̇ẇ_{1}^{′−1}. We shall denote both by μ̃.

By 3.24, applied with L_{∘} for L, first with G and after with L for G we get the following equalities:

R_{L_{∘}^{v̇ẇF} /F′}^{G^{F} /F′}(Sh_{F′/v̇ẇF} μ̃) = q_{F′}^{l(w_{∘}^{G})−l(w_{∘}^{L∘})+ \frac{l(vw)−l̃(y_{G})}{2}} ∑_{χ̃∈Irr(H_{G})^{T_{w_{1}F}}} χ̃(T_{y_{G}}T_{w_{1}F}) Sh_{F′/F} ρχ̃

(1),

for g ∈ G^{F′}, and:

R_{L_{∘}^{v̇ẇF} /F′}^{L^{ẇF} /F′}(Sh_{F′/v̇ẇF} μ̃) = q_{F′}^{l(w_{∘}^{L})−l(w_{∘}^{L∘})+ \frac{l(v)−l̃(y_{L})}{2}} ∑_{ψ̃∈Irr(H_{L})^{T_{yw_{1}F}}} ψ̃(T_{y_{L}}T_{yw_{1}F})Sh_{F′/ẇF} ρψ̃

(2),

for l ∈ L^{F′}.

Let us extend by semi-linearity the notation ρχ̃ to any T_{w_{1}F}^{−1}-class function χ̃ on H_{G}. We will prove:

Theorem 9.2.

R_{L^{ẇF} /F′}^{G^{F} /F′}(Sh_{F′/ẇF} ρψ̃) = q_{F′}^{l(w_{∘}^{G})−l(w_{∘}^{L})+1/2(l(w)−l̃(y))} Sh_{F′/F} ρ_{T_{w_{1}F} −Ind(ψ̃)}

where T_{w_{1}F} − Ind(ψ̃) is the T_{w_{1}F}^{−1}-class function on H_{G} defined by:

T_{w_{1}F} − Ind(ψ̃)(T) = Trace(T T_{w_{1}F}^{−1} | [ψ̃] ⊗_{H_{L} ><<T_{yw_{1}F} >} H_{G} ><<T_{w_{1}F} >).

Proof: If we apply to equation (2) the functor R_{L^{ẇF} /F′}^{G^{F} /F′}, we get:

R_{L_{∘}^{v̇ẇF} /F′}^{G^{F} /F′}(Sh_{F′/v̇ẇF} μ̃) =q_{F′}^{l(w_{∘}^{L})−l(w_{∘}^{L∘})+1/2(l(v)−l̃(y_{L}))}

∑_{ψ̃∈Irr(H_{L})^{T_{yw_{1}F}}} ψ̃(T_{y_{L}}T_{yw_{1}F})R_{L^{ẇF} /F′}^{G^{F} /F′}(Sh_{F′/ẇF} ρψ̃). (3)

So we get the equality of right hand sides of equations (1) and (3). We then use 6.9 and 6.11 to invert this equation in the Frobenius algebra H_{L}, and we get:

R_{L^{ẇF} /F′}^{G^{F} /F′}(Sh_{F′/ẇF} ρψ̃) = d_{ψ}(q_{F′})P_{H_{L}}(q_{F′})^{−1}ψ(1)^{−1} ∑_{y_{L}∈W_{L}(δ)} ∑_{χ̃∈Irr(H_{G})^{T_{yw_{1}F}}}

q_{F′}^{l(w_{∘}^{G})−l(w_{∘}^{L})+1/2(l(vw)−l(v)−l̃(y_{G})+l̃(y_{L}))−l̃(y_{L})}ψ̃(T_{y_{L}}^{−1}T_{yw_{1}F}^{−1})χ̃(T_{y_{G}}T_{w_{1}F}) Sh_{F′/F} ρχ̃

We have l̃(y_{L}y) = l̃(y_{L})+l̃(y) (see proof of 9.1), and furthermore since ^{wF}K = K, we have w^{−1}K = ^{F}K ⊂ Σ^{+}, so for v ∈ W_{L}, we have l(vw) = l(v) + l(w). So, we get:

R_{L^{ẇF} /F′}^{G^{F} /F′}(Sh_{F′/ẇF} ρψ̃) = d_{ψ}(q_{F′})P_{H_{L}}(q_{F′})^{−1}ψ(1)^{−1} ∑_{y_{L}∈W_{L}(δ)} ∑_{χ̃∈Irr(H_{G})^{T_{w_{1}F}}}
q_{F′}^{l(w_{∘}^{G})−l(w_{∘}^{L})+1/2(l(w)−l̃(y))−l̃(y_{L})}ψ̃(T_{y_{L}}^{−1}T_{yw_{1}F}^{−1})χ̃(T_{y_{L}}T_{yw_{1}F}) Sh_{F′/F} ρχ̃.

(3′)

To prove theorem 9.2 we have to prove that:

d_{ψ}(q_{F′})P_{H_{L}}(q_{F′})^{−1}ψ(1)^{−1} ∑_{y_{L}∈W_{L}(δ)} ∑_{χ̃∈Irr(H_{G})^{T_{w_{1}F}}} q_{F′}^{−l̃(y_{L})}ψ̃(T_{y_{L}}^{−1}T_{yw_{1}F}^{−1})

χ̃(T_{y_{L}}T_{y}T_{w_{1}F})χ̃(T T_{w_{1}F}^{−1}) = Trace(T T_{w_{1}F}^{−1} | [ψ̃] ⊗_{H_{L} ><<T_{yw_{1}F} >} H_{G} ><<T_{w_{1}F} >) (4)

Lemma 9.3. Let H be a semi-simple algebra and <γ> a cyclic group acting on H, then, for a, b ∈ H:

Trace(x ↦ γaxbγ^{−1} | H) = ∑_{χ̃} χ̃(γa)χ̃(bγ^{−1})

where χ̃ runs over one extension of each γ-invariant character of H to H ><<γ>.

Proof: H admits a decomposition H = ∑_{χ∈Irr(H)}[χ] ⊗ [χ]^{∨}. The conjugation by γ permutes the isotypic components in this decomposition, so only the γ-invariant isotypic components contribute to the trace of x ↦ γaxbγ^{−1}. When χ is γ-invariant, we are reduced to the case of a matrix algebra where a direct computation gives the result.

As χ̃ is a class function the lemma implies that we have also

Trace(x ↦ aγxbγ^{−1} | H) = ∑_{χ̃} χ̃(aγ)χ̃(bγ^{−1}).

Using this last formula with γ = (T_{y}T_{w_{1}F})^{−1}, a = T_{y}T, and b = T_{y_{L}} the LHS of (4) can be rewritten:

d_{ψ}P_{H_{L}}^{−1}ψ(1)^{−1} ∑_{y_{L}} q_{F′}^{−l̃(y_{L})}ψ̃(T_{y_{L}}^{−1}T_{yw_{1}F}^{−1}) Trace(x ↦ T_{y}T (T_{y}T_{w_{1}F})^{−1}xT_{y_{L}}T_{y}T_{w_{1}F} | H_{G})
= Trace(f ∘ g | H_{G} ⊗_{C} [ψ̃])

where f is the map

x ⊗ m ↦ T_{y}T^{(T_{y}T_{w_{1}F})^{−1}}x ⊗ T_{yw_{1}F}^{−1}m

and g is the map

x ⊗ m ↦ d_{ψ}P_{H_{L}}^{−1}ψ(1)^{−1} ∑_{y_{L}} q_{F′}^{−l̃(y_{L})}xT_{y_{L}} ⊗ T_{y_{L}}^{−1}m.

lemma 9.4. Let H be a symmetric algebra with defining bilinear form λ and dual bases (e_{i}) and (f_{i}) and let H′ ⊃ H be a semi-simple algebra. Let [ψ] be a representation of H with character ψ, and put λ_{ψ} = ∑_{i} ψ(f_{i})ψ(e_{i}). Then the operator p on H′ ⊗_{C} [ψ] defined by x ⊗ m ↦ λ_{ψ}^{−1} ∑_{i} xe_{i} ⊗ f_{i}m is a projector on H′ ⊗_{H} [ψ].

Proof: We first prove that p is a projector by applying it twice. If we recall that, by definition, (t | e_{i}) = λ(tf_{k}) where (t | e_{i}) is the coefficient of t ∈ H on e_{i}, and similarly (t | f_{i}) = λ(tf_{i}), we get:

λ_{ψ}^{−2} ∑_{i,j} xe_{i}e_{j} ⊗ f_{j}f_{i}m = λ_{ψ}^{−2} ∑_{i,j,k,l} xe_{k} ⊗ f_{l}mλ(e_{i}e_{j}f_{k})λ(e_{l}f_{j}e_{i})

= λ_{ψ}^{−2} ∑_{k,l} λ(e_{l} ∑_{j} f_{j}e_{j}f_{k})xe_{k} ⊗ f_{l}m

= λ_{ψ}^{−2} ∑_{k} xe_{k} ⊗ (∑_{j} f_{j}e_{j})f_{k}m

So we get p^{2} = p from the:

lemma 9.5. The element ∑_{j} f_{j}e_{j} acts by the scalar λ_{ψ} on [ψ].

Proof: If φ is a character of H, let us define φ^{∗} by λ(φ^{∗}x) = φ(x) for any x ∈ H. Then we have for any k:

λ(∑_{j} f_{j}e_{j}f_{k}) = ∑_{j}(f_{j}f_{k} | f_{j}) = Trace(x ↦ xf_{k} | H)

= ∑_{φ∈Irr(H)} φ(f_{k})φ(1) = ∑_{φ∈Irr(H)} λ(φ(1)φ^{∗}f_{k})

so ∑_{j} f_{j}e_{j} = ∑_{φ∈Irr(H)} φ^{∗}φ(1). Now φ^{∗} is a central element since:

(φ^{∗}e_{i} | e_{j}) = λ(φ^{∗}e_{i}f_{j}) = φ(e_{i}f_{j}) = φ(f_{j}e_{i}) = λ(φ^{∗}f_{j}e_{i})

= λ(e_{i}φ^{∗}f_{j}) (since H is symmetric) = (e_{i}φ^{∗} | e_{j})

so the trace of φ^{∗} on [ψ] is ψ(φ^{∗})/ψ(1). But, since λ(φ^{∗}f_{j}) = φ(f_{j}), we have φ^{∗} = ∑_{j} φ(f_{j})e_{j} so ψ(φ^{∗}) = ∑_{j} φ(f_{j})ψ(e_{j}) = δ_{ψψ′}λ_{ψ}. Finally we get:

Trace(∑_{j} f_{j}e_{j} | [ψ]) = Trace(∑_{φ} φ^{∗}φ(1) | [ψ]) = ∑_{φ} δ_{ψφ}λ_{ψ}ψ(1)/φ(1) = λ_{ψ}

whence the lemma.

The kernel of the projection: H′ ⊗_{C} [ψ] → H′ ⊗_{H} [ψ] is the span of the elements xh ⊗ m − x ⊗ hm, so next we check that elements of the form xh ⊗ m − x ⊗ hm are in the kernel of p; we have:

\begin{aligned}p(xe_{k} ⊗ m) & = ∑_{i} xe_{k}e_{i} ⊗ f_{i}m = ∑_{i,l} xe_{l} ⊗ f_{i}mλ(e_{k}e_{i}f_{l}) \\ & ∑_{i,l} xe_{l} ⊗ f_{i}mλ(e_{i}f_{l}e_{k}) = ∑_{l} xe_{l} ⊗ f_{l}e_{k}m = p(x ⊗ e_{k}m) \end{aligned}

Finally we check that on the space H′ ⊗_{C} [ψ]/<xh ⊗ m − x ⊗ hm> the projector p induces the identity map:

\begin{aligned}λ_{ψ}^{−1} ∑_{i} xe_{i} ⊗ f_{i}m & = λ_{ψ}^{−1}(∑_{i} xe_{i} ⊗ f_{i}m − ∑_{i} x ⊗ e_{i}f_{i}m) + λ_{ψ}^{−1}x ⊗ ∑_{i} e_{i}f_{i}m \\ & = λ_{ψ}^{−1}x ⊗ ∑_{i} e_{i}f_{i}m modulo <xh ⊗ m − x ⊗ hm> \end{aligned}

which ends the proof since ∑_{i} e_{i}f_{i} acts by λ_{ψ} on [ψ].

From lemma 9.4 we see that g is a projector on H_{G} ⊗_{H_{L}} [ψ̃], so the LHS of (4) is equal to: Trace(f | H_{G} ⊗_{H_{L}} [ψ̃]). We now get theorem 9.2 from:

Lemma 9.6.

H_{G} ><<T_{y}T_{w_{1}F} > ⊗_{H_{L} ><<T_{yw_{1}F} >} [ψ̃] ≃ H_{G} ⊗_{H_{L}} [ψ̃]

and the action of T_{yw_{1}F} on the left hand side corresponds by this isomorphism to the action on H_{G} ⊗_{H_{L}} [ψ̃] defined by x ⊗ m ↦ ^{T_{yw_{1}F}} x ⊗ T_{yw_{1}F} m.

Proof: The lemma is easy to check if we notice that the natural application:

H_{G} ⊗_{C} [ψ̃] ↪ H_{G} ><<T_{w_{1}F} > ⊗_{C} [ψ̃] → H_{G} ><<T_{w_{1}F} > ⊗_{H_{L} ><<T_{yw_{1}F} >} [ψ̃]

factors through H_{G} ⊗_{H_{L}} [ψ̃].

As in chapter 7 we consider an element s of T^{∗} and a split Frobenius endomorphism F_{∘}.

Theorem 9.7. We assume that 7.2 and 7.3 hold for L_{∘} for s and for δ, then:

R_{L^{ẇF}}^{G^{F}} (Sh^{0} ρψ̃) = Sh^{0} ρ_{Ind}{W_{G}(δ) ><<w_{1}F > \atop W_{L}(δ) ><<yw_{1}F >} ψ̃

where Sh^{0} is defined as in 7.4.

Proof: We put F′ = F_{∘}^{m}, with m multiple of d′ (cf. 7.2 (iii)). By 7.4, we have Sh_{F′/ẇF} ρ_{ψ̃⊗θ_{δ,m}} ∈ C(L^{F′}, s) so we can apply 5.4 and we have:

R_{L^{ẇF} /F′}^{G^{F} /F′}(Sh_{F′/ẇF} ρ_{ψ̃⊗θ_{δ,m}}) =

∑_{ρ̃∈E(G^{F} /F′,s^{−1})} λF′ ; ρ̃ ρ̃ ∑_{h∈IN} ⟨ (H_{c}^{∗}(S) ⊗ ψ_{L,F′}Sh_{F′/ẇF} ρ_{ψ̃⊗θ_{δ,m}})^{L^{ẇF} ,(h)}, ρ ⟩q_{F′}^{h/2} (5)

Lusztig functor and Shintani descent F.Digne and J.Michel And by 7.4 the right-hand side of the above equality is equal to:

∑_{ρ̃∈E(G^{F} /F′,s^{−1})} λF′ ; ρ̃ ρ̃ ∑_{h∈IN} ⟨ (H_{c}^{∗}(S) ⊗ λ_{δ,m}^{−1} Sh^{0} ρψ̃)^{L^{ẇF} ,(h)}, ρ ⟩q_{F′}^{h/2}  (6)

But by (3′) in the proof of 9.2 the left-hand side of (5) is equal to:

d_{ψ⊗θ_{δ,m}}(q_{F′})P_{H_{L}}(q_{F′})^{−1}(ψ ⊗ θ_{δ,m})(1)^{−1} ∑_{y_{L}∈W_{L}(δ)} q_{F′}^{l(w_{∘}^{G})−l(w_{∘}^{L})+ \frac{l(w)−l̃(y)}{2} −l̃(y_{L})}

∑_{χ̃∈Irr(H_{G})^{T_{w_{1}F}}} (ψ̃ ⊗ θ_{δ,m})(T_{y_{L}}^{−1}T_{yw_{1}F}^{−1})(χ̃ ⊗ θ_{δ,m})(T_{y_{L}}T_{yw_{1}F}) Sh_{F′/F} ρ_{χ̃⊗θ_{δ,m}}. (7)

It is clear that

d_{ψ⊗θ_{δ,m}} = d_{ψ},
(ψ ⊗ θ_{δ,m})(1) = ψ(1),
(ψ̃ ⊗ θ_{δ,m})(T_{y_{L}}^{−1}T_{yw_{1}F}^{−1}) = θ_{δ,m}(y_{L}^{−1}(yw_{1}F)^{−1})ψ̃(T_{y_{L}}^{−1}T_{yw_{1}F}^{−1})

and

(χ̃ ⊗ θ_{δ,m})(T_{y_{L}}T_{yw_{1}F}) = θ_{δ,m}(y_{L}yw_{1}F)χ̃(T_{y_{L}}T_{yw_{1}F}).

So, applying 7.4 to Sh_{F′/F} ρ_{χ̃⊗θ_{δ,m}}, and using that ψ_{G,F′}^{−1}ρ = λF′ ; ρ̃ ρ̃, expression (7) is equal to:

d_{ψ}(q_{F′})P_{H_{L}}(q_{F′})^{−1}ψ(1)^{−1} ∑_{y_{L}∈W_{L}(δ)} ∑_{χ̃∈Irr(H_{G})^{T_{w_{1}F}}} q_{F′}^{l(w_{∘}^{G})−l(w_{∘}^{L})+1/2(l(w)−l̃(y))−l̃(y_{L})}

ψ̃(T_{y_{L}}^{−1}T_{yw_{1}F}^{−1})χ̃(T_{y_{L}}T_{yw_{1}F})λ_{δ,m}^{−1} ∑_{ρ∈E(G^{F} ,s^{−1})} c_{ρ,χ̃}λF′ ; ρ̃ ρ̃. (8)

Identifying the coefficients of ρ̃ in (6) and (8), we get for any ρ ∈ E(G^{F} , s^{−1}):

∑_{h∈IN} ⟨ (H_{c}^{∗}(S) ⊗ Sh^{0} ρψ̃)^{L^{ẇF} ,(h)}, ρ ⟩q_{F′}^{h/2} = d_{ψ}(q_{F′})P_{H_{L}}(q_{F′})^{−1}ψ(1)^{−1}

∑_{y_{L}∈W_{L}(δ)} ∑_{χ̃∈Irr(H_{G})^{T_{w_{1}F}}} q_{F′}^{l(w_{∘}^{G})−l(w_{∘}^{L})+1/2(l(w)−l̃(y))−l̃(y_{L})}ψ̃(T_{y_{L}}^{−1}T_{yw_{1}F}^{−1})χ̃(T_{y_{L}}T_{yw_{1}F})c_{ρ,χ̃}.

Taking the limit of both sides for m = 0 we get ⟨ (H_{c}^{∗}(S) ⊗ Sh^{0} ρψ̃)^{L^{ẇF}} , ρ ⟩ =

|W_{L}(δ)|^{−1} ∑_{y_{L}∈W_{L}(δ)} ∑_{χ̃∈Irr(W_{G}(δ))^{w_{1}F}} ψ̃(y_{L}^{−1}(yw_{1}F)^{−1})χ̃(y_{L}yw_{1}F)c_{ρ,χ̃}.

Now, for any character ψ ∈ Ĥ_{L}^{γ_{L}}, let χ_{ψ} be the isotypic component of type ψ in the restriction of χ to H_{L}. Then, by lemma 9.1, the component χ_{ψ} is normalized by T_{y}T_{γ_{G}}. So, if we write χ_{ψ} = [ψ] ⊗ M_{χ,ψ}, the element T_{y}T_{γ_{G}} acts on χ_{ψ} by an operator of the form T_{γ_{L}} ⊗ X, and T_{y_{L}} acts by T_{y_{L}} ⊗ 1. Therefore:

Trace(T_{y_{L}}T_{y}T_{γ_{G}}|χ) = ∑_{ψ∈Ĥ_{L}^{γ_{L}}} ψ̃(T_{y_{L}}T_{γ_{L}}) Trace(X|M_{χ,ψ}).

Applying this and 6.9, 6.11 in the preceding calculations, we get: Theorem 9.2. With the preceding notations and hypotheses, we have:

(R_{L^{ẇF} /F′}^{G^{F} /F′} ∘ Sh_{F′/ẇF})(ρψ̃) = q_{F′}^{l(w_{∘}^{G})−l(w_{∘}^{L})+1/2(l(w)−l̃(y))} ∑_{χ̃∈Ĥ_{G}^{γ_{G}}} Trace(X|M_{χ,ψ}) Sh_{F′/F} ρχ̃.

To simplify the expression in the above theorem we use:

Lemma 9.3. Let e_{ψ} be the element d_{ψ}(q_{F′})P(q_{F′})^{−1} ∑_{y_{L}∈W_{L}(δ)} q_{F′}^{−l̃(y_{L})}ψ(T_{y^{−1}})T_{y_{L}} of H_{L}, then, for any z ∈ W_{L}(δ) such that ψ̃(T_{z}T_{γ_{L}}) ≠ 0, we have:

Trace(X|M_{χ,ψ}) = \frac{χ̃(T_{z}T_{y}T_{γ_{G}}e_{ψ})}{ψ̃(T_{z}T_{γ_{L}})}.

Proof: The idempotent e_{ψ} is the projection operator on the isotypic part of type ψ, so that:

Trace(T_{z}T_{y}T_{γ_{G}}|χ_{ψ}) = χ(T_{z}T_{y}T_{γ_{G}}e_{ψ}).

As the operator T_{z}T_{y}T_{γ_{G}} acts by T_{z}T_{γ_{L}} ⊗ X on χ_{ψ} = [ψ] ⊗ M_{χ,ψ}, its trace is equal to ψ̃(T_{z}T_{γ_{L}}) Trace(X|M_{χ,ψ}), whence the result. Remark 9.4: Such an element z ∈ W_{L}(δ) as in 9.3 always exists because the character ψ is irreducible, so there must exist an element h ∈ H_{L} which acts as T_{γ_{L}}^{−1} on [ψ]. So, ψ̃(hT_{γ_{L}}) = dimψ ≠ 0, and there is a generator T_{z} such that ψ̃(T_{z}T_{γ_{L}}) is not zero.

We now come to the main result of this section. We return to the situation of sections 5 and 7, where F′ is equal to the m-th power of a split Frobenius endomorphism F_{∘} defining a rational structure over the prime field. We assume 7.2 to be true for G, L_{∘} and δ; this implies that 4.1 is true for L and δ because of the natural inclusions of W_{L}(δ) in W_{G}(δ) and of H_{L} in H_{G}. Definition 9.5. We define the operator γ − Ind, from the space of γ_{L}-class functions on W_{L}(δ) to the space of γ_{G}-class functions on W_{G}(δ) by the formula:

γ − Ind(ψ̃)(zγ_{G}) = |W_{L}(δ)|^{−1} ∑_{{ h∈W_{G}(δ)|^{h}(zγ_{G})∈W_{L}.γ_{L} }} ψ(^{h}(zγ_{G})),

where ψ is any γ_{L}-invariant irreducible character of W_{L}(δ), and ψ̃ an extension of ψ to W_{L}(δ) ><<γ_{L}>.

We extend by semi-linearity the notation ρχ̃ to define ρ_{f} for any γ_{G}-class function f on H_{G} (semi-linearity because the definition of ρχ̃ is semi-linear with respect to multiplication of χ̃ by appropriate roots of unity (cf.3.25)). With these notations we have: Theorem 9.6. Let ψ be an irreducible, γ_{L} -invariant character of H_{L}, and let ψ̃ be an extension of ψ to the algebra H_{L} ><<γ_{L}>. Then we have:

R_{L^{ẇF}}^{G^{F}} Sh_{L}^{o} ρψ̃ = Sh_{G}^{o} ρ_{γ−Ind(ψ̃)}.

Proof: Consider the expression :

⟨R_{L^{ẇF} /F′}^{G^{F} /F′}Sh_{F′/ẇF} ρψ̃, ρ̃⟩_{G^{F} .F′}

where F′ is some power F_{∘}^{m} with m multiple of d′. Then by 7.4 we have Sh_{F′/ẇF} ρψ̃ ∈ E(L^{ẇF} /F′, s) so we can apply 5.4 and we get for the above expression:

λm/d ; ρ̃ ∑_{h∈IN} q_{F′}^{h}/2⟨(H_{c}^{∗}(S) ⊗ ψ_{L,F′}Sh_{F′/ẇF} ρψ̃)^{L^{ẇF} ,(h)}, ρ⟩_{G^{F}}  (1).

By 9.2 and 9.3, the same expression is equal to:

q_{F′}^{l(w_{∘}^{G})−l(w_{∘}^{L})+1/2(l(w)−l̃(y))} ∑_{χ̃∈Ĥ_{G}^{γ_{G}}} \frac{χ̃(T_{z}T_{y}T_{γ_{G}}e_{ψ})}{ψ̃(T_{z}T_{γ_{L}})}⟨Sh_{F′/F} ρχ̃, ρ̃⟩_{G^{F} .F′}  (2).

We now apply theorem 7.4 in L to transform expression (1), and the same theorem 7.4 in G to transform expression (2). We get for m multiple of M_{∘} (cf 7.2 (iii)):

λm/d ; ρ̃ ∑_{h∈IN} q_{F′}^{h/2}⟨(H_{c}^{∗}(S) ⊗ Sh_{L}^{o} ρψ̃)^{L^{ẇF} ,(h)}, ρ⟩_{G^{F}}

= q_{F′}^{l(w_{∘}^{G})−l(w_{∘}^{L})+1/2(l(w)−l̃(y))} ∑_{χ̃∈Ĥ_{G}^{γ_{G}}} \frac{χ̃(T_{z}T_{y}T_{γ_{G}}e_{ψ})}{ψ̃(T_{z}T_{γ_{L}})}λm/d ; ρ̃⟨Sh^{o} ρχ̃, ρ⟩_{G^{F}} .

Taking “limits for m = 0”, we get:

⟨R_{G^{F}}^{L^{ẇF}} Sh_{L}^{o} ρψ̃, ρ⟩_{G^{F}} = ∑_{χ̃∈C[W_{G}(δ)]^{∧γ_{G}}} \frac{χ̃(zyγ_{G}e_{ψ})}{ψ̃(zγ_{L})}⟨Sh_{G}^{o} ρχ̃, ρ⟩_{G^{F}} ,

where we have written χ, ψ, etc... for the characters of the Coxeter groups specializing the characters of the formal Hecke algebra denoted in the same way.

To get the theorem, it remains only to prove the following lemma: Lemma 9.7. For any extension ψ̃ of an irreducible γ_{L}-invariant character ψ of W_{L}(δ), we have:

γ − Ind(ψ̃) = ∑_{χ̃∈W_{G}(δ)^{∧γ_{G}}} \frac{˜\overline{χ}(zyγ_{G}e_{ψ})}{ψ̃(zγ_{L})}χ̃.

Proof: We know (cf. [DM]II, 2.9) that:

∑_{χ̃∈C[W_{G}(δ)]^{∧γ_{G}}} χ̃(xγ_{G})˜\overline{χ}(yγ_{G}) = #\{ h ∈ W_{G}(δ) | xγ_{G} = hyγ_{G}h^{−1} \},

for any x and y in W_{G}(δ). So, the right hand side of the equality in lemma 9.7, evaluated at xγ_{G}, where x is in W_{G}(δ), is equal to:

ψ̃(zγ_{L})^{−1} ∑_{t∈W_{G}(δ)}<zyγ_{G}\overline{e}_{ψ} | tγ_{G}>#\{ h | zγ_{G} = htγ_{G}h^{−1} \},

where we have denoted by <α | tγ_{G}> the coefficient of the element α ∈ C[W_{G}(δ) ><<γ_{G}>] on the element tγ_{G}. The idempotent e_{ψ} being γ_{G}-invariant, the coefficient <zyγ_{G}\overline{e}_{ψ} | tγ_{G}> is equal to <\overline{e}_{ψ} | ty^{−1}z^{−1}>, and thus equal to ψ(1)|W_{L}(δ)|^{−1}\overline{ψ}(zyt^{−1}) if yt^{−1} is in W_{L}(δ) and to 0 if not. So the right hand side of 9.7 becomes:

ψ(1)ψ̃(zγ_{L})^{−1}|W_{L}(δ)|^{−1} ∑_{\{ t∈W_{G}(δ)|yt^{−1}∈W_{L}(δ) \}} \overline{ψ}(zyt^{−1})#\{ h | tγ_{G} = hxγ_{G}h^{−1} \}.

We can write this expression as:

ψ(1)|W_{L}(δ)|^{−1} ∑_{\{ h∈W_{G}(δ)|γ_{L}^{h}(xγ_{G})^{−1}∈W_{L}(δ) \}} \frac{\overline{ψ}(zγ_{L}^{h}(xγ_{G})^{−1})}{ψ̃(zγ_{L})},

because y = γ_{L}γ_{G}^{−1}. This last expression does not depend on z ∈ C[W_{L}(δ)], provided that ψ̃(zγ_{L}) ≠ 0. So we can choose an element z such that zγ_{L} acts by the identity on [ψ]. With this particular choice, the expression becomes:

|W_{L}(δ)|^{−1} ∑_{{ h∈W_{G}(δ)|^{h}(xγ_{G})∈W_{L}(δ) }} ψ̃(^{h}(xγ_{G})),

which is equal to γ − Ind(ψ̃) by definition. Whence the lemma and the theorem.

39 bis Which by summing over ρ ∈ E(G^{F} , s^{−1}) gives the theorem since

(H^{∗}(S) ⊗ Sh^{0} ψ̃)^{L^{ẇF}} = R_{L^{ẇF}}^{G^{F}} (Sh^{0} ρψ̃)

and

|W_{L}(δ)|^{−1} ∑_{y_{L}∈W_{L}(δ)} ∑_{χ̃∈Irr(W_{G}(δ))^{w_{1}F}} ψ̃(y_{L}^{−1}(yw_{1}F)^{−1})χ̃(y_{L}yw_{1}F)χ̃ = Ind_{W_{L}(δ) ><<yw_{1}F >}^{W_{G}(δ) ><<w_{1}F >} ψ̃.

10. References.

[BT] A. Borel and J. Tits, “Groupes réductifs”, Publications Mathématiques de l’IHES 27 (1965) 55–160.

[DM1] F. Digne et J. Michel, “Descentes de Shintani des caractères d’un groupe de Chevalley fini”, C.R.A.S. t.291(17 novembre 1980), 571–574.

[DM] F. Digne and J. Michel, “Fonctions L des variétés de Deligne-Lusztig et descente de Shintani”, Bulletin de la S.M.F., mémoire n^{∘}20 113, 1985.

[HK] R. Howlett and R. Kilmoyer, “Principal series representations of finite groups with split BN pairs”, Communications in algebra, 8(6) (1980) 543–583.

[HL] R. Howlett and G. Lehrer, “Induced cuspidal representations and generalized Hecke rings”, Inventiones Math.58 (1980), 37–64.

[L] G. Lusztig, “Characters of reductive groups over a finite field”, Annals of Math. Studies 107, Princeton university press 1984.

[S] T. Shoji, “Some generalization of Asai’s result for classical groups”, Advanced Studies in Pure Math. 6 (1985), 207–229.

[S2] T. Shoji, “Shintani descent for exceptional groups over a finite field”, Proc. of Symp. in Pure Math. 47 (1987), 297–303.