Shintani descent F.Digne

Shintani descent and L functions of Deligne-Lusztig varieties

by François Digne

1. Shintani descent.

First we recall some notations and definitions. We consider the following situation: Let G be a finite group and <F > a cyclic group acting on G. We denote by j the injective map g ↦ gF from the set G into the set G ><<F >. We shall denote by Irr(G) the set of irreducible characters of G (on the complex number field).

Definition 1.1.

a) We call F -conjugacy classes (or F -classes) of G the sets j^{−1}(c) where c runs over the conjugacy classes of G ><<F > whose inverse images by j are nonempty.

b) We call F -class functions on G the functions of type f ∘ j where f is a class function on G ><<F >.

c) If g is an element of G, we call F -centralizer of g the group C_{G}(j(g)), which will be denoted by F C_{G}(g).

With these notations we have:

Proposition 1.2. The set { j ∘ χ̃ | χ ∈ Irr(G)^{F} }, where χ̃ is an arbitrary extension of χ to the semi-direct product G ><<F >, is a basis of the space of F -class functions. This basis is orthonomal for the usual scalar product ⟨ f_{1}, f_{2} ⟩ = |G|^{−1} ∑_{g∈G} f_{1}(g)\overline{f_{2}(g)}.

We shall often identify f ∘j and the restriction of f to the coset G.F in G ><<F >, and, in the following, the notation χ̃ will always have a meaning similar to the one stated in 1.2.

Let us give now the general setting for defining Shintani descent.

Definition 1.3. Let G be a group and F be an automorphism of (the abstract group) G. We say that these data satisfy Lang’s property if G^{F} is finite and if any element g ∈ G can be written x^{−1}.^{F}x for some x ∈ G.

One example of this situation is the case of a connected algebraic group defined over the algebraic closure F of a finite field F_{q}, if we take for F a Frobenius endomorphism defining on the group a rational structure over F_{q}. In this case G is the group of points over F of the algebraic group, and Lang’s property comes from the well known Lang’s theorem.

We then define Shintani descent:

Theorem 1.4. Let G be a group and F and F′ be two commuting automorphisms of G such that Lang’s property is satisfied for both F and F′. Then

(i) For any y ∈ G, the element y^{−1}.^{F}y is in G^{F′} if and only if the element y.^{F′}y^{−1} is in G^{F}. (ii) The correspondance:

\begin{aligned}y^{−1}.^{F}y & ⟷ y.^{F′}y^{−1} \\ G^{F′} & ⟷ G^{F} \end{aligned}

defines a bijection from the set of F -classes of G^{F′} to the set of F′-classes of G^{F}.

This theorem follows from an easy calculation.

Definition 1.5. We shall denote by N_{F′/F} the map defined in theorem 1.4 and we shall call Shintani descent from G^{F′} to G^{F} the transpose of N_{F/F′}. It will be denoted by Sh_{F′/F}.

We shall sometimes use the notation N_{F′/F} (x) to denote an arbitrary element of the F′-class image of the F -class of x.

We have:

Proposition 1.6. The Shintani descent is an isometry for the usual scalar product (cf. 1.2).

Sketch of the proof: An element x of G^{F′} F -centralizes the element y^{−1}.^{F}y ∈ G^{F′} if and only if yxy^{−1} is in G^{F}. As yxy^{−1} clearly F′-centralizes y.^{F′}y^{−1}, we have defined a bijection between F C_{G^{F′}} (x) and F′C_{G^{F}} (N_{F′/F} (x)), so these two groups have the same number of elements, whence the result.

Remark 1.7. The definition of Shintani descent that we gave above differs from the original one in [Sh] and in [K1] which was in fact i ∘ Sh_{F′/F}, where i is the transpose of the inverse map on the group. We shall see later what are the advantages of each of these definitions (moreover in [Sh] and [K1] the only case considered is the case when F′ is a power of F).

We consider now the case when G is the group GL_{n}(F), where F is the algebraic closure of the finite field F_{q}. On this group we can define two Frobenius endomorphisms such that the group of fixed points is either GL_{n}(F_{q}) or U_{n}(F_{q^{2}}). We shall denote either of these Frobenius endomorphisms by F. With these notations we can now state Shintani’s ([Sh]) and Kawanaka’s ([K1]) results.

Theorem 1.8 (Shintani). Assume that F is such that G^{F} is equal to GL_{n}(F_{q}), then for any irreducible character χ of G^{F^{m}} invariant by F, there exists an extension χ̃ of χ to the semi-direct product G^{F^{m}} ><<F > such that χ̃ ∘ N_{F^{m}/F}^{−1} is an irreducible character of G^{F}.

Theorem 1.9 (Kawanaka). Assume that m is prime to the characteristic and that F is such that G^{F} is equal to U_{n}(F_{q^{2}}), then for any irreducible character χ of G^{F^{m}} invariant by F, there exists an extension χ̃ of χ to the semi-direct product G^{F^{m}} ><<F > such that χ̃ ∘ N_{F^{m}/F}^{−1} is an irreducible character of G^{F} up to sign.

Remark 1.10. Let us return to the general case and assume that F = F′. Then for x ∈ G, the elements x^{−1}.^{F}x and ^{F}x.x^{−1} are conjugate in G, but not necessarily in G^{F}. So the map i ∘ Sh_{F/F} is not in general the identity. It is the identity in the GL_{n} and U_{n} cases because in these groups geometric conjugacy implies rational conjugacy. This will not be the case in other groups, and the study of Sh_{F/F} give a key for studying Sh_{F^{m}/F} for any m.

For instance, we have the following result:

Theorem 1.11 ([DM2], [DM6] and [A1], [A2]). If ρ_{o} ∈ Irr(G^{F^{m}}) is a unipotent (cf. below, section 2), F -invariant character in the principal series, and if ρ̃_{o} is an extension of ρ_{o} to G^{F^{m}} ><<F >, then there exist roots of unity λ_{ρ} such that

Sh_{F^{m}/F} ρ̃_{o} = ∑_{ρ∈Irr(G^{F})} λ_{ρ}^{m}c_{ρ,ρ̃_{o}}ρ,

for any m such that F^{m} is split, where the coefficients c_{ρ,ρ̃_{o}} are independant of m.

Moreover, with the notations of next section, these coefficients are given by c_{ρ,ρ̃_{o}} = |W |^{−1} ∑_{w∈W} ⟨ R_{T^{wF}}^{G^{F}} , ρ ⟩χ̃((wF)^{−1}) where the F -invariant character χ of W corresponds to the character ρ_{o} of the principal series, and its extension χ̃ to W ><<F > corresponds to ρ̃_{o}.

This result shows in particular that if F is split, the knowledge of Sh_{F/F} gives results about Sh_{F^{m}/F} for any m. There is a precise study of N_{F/F} in chapter 4 of [DM6], where N_{F/F} is computed for any semi-simple group of type A, B, C, D and G_{2}, split or non split.

Asai has studied Sh_{F/F} in many cases (cf. [A3]–[A7]), and we now state his main results:

Theorem 1.12. For all reductive algebraic groups with connected center, of type B_{n}, C_{n}, D_{n} or ^{2}D_{n}, the map Sh_{F/F} is explicitly known. More precisely,

(i) i ∘ Sh_{F/F} commutes with Lusztig ’s parametrization of characters. (ii) If the group is split, for any family of unipotent characters of G^{F} parametrized by a set M(Γ), associated

to the finite group Γ (with Lusztig’s notations, cf. [L]), we have:

Sh_{F/F} (λ_{x}ρ_{x}) = ∑_{y∈Γ}{x, y}λ_{y}ρ_{y},

where ρ_{y} is the character of the family associated to the element y ∈ M(Γ), where {x, y} is the Fourier coefficient defined in [L], and where λ_{y} is a root of unity associated to y (λ_{y} = χ(γ^{−1})/χ(1) if y = (γ, χ)).

For type ^{2}D_{n}, Asai got an analogous formula (cf. [A6]). For split exceptionnal types, Asai (cf. [A5]) got part (ii) of this result (with the exception of the so-called exceptionnal families of unipotent characters for type E_{7} and E_{8}). In [A3]–[A5] and [A7], Asai also got similar formulas for Sh_{F_{o}/F} where F_{o} is a split Frobenius endomorphism on G and where F and F_{o} define rational structures over the same finite field.

As for methods, Asai uses Lusztig’s results, L-functions methods (see section 2) and Kawanaka’s results that we now state:

Theorem 1.13 ([K2]). Let G be a symplectic or special orthogonal group; there exists an integer M depending only on the type of G such that if m is prime to M and to the characteristic and if r is and integer inverse to m modulo M, then for any F -invariant irreducible character of G^{F^{m}}, the class function (i ∘ Sh_{F/F})^{−r}(i ∘ Sh_{F^{m}/F})(χ̃) is an irreducible character of G^{F} up to multiplication by ± an m-th root of unity.

The map (i ∘ Sh_{F/F})^{−r}(i ∘ Sh_{F^{m}/F}) was later generalized by Kawanaka ([K3]) to non connected groups, and an analogous result to 1.13 is proved in that paper (see also [D] for a simpler definition of Kawanaka’s map). Mac Govern ([MG]) proved the same result for groups of type G_{2}.

Shoji has proved the following result about Shintani descent:

Theorem 1.14 ([Sho]). For some classical groups with connected center, there exists an integer M such that for any m multiple of M, and any χ ∈ Irr(G^{F^{m}})^{F} there exists a root of unity λχ̃ such that

i ∘ Sh_{F^{m}/F} (λχ̃χ̃) = R_{x},

where χ is parametrized by x ∈ M(Γ), with Lusztig’s notations, and where R_{x} is the “Fourier transform of χ” as defined in [L], 4.24.1.

(The phrase “some classical groups” in the preceding statement depends on the existence of an element s as in the hypothesis of lemma 2.17 of [Sho] being in the fixed maximal torus and being fixed by F).

Let us now do some remarks about the various definitions of descent (or lifting). In view of Kawanaka’s and Shoji’s results, it seems that the good definition of Shintani descent should be i ∘ Sh_{F′/F}. But there is some advantages in considering Sh_{F′/F}. The first one is the complete symmetry between the two Frobenius endomorphisms. The second one is that Sh_{F′/F} and Sh_{F/F′} are inverse maps. Besides, we have the following easy to prove result:

Lemma 1.15. Let F_{1}, F_{2} and F_{3} be three pairwise commuting automorphisms of a group G, each one satisfying Lang’s property, and such that F_{3} = F_{1} ∘ F_{2}. Then we have:

Sh_{F_{3}/F_{1}} = Sh_{F_{2}/F_{1}} ∘ Sh_{F_{3}/F_{2}} ∘i.

This shows that “i appears always somewhere”. Let us remark that the lemma makes sense (i.e., under the assumptions of the lemma, F_{3}-classes in G^{F_{2}} are also F_{1}-classes). In the same spirit, one can try to compose the map Sh_{F^{m}/F^{n}} with Sh_{F^{n}/F^{r}}, where r is the remainder of m modulo n, and then to do it again, following the Euclid algorithm. Let us call G_{m,n} the resulting map, from the set of F^{n}-class functions on G^{F^{m}} to the set of class functions on G^{F^{gcd(m,n)}}, composed with (i)^{k} where k is the number of steps in the Euclid algorithm. In this situation, Gyoja has proved:

Theorem 1.16 ([G]). Let G be a connected reductive algebraic group, and F a Frobenius endomorphism defining on G a rational structure over F_{q}; assume that p and q are large enough, and that m is prime to p; then, for any Deligne-Lusztig character of the form χ_{m} = R_{T^{F^{m}}}^{G^{F^{m}}} (θ ∘ Norm_{F_{q^{m}} /F_{q}}), where θ is a character of T^{F} (here T is an arbitrary rational maximal torus), there exists an extension χ̃_{m} of χ_{m} such that G_{m,n}(χ̃_{m}) is equal to χ_{(gcd(m,n))} for any n.

In particular this theorem implies:

Corollary 1.17. With the preceding notations, we have i ∘ Sh_{F^{m}/F} (χ̃_{m}) = R_{T^{F}}^{G^{F}} (θ).

2. Deligne-Lusztig varieties.

In this section we shall assume to have chosen an embedding of the field Q̄_{l} into the complex numbers. Let G be a connected reductive algebraic group over F, with a rational structure over F_{q} defined by a

Frobenius endomorphism F as above. Let (T, B) be a pair consisting of a rational maximal torus embedded in a rational Borel subgroup of G. We denote by U the unipotent radical of B. Then, following Deligne and Lusztig [DL], we define, for any ẇ ∈ N_{G}(T) representative of an element w ∈ W the variety

Yẇ = { x ∈ G | x^{−1}.^{F}x ∈ UẇU }.

On this variety acts the group G^{F} × T^{wF}, if we define the action of (g, t) by x ↦ gxt^{−1}. So the alternating sum of the spaces of l-adic cohomology with compact support defines a virtual G^{F} × T^{wF} -module. We shall denote this alternating sum by H_{c}^{∗}(Yẇ, Q̄_{l}). This module can be used to define Deligne-Lusztig induction from T^{wF} -modules to G^{F} -modules, namely, we can associate to any irreducible character θ of T^{wF} the G^{F} -virtual module R_{T^{wF}}^{G^{F}} (θ) = H_{c}^{∗}(Yẇ, Q̄_{l})_{θ}, where this notation means the T^{wF} isotypic part of type θ of the cohomology. The (virtual) character of this module is called a Deligne-Lusztig character. Deligne and Lusztig have shown that any irreducible character of G^{F} is a component of some Deligne-Lusztig character. In particular:

Definition 2.1. The irreducible characters of G^{F} appearing in Deligne-Lusztig characters with θ = 1 are called unipotent characters.

Let us now fix some w ∈ W and some irreducible character θ of T^{wF}. Let k be an integer such that F^{k} fixes w (and then acts on T^{wF}) and fixes θ. For any g ∈ G^{F}, we put:

N_{Yẇ}^{m}(g, θ) = Trace(gF^{km} | H_{c}^{∗}(Yẇ)_{θ}).

Then we have:

Proposition 2.2. The sum of the formal series ∑_{m≥1} N_{Yẇ}^{m}(g, θ)X^{m} is a rational fraction whose value at infinity is the opposite of Trace(g | R_{T^{wF}}^{G^{F}} (θ)).

Note that the above proposition remains true if we replace F^{k} with any Frobenius endomorphism commuting with F, normalizing T, B, and fixing g and θ.

This proposition, that appears in the Deligne-Lusztig paper [DL] shows the interest of studying such formal series, that are the so-called L- functions of the varieties Yẇ. These L-functions are related to Shintani descent as can be seen in the following result:

Theorem 2.3. Let F and F′ be two commuting Frobenius endomorphisms on G. Assume that both F and F′ fix a pair (T, B) as above. Consider an element w ∈ W^{F′} and let ẇ be a representative of w in N_{G}(T)^{F′}. If θ is an F′-fixed linear character of T^{wF}, then:

Trace(gF′ | H_{c}^{∗}(Yẇ)_{θ}) = Sh_{F′/F} (Trace(gτẇF | Ind_{B^{F′}}^{G^{F′}} (Θ))),

where Θ is the trivial extension to B^{F′} of Sh_{wF/F′}(θ), and where τẇ is an operator commuting with G^{F′}

on the induced representation (in fact the endomorphism algebra H = End_{G^{F′}} (Ind_{B^{F′}}^{G^{F′}} (Θ)) is a “generalized Hecke algebra” constructed on the group W (Θ) = { y ∈ W | ^{y}Θ = Θ }, and w can be written w = yw_{1} where w_{1}F defines “diagram” automorphisms of W (Θ) and of H, and where y is in W (Θ). The operator τẇ decomposes as T_{y}Ψ where T_{y} is the generator of H corresponding to y, and where Ψ ∘ F induces on H the automorphism defined by w_{1}F).

This formula is proved when θ is the trivial character in [A1], [A2], and [DM1], [DM6]. It has been generalized to a general θ by Lusztig in [L], 2.10, when F′ is split and the center of G is connected, and can be generalized to any F′ and to the non connected center case ([DM7]). We now give, in the easier case of a trivial θ, the following consequence of 2.3:

Corollary 2.4. With the notations of 2.3, if θ is the trivial character, we have :

Trace(gF′ | H_{c}^{∗}(Yẇ)_{1}) = ∑_{χ∈Irr(H)^{F}} χ̃(T_{w}F)(Sh_{F′/F} ρχ̃)(gF),

where ρ_{χ} is the character of G^{F′} corresponding to the character χ of the endomorphism algebra, and where ρχ̃ is an extension of ρ_{χ} defined by the arbitrarily chosen extension χ̃ of χ to the semi-direct product H ><<F >.

Remark. In the above corollary, if the extension χ̃ is chosen such that F acts with finite order in the corresponding representation, then the extension ρχ̃ has the same property.

Formula 2.4 is one of the main tools in Asai’s and Digne-Michel’s papers, and its generalization to the case of any F′-invariant character θ of T^{wF} is proved and used in chapter 3 of Lusztig’s book ([L], 3.6.3), in the connected center case, assuming F′ to be split. When T is replaced with a standard Levi subgroup (Lusztig variety), 2.4 has been generalized in [Sho] (connected center case) and in [DM7] (if the center of G is non necessarily connected).

3. Concluding remarks.

In order to use these methods and results, one needs:

a) A good knowledge of the endomorphism algebras that appear (and which, in the non connected center case, are more general than Hecke algebras).

b) Results on eigenvalues of Frobenius endomorphisms on the cohomology (many such results appear in Lusztig’s book, see e.g., [L], chapter 2). Using a), b) and the general properties of Shintani descent one can

1) prove general results on Shintani descent,

2) find the decomposition of Deligne-Lusztig characters (cf. [L]) and of Lusztig’s generalized induction (cf. [Sho]). Many problems are still open in these directions, such as:

– Compute Sh_{F′/F} in general (F′ non necessarily split).

– Get general results on the eigenvalues of a non split Frobenius endomorphism on the cohomology of Deligne-Lusztig’s or Lusztig ’s varieties.

References.

[A1] T. Asai On the zeta function of the varieties X(W) of the split classical groups and the unitary groups. Osaka Math. Journal 20 (1983) pp. 21–32.

[A2] T. Asai Lifting and the zeta functions of the Deligne-Lusztig varieties. Comm. in Algebra 11 (1983) pp. 2187–2201.

[A3] T. Asai Unipotent class functions of split special orthogonal groups SO_{2n}^{+} over finite fields. Comm. in Algebra 12 (1984) pp. 517–615,

[A4] T. Asai The unipotent class functions of the symplectic and odd orthogonal groups over finite fields. ibid. pp. 617–645.

[A5] T. Asai Unipotent class functions of exceptionnal groups over finite fields. ibid. pp. 2729–2857.

[A6] T. Asai The unipotent class functions of non-split special orthogonal groups. Comm. in Algebra 13 (1985) pp. 845–924.

[A7] T. Asai On the twisting operators on the finite classical groups. Preprint

[D] F. Digne Descente de Shintani et blocs selon N. Kawanaka. Séminaire sur les groupes finis III. To appear in Publications Mathématiques de Paris VII.

[DL] P. Deligne et G. Lusztig Characters of reductive groups over finite fields. Annals of Math. 103 (1976) pp. 103–161.

[DM1] F. Digne et J. Michel Nombre de points rationnels des variétés de Deligne–Lusztig et caractères de l’algèbre de Hecke. CRAS. t. 287 (nov. 1978) pp. 811–814.

[DM2] F. Digne et J. Michel Descente de Shintani des caractères d’un groupe de Chevalley fini. CRAS. t. 291 (nov. 1980) pp. 571–574.

[DM3] F. Digne et J. Michel Descente de Shintani des caractères de Deligne-Lusztig. CRAS. t. 291 (dec. 1980) pp. 651–653.

[DM4] F. Digne et J. Michel Décomposition des descentes de Shintani. Séminaire sur les groupes finis II Publications Mathématiques de Paris VII 17 (1985) pp. 124–159.

[DM5] F. Digne et J. Michel Foncteur de Lusztig et fonctions de Green généralisées CRAS. t. 297 (sept. 1983) pp. 89–92.

[DM6] F. Digne et J. Michel Fonctions L des variétés de Deligne-Lusztig; descente de Shintani. Mémoires de la SMF 20 (1985).

[DM7] F. Digne et J. Michel Decomposition of Lusztig ’s induction and Shintani descent for groups with

non connected center. unpublished.

[G] A. Gyoja Liftings of irreducible characters of finite reductive groups.Osaka Journal of Math. 16 (1979) pp. 1–30.

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

[K1] N. Kawanaka On the irreducible characters of the finite unitary groups. J. of Math. Society of Japan 29 (1977)pp. 425–450.

[K2] N. Kawanaka Lifting of irreducible characters of finite classical groups I. J. of the Faculty of Science of the University of Tokyo 28 (1982) pp. 851–861.

[K3] N. Kawanaka Lifting of irreducible characters of finite classical groups II. J. of the Faculty of Science of the University of Tokyo 30 (1984) pp. 499–516.

[L] G. Lusztig Characters of reductive groups over a finite field. Annals of Math. Studies 107 Princeton

University press (1984).

[MG] K. McGovern On the lifting theory of finite groups of Lie type. Pacific Journal of Math. Vol 113 No 2 pp. 383–394.

[Sh] T. Shintani Two remarks on irreducible characters of finite general linear groups. J. Math. Soc. of Japan 28 (1976) pp. 396–414.

[Sho] T. Shoji Some generalisation of Asai’s result for classical groups. Advanced Studies in Pure Mathematics 6 1985 Algebraic groups and related topics pp. 207–209.