Regular unipotent elements ´El´ements unipotents r´eguliers

Regular unipotent elements El´ ´ ements unipotents r´ eguliers

C´ edric Bonnaf´ e

University of Chicago, Eckhart Hall, 5734 South University Avenue,

CHICAGO - IL 60637

Abstract : We construct a map from the set of regular unipotent classes of a finite reductive group to the corresponding set in a Levi subgroup. A theorem of Digne, Lehrer and Michel on Lusztig restriction of characteristic functions of such classes gives in particular another (not explicit) construction of such a map. We conjecture these two maps coincide.

R´esum´e : Nous construisons une application entre l’ensemble des classes unipotentes r´eguli`eres d’un groupe r´eductif fini et l’ensemble correspondant dans un de ses sous-groupes de Levi. Cela permet de pr´eciser conjecturalement un th´eor`eme de Digne, Lehrer et Michel sur la restriction de Lusztig des fonctions caract´eristiques de ces classes.

Version fran¸ caise abr´ eg´ ee

Soit Gun groupe r´eductif connexe d´efini sur une clˆoture alg´ebrique d’un corps fini et soit F : G → G un endomorphisme de Frobenius munissant G d’une structure rationnelle sur un corps fini. SoitL un sous-groupe de Levi F-stable d’un sous-groupe paraboliqueP (non n´ecessairement F-stable) deG. Dans [1], conjecture 5.2, Digne, Lehrer et Michel font la conjecture suivante : Conjecture 1 : Soit uun ´el´ement unipotent r´egulier de G^F. Alors il existe un ´el´ement unipotent r´egulier v dans L^F tel que ^∗R^GL⊂Pγ_u^G = γ_v^L o`u γ_u^G est la fonction caract´eristique de la classe de G^F-conjugaison de umultipli´ee par |CG^F(u)|(idem pourγ_v^L).

Dans [1], ils prouvent que cette conjecture a lieu lorsque P est F-stable (proposition 5.3) et lorsque le centre de Gest connexe (proposition 5.4). Dans ces deux cas, l’´el´ementv est d´etermin´e explicitement. Dans [2], ils prouvent que la conjecture a lieu lorsque p est bon et q assez grand (th´eor`eme 3.7). Cependant, dans ce dernier cas, l’´el´ement v n’est pas explicit´e. Le probl`eme est mˆeme plus s´erieux : comment d´efinir a priori un ´el´ement v dont la classe de L^F-conjugaison ne d´epend que de la classe deG^F-conjugaison deu?

Si on note Reg_uni(G^F) l’ensemble des classes deG^F-conjugaison d’´el´ements unipotents r´eguliers de G^F, le but de cette note est de construire a priori, lorsquep est bon pourG, une application explicite res^GL : Reg_uni(G^F)−→Reg_uni(L^F).

Conjecture 2 : Supposons p bon pour G. Soit u un ´el´ement unipotent r´egulier de G^F et soit u^L= res^GLu. Alors ^∗RL⊂P^G γ_u^G=γ_u^LL.

L’application res^GL est construite en utilisant des m´ethodes inspir´ees de la th´eorie des faisceaux- caract`eres, th´eorie utilis´ee dans [2] pour prouver la conjecture 1 lorsquepest bon et qassez grand.

C’est pourquoi la conjecture 2 semble raisonnable. Cependant, mˆeme sous ces hypoth`eses, nous ne savons pas d´emontrer cette conjecture un peu plus pr´ecise.

L’application res^GL ainsi construite v´erifie quelques propri´et´es : elle ne d´epend pas dePet elle est transitive, c’est-`a-dire

res^LM◦res^GL = res^GM

si M est un sous-groupe de Levi F-stable d’un sous-groupe parabolique de L. D’autre part, si z∈Z(G) et sil∈Lv´erifiel⁻¹F(l) =z, alors, pour toutU ∈Reg_uni(G^F), ^lU ∈Reg_uni(G^F) et

res^GL lU =^l(res^GL U).

Pour pouvoir avoir une chance de satisfaire la conjecture 2, il est heureux que l’application res^GL

v´erifie ces propri´et´es.

Au cours de cette note, nous d´emontrons le r´esultat suivant :

Soitvun ´el´ement unipotent deLtel que sa classe deL-conjugaison supporte un syst`eme local cuspidal (cf. [3]). Alors le morphisme de groupesC<sup>L</sup>(v),→C<sup>G</sup>(v) induit un isomorphisme A<sup>L</sup>(v)'A<sup>G</sup>(v) o`uA^L(v) =C^L(v)/CL^◦(v)et idem pourA^G(v).

Ce r´esultat n’est pas n´ecessaire pour construire l’application res^GL : il ne sert qu’`a simplifier la preuve de la proposition 1.4 qui pourrait ˆetre obtenue en v´erifiant au cas par cas. Cependant, il est vrai en toute g´en´eralit´e (c’est-`a-dire dans le cas o`uvn’est pas forc´ement r´egulier et mˆeme lorsque la caract´eristique du corps n’est pas bonne pourG). Il g´en´eralise un r´esultat de Lusztig qui montrait la mˆeme propri´et´e lorsque la caract´eristique du corps est assez grande [4]. Il pourrait d’autre part permettre de lever l’ambiguit´e sur la d´efinition des fonctions caract´eristiques de faisceaux-caract`eres cuspidaux ce qui donnerait un point d’appui pour attaquer la conjecture 2.

1. Preliminaries in general reductive groups

LetGbe a connected reductive group over an algebraically closed fieldFof characteristicp>0.

We denote by Z(G) the center of G. If g ∈G,A^G(g) will denote the finite group C^G(g)/CG^◦(g).

TheG-conjugacy class ofgwill be denoted by (g), or (g)^Gif the group is not clear from the context.

Cuspidal local systems. LetPbe a parabolic subgroup ofGand letLbe a Levi subgroup ofP.

We fix a unipotent elementvin L. Recall the following result :

Proposition 1.1 (Lusztig [3]). Assume that(v)^L supports a cuspidal local systemE. Then : (a) v is a distinguished unipotent element of L, that is,v is not contained in a Levi subgroup of a proper parabolic subgroup of G.

(b) N^G(L)stabilizes (v)^L andE.

(c) N^G(L)is a reflection group.

(d) All parabolic subgroups ofGhavingL as Levi subgroup are conjugate (under N^G(L)).

Corollary. Assume that(v)^L supports a cuspidal local system. Then the morphismC^L(v),→C^G(v) induces an isomorphism A^L(v)'A^G(v).

Remark - This corollary is proved by Lusztig in [4] for sufficiently large p ; his proof uses the classification of cuspidal local systems. Our result generalizes slightly Lusztig’s one and our proof does not use the classification.

Proof - It is well-known that the morphismA^L(v)→ A^G(v) is injective. So we consider A^L(v) as a subgroup of A^G(v). Let g ∈C^G(v), then ^gZ(L)^◦ is a maximal torus of CG^◦(v) (becausev is distinguished inL by proposition 1.1, (a)) so there exists h∈CG^◦(v) such that ^gZ(L)^◦ = ^hZ(L)^◦, henceh⁻¹g∈N^G(L). This proves that the groupA^L(v) is normal in A^G(v) and that the character ζ is invariant under A^G(v) (cf. proposition 1.1, (b)). By Mackey formula, it is then sufficient to show that Ind^A_A^G_L(v)^(v)ζis irreducible.

Letχbe an irreducible character ofAG(v). In [3], Lusztig attached to the pair (v, χ) a quadruple (M, w, ψ, ρ), well-defined up toG-conjugacy, whereM is a Levi subgroup of a parabolic subgroup ofG,wis a unipotent element ofM,ψis an irreducible character ofA^M(w) andρis an irreducible character ofN^G(M)/M. Moreover, the quadruple (M, w, ψ, ρ) determines (v, χ) up to conjugacy.

According to [3], proposition 9.5, (v, χ) is associated to a pair of the form (L, v, ζ, ρ) if and only if ρ = sgn is the sign character of N^G(L)/L (N^G(L)/L is a reflection group by proposition 1.1, (c)). But by [5], 1.4, (IV), (v, χ) is associated to a pair of the form (L, v, ζ, ρ) if and only ifχis an irreducible component of Ind^A_A^G_L(v)^(v)ζ. That proves that Ind^A_A^G_L(v)^(v)has only one irreducible component occuring with some multiplicitym. By [5], 1.2, (II),m= sgn(1) = 1.

Regular unipotent class. IfLis a Levi subgroup of a parabolic subgroup ofG, then the injective morphismZ(G),→Z(L) induces a surjective morphismh^L:Z(G)/Z(G)^◦ →Z(L)/Z(L)^◦. If there is any ambiguity, we will denote this map byh^GL. The groupGis said to be cuspidalifh^L is not an isomorphism for every Levi subgroupsL of proper parabolic subgroups ofG.

Proposition 1.2. Let L be a cuspidal Levi subgroup of a parabolic subgroup ofG. Then :

(a) IfM is a cuspidal Levi subgroup of a parabolic subgroup of GandKerh^L= Kerh^M, then L andMare G-conjugate.

(b) Lsatisfies properties (c) and(d)of proposition 1.1.

Proof -LetB be a Borel subgroup ofGand letTbe a maximal torus ofB. We denote by Φ the root system ofGrelative toTand by ∆ the basis of Φ associated to B. IfI⊂∆, we denote byLI

the Levi subgroup of the parabolic subgroup ofGassociated toI. We recall the following

Lemma 1.3 ([1], 1.5). Let I andJ be two subsets of∆. ThenKerh^LI∩J = (Kerh^LI).(Kerh^LJ).

We may assume thatL = LI and M = LJ for some subsets I and J of ∆. By lemma 1.3, the map Z(LI)/Z(LI)^◦ → Z(LI∩J)/Z(LI∩J)^◦ is an isomorphism. Since LI is cuspidal, we have I∩J =I so I⊂J. By the same argument, we haveJ⊂I. So I=J. As in [3], 9.4, the proposition follows immediately from this fact.

Proposition 1.4. Assume that pisgoodfor G. LetL be a cuspidal Levi subgroup of a parabolic subgroup ofGand letv be a regular unipotent element of L. Then A^L(v) =A^G(v).

Proof - The groupA^L(v) is isomorphic to Z(L)/Z(L)^◦ because pis good forG. Without loss of generality, we may assume thatGis semisimple, simply connected, and almost simple.

•Case 1 : The groupZ(L)/Z(L)^◦ is cyclic. In this case, there exists an injective linear character ζ:A^L(v)→Q_`^×. The local system on (v)^L associated to ζis cuspidal because Lis cuspidal so the result follows from the corollary to proposition 1.1.

•Case 2 : The groupZ(L)/Z(L)^◦ is not cyclic. In this case,Gis of typeD2n, withn>2 and Z(L)/Z(L)^◦ 'Z(G)'Z/2Z×Z/2Z. Moreover, L is of typeA1× · · · ×A1 ((n+ 1) times) : this determinesL up toG-conjugacy. The classification being well-known in this case (becausepis odd or, equivalently, good), one can check thatA^G(v)'Z(G).

IfKis a subgroup ofZ(G)/Z(G)^◦, we will denote byL(K) orL^G(K) the set of Levi subgroups Lof a parabolic subgroup ofGsuch that Kerh^L⊂Kand byLmin(K) orL^G_min(K) the set of minimal elements ofL(K). It follows immediately from lemma 1.3 that

Lemma 1.5. Two elements of Lmin(K) areG-conjugate.

2. Reductive groups over finite fields

Until the end of this note, we assume that Fis an algebraic closure of a finite field (thusp >0) and thatGis endowed with a Frobenius endomorphismF :G→G. Ifg∈G^F, we denote by [g] or [g]G^F theG^F-conjugacy class ofgand byγ_g^Gthe charateristic function of [g] multiplied byCG^F(g).

Proposition 2.1. Let M and M⁰ be two F-stable Levi subgroups of (non necessarily F-stable) parabolic subgroups ofGwhich are geometrically conjugate and letube a unipotent element ofM^F. Assume the following conditions holds :

(a) uis a distinguished element of M, (b) N^G(M)stabilizes the class (u)^M, and (c) A^M(u) =A^G(u).

Then (u)G^F ∩M⁰ is a singleM^0F-conjugacy class.

Proof - Let g be an element of G such that M⁰ = ^gM. Then n = g⁻¹F(g) ∈ N^G(M). Since N^G(M) stabilizes (u)^M and A^M(u) is isomorphic to A^G(u), there exists m∈ M such that nm∈ N^G(M)∩CG^◦(u). From Lang’s theorem, there existsx∈Msuch thatx⁻¹nF(x)n⁻¹=nmn⁻¹. Let h=gx. Then h⁻¹F(h) =nm andM⁰ = ^hM. Let u⁰ = ^hu. Then u⁰ ∈M⁰ and u⁰ ∈ (u)G^F since h⁻¹F(h)∈ CG^◦(u). That proves that (u)^GF ∩M⁰ is not empty. To prove proposition 2.1, it now suffices to prove the following

Lemma. (u)G^F ∩M is a singleM^F-conjugacy class.

Proof of lemma - Letu⁰ be an element of (u)G^F ∩M. Then, there exists an element g ∈ G^F such thatu⁰=^gu. Sinceuis distinguished inM,Z(M)^◦ and^g−1Z(M)^◦are maximal tori ofCG^◦(u).

Thus there exists y ∈CG^◦(u) such that ^yZ(M)^◦ = ^g−1Z(M)^◦. Let h=gy. Then h∈G, u⁰ = ^hu and ^hZ(M)^◦=Z(M)^◦, that is h∈N^G(M). Moreover,h⁻¹F(h) =y⁻¹F(y)∈CG^◦(u).

By hypothesis, N^G(M) stabilizes the class (u)^M. Hence, there exists m ∈ M such that u⁰ =

mu. Let x = h⁻¹m. We have x ∈ C^G(u) and (hx)⁻¹F(hx) = m⁻¹F(m). Hence m⁻¹F(m) = (yx)⁻¹F(yx).

SinceA^M(u) is isomorphic toA^G(u), there existsa∈C^M(u) andb∈CG^◦(u) such thatyx=ba.

Thusm⁻¹F(m) =a⁻¹(b⁻¹F(b))aa⁻¹F(a). This implies thata⁻¹(b⁻¹F(b))abelongs toCG^◦(u)∩M= CM^◦ (u) and thatb⁻¹F(b) also belongs to CM^◦ (u). It results from Lang’s theorem that there exists

c ∈ CM^◦ (u) such that b⁻¹F(b) = c⁻¹F(c). So m⁻¹F(m) = (ca)⁻¹F(ca), and ca ∈ C^M(u). Let m⁰=ma⁻¹c⁻¹. ThenF(m⁰) =m⁰ andu⁰=^m0u.

From now on, we make the following Hypothesis : pis good forG.

We will denote by Reg_uni(G^F) the set of G^F-classes of regular unipotent elements. If U ∈ Reg_uni(G^F) andz∈H¹(F,Z(G)) we denote by Uz the class ^gU whereg∈Gis such thatg⁻¹F(g) is inZ(G) and representsz. Then Reg_uni(G^F) ={Uz|z∈H¹(F,Z(G))}, becausepis good forG.

LetLbe an F-stable Levi subgroup of a parabolic subgroupPofG. Digne, Lehrer and Michel [1] made the following conjecture :

Conjecture 1 : Let u be a regular unipotent element of G^F. There exists a regular unipotent elementv∈L^F such that ^∗R^GL⊂P(γ^G_u) =γ_v^L where ^∗R^GL⊂P is the Lusztig restriction.

They proved this conjecture in the following three cases (cf.[1] and [2]) :

A. Whenever P is F-stable (cf. [1], proposition 5.3). In this case,v is obtained as follows : the intersection of the G^F-conjugacy class of u with P^F is a single P^F-conjugacy class and v is an element of the projection (fromP^F to L^F via the canonical map) of this class. Moreover, the L^F- conjugacy class ofv does not depend onP. We denote byρ^GL : Reg_uni(G^F)→Reg_uni(L^F) the map obtained in this way. We have, for allU ∈Reg_uni(G^F) andz∈H¹(F,Z(G)),

(?) ρ^GL(Uz) = (ρ^GL(U))_h⁰_L(z)

whereh⁰L:H¹(F,Z(G))→H¹(F,Z(L)) is the canonical surjective morphism induced byh^L. B.Whenever the center of L is connected (cf. [1], proposition 5.4). In this case, there is only one L^F-conjugacy class of regular unipotent elements inL^F...

C.Whenever qis large enough (cf.[2], theorem 3.7) : the proof of this fact is particularly difficult and involves character sheaves theory. In this last case, the existence of v has been proved but it has not been explicitly described.

The purpose of this note is to make conjecture 1 precise, that is, to construct a map res^GL : Reg_uni(G^F)→Reg_uni(L^F).

The setL^L_min({1}) is a single conjugacy class underL(cf.lemma 1.5) and isF-stable because the trivial subgroup{1} ofZ(L)/Z(L)^◦ isF-stable. Hence, there is anF-stable element in L^L_min({1}).

Moreover, by proposition 1.2, (b), we can choose en element M of L^L_min({1}) such that M is an F-stable Levi subgroup of anF-stable parabolic subgroupQofL. Becauseh^LM is an isomorphism, the mapρ^LM: Reg_uni(L^F)→Reg_uni(M^F) is bijective (cf.(?)).

Moreover,M ∈ L^G_min(Kerh^L). Hence, by the same argument, M is conjugate, in G, to anF- stable Levi subgroupM0of anF-stable parabolic subgroupQ0 ofG. LetU ∈Reg_uni(G^F) and let v∈ρ^GM₀U. By propositions 1.4 and 2.1, [v]G^F ∩M is a singleM^F-conjugacy class. We define :

res^GL(U) = (ρ^LM)⁻¹([v]G^F ∩M).

Then res^GL(U) is well-defined becauseMandM0are unique up toL^F-conjugacy andG^F-conjugacy respectively. So we have constructed a map res^GL : Reg_uni(G^F)→Reg_uni(L^F).

Conjecture 2 : Let u be a regular unipotent element of G^F and let u^L ∈ res^GL([u]G^F). Then

∗RL⊂P^G (γ_u^G) =γ_u^LL.

Proposition 2.2. (a) res^GL is independent ofP.

(b) IfPisF-stable, then res^GL =ρ^GL.

(c) IfU ∈Reg_uni(G^F), thenres^GLUz= (res^GL U)h⁰L(z). In particular, res^GL is surjective.

(d) Let M be an F-stable Levi subgroup of a parabolic subgroup of G. Assume that each regular unipotent element v of L satisfies A^L(v) ' A^G(v) and that M is G-conjugate to L. If U ∈Reg_uni(G^F)andv∈resL^G(U), then res^GM(U) = [v]G^F ∩M (cf. proposition 2.1).

(e) LetMbe anF-stable Levi subgroup of a parabolic subgroup ofL. ThenresM^G = res^LM◦res^GL. (a), (b) and (c) are easy but (d) and (e) are quite technical. Proofs of these facts will appear in a forthcoming paper.

Remark - If Gis a rational Levi subgroup of a parabolic subgroup of a split reductive group all components of which are of type A, then each F-stable Levi subgroupL of a parabolic subgroup of G is G-conjugate to an F-stable Levi subgroup M of an F-stable parabolic subgroup of G.

Moreover we have A^L(v)'A^G(v) for all regular unipotent v of L because the natural morphism Z(G) → A^G(v) is surjective. Hence the map res^GL is described well by properties (b) and (d) of res^GL given in proposition 2.2.

References

[1] Digne F., Lehrer G. and Michel J., 1992. The characters of the group of rational points of a reductive group with non-connected centre,J. reine angew. Math., 425, p. 155-192.

[2] Digne F., Lehrer G. and Michel J., 1997.On Gel’fand-Graev characters of reductive groups with non-connected centre,J. reine. angew. Math., 491, p. 131-147.

[3] Lusztig G., 1984. Intersection cohomology complexes on a reductive group,Invent. Math., 75, p. 205-272.

[4] Lusztig G., 1995.Study of perverse sheaves arising from graded Hecke algebras,Adv. in Math., 112, p. 147-217.

[5] Spaltenstein N., 1985. On the generalized Springer correspondence for exceptional groups,Adv.

Stud. in Pure Math., 6, p. 317-338.