Éléments réguliers et représentations de Gelfand-Graev des groupes réductifs non connexes

Abstract

Résumé. -Soient G un groupe algébrique réductif connexe défini sur Fq et F l'endomorphisme de Frobenius correspondant. Soit σ un automorphisme rationnel quasicentral de G. Nous construisons ci-dessous l'équivalent des représentations de GelfandGraev du groupe G F = G F · σ , lorsque σ est unipotent et lorsqu'il est semi-simple. Nous montrons de plus que ces représentations vérifient des propriétés semblables a celles vérifiées par les représentations de Gelfand-Graev dans le cas connexe en particulier par rapport auxéléments réguliers. Abstract (Regular Elements and Gelfand-Graev Representations for Disconnected Reductive Groups)Let G be a connected reductive group defined over Fq and let F be the corresponding Frobenius endomorphism. Let σ be a quasi-central automorphism of G, which means that σ is quasi-semi-simple (i.e. σ stabilises (T ⊂ B) where T is a maximal torus included in a Borel subgroup B of G) and dim(G σ ) > dim(G σ ) for any quasi-semisimple automorphism σ = σ • ad(g), where ad(g) is the conjugation by g for all g ∈ G.We suppose also that σ is rational.We define in this article Gelfand-Graev representations for the group G F = G F · σ when σ is unipotent and when it is semi-simple, which extend the σ-stable GelfandGraev representations for connected reductive groups.

On semisimple classes and semisimple characters in finite reductive groups

“…for any i ≥ 0, where E(ρ s,1 ) denotes an extension of ρ s,1 to H F ′ (such extensions exist by [19, 11.22]). Now, [22,Proposition 8.1]…”

“…In [22], Sorlin develops a theory of Gelfand-Graev characters for disconnected groups when σ is semisimple or unipotent. These characters are extensions of some σ-stable Gelfand-Graev characters of G F to H F ; see [22, §5].…”

“…Remark 3.9. The character E(Γ 1 ) defined in Equation ( 35) is a Gelfand-Graev character of H F in the sense of [22], because the linear character E(φ 1 ) defined in Equation ( 34) is regular [22,Définition 4.1]). Note that by [8, 12.2.3] the graph automorphism σ that we consider here always satisfies the condition (RS) defined in [22, Notation 2.1].…”

“…But S E ≃ S 3 and the centralizer of ρ(σ) in S E has order 2. Thus, there is an A χ -stable character χ in E. By [22,Proposition 8.1], because σ is semisimple). Then E( χ) has a non-zero value on at least U 1 or U −1 .…”

“…Digne-Michel [15] and Malle [21] developed a theory of Deligne-Lusztig characters for finite disconnected reductive groups. Using this theory, Sorlin [22] constructed extensions of Gelfand-Graev characters of G F to G F ⋊ σ , where σ is a quasi-central semisimple or unipotent automorphism of G. We will use results of [22] in order to prove that, under certain assumptions, the semisimple characters of G F are extendible to their inertia groups in the full automorphism group.…”