SMF

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

Regular Elements and Gelfand-Graev Representations for Disconnected Reductive Groups

Karine Sorlin
Éléments réguliers et représentations de Gelfand-Graev des groupes réductifs non connexes
     
                
  • Année : 2004
  • Fascicule : 2
  • Tome : 132
  • Format : Électronique
  • Langue de l'ouvrage :
    Français
  • Class. Math. : 20C33, 20G05
  • Pages : 157-199
  • DOI : 10.24033/bsmf.2463
Soient G un groupe algébrique réductif connexe défini sur Fq et F l'endomorphisme de Frobenius correspondant. Soit σ un automorphisme rationnel quasi-central de G. Nous construisons ci-dessous l'équivalent des représentations de Gelfand-Graev du groupe ˜GF=GFσ, lorsque σ est unipotent et lorsqu'il est semi-simple. Nous montrons de plus que ces représentations vérifient des propriétés semblables à celles vérifiées par les représentations de Gelfand-Graev dans le cas connexe en particulier par rapport aux éléments réguliers.
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 (TB) where T is a maximal torus included in a Borel subgroup B of G) and dim(Gσ)>dim(Gσ) for any quasi-semi-simple automorphism σ=σad(g), where ad(g) is the conjugation by g for all gG. We suppose also that σ is rational. We define in this article Gelfand-Graev representations for the group ˜GF=GFσ when σ is unipotent and when it is semi-simple, which extend the σ-stable Gelfand-Graev representations for connected reductive groups. Let T be a σ-stable rational maximal torus of G included in a σ-stable rational Borel subgroup of G. Let U be the unipotent radical of B. In the connected reductive case, Gelfand-Graev representations of GF are obtained by inducing an irreducible linear character of UF which is called a regular character. We define a regular character of UFσ as the extension of a σ-stable regular character of UF. When σ is unipotent, σ-stable Gelfand-Graev representations of GF are obtained by inducing σ-stable regular characters of UF. In this case, we define Gelfand-Graev representations of GFσ as the representations obtained by inducing regular characters of UFσ. When σ is semi-simple, the definition of Gelfand-Graev representations is more complicated. Gelfand-Graev representations of GFσ have similar properties to Gelfand-Graev representations of GF. They are multiplicity free and their Harish-Chandra restrictions to a rational σ-stable Levi subgroup included in a rational σ-stable parabolic subgroup still are Gelfand-Graev representations. We say that an element of Gσ is regular if the dimension of its centralizer in G is minimal among all elements of Gσ. The dual of any Gelfand-Graev representation of GFσ is zero outside regular unipotent elements of GFσ when σ is unipotent (resp. outside regular pseudo-unipotent elements of GFσ, i.e. conjugates under G of regular elements of Uσ, when σ is semi-simple). Moreover, Gelfand-Graev representations can be used to calculate the average value of irreducible characters of GFσ on the set of GF- es of regular unipotent (resp. pseudo-unipotent) elements of GFσ if σ is unipotent (resp. semi-simple). When σ is semi-simple, the characteristic can be chosen good for (Gσ)0 and we can get the exact values of irreducible characters of GFσ on GF- es of regular pseudo-unipotent elements of GFσ.
Groupes réductifs finis, groupes algébriques non connexes
Finite reductive groups, disconnected algebraic groups


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