Éléments réguliers et représentations de Gelfand-Graev des groupes réductifs non connexes  [ Regular Elements and Gelfand-Graev Representations for Disconnected Reductive Groups ]
Bulletin de la Société Mathématique de France, Volume 132 (2004) no. 2, p. 157-199

Let G be a connected reductive group defined over 𝔽 q 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(G σ )>dim(G σ ' ) for any quasi-semi-simple automorphism σ ' =σ ad (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 G ˜ F =G F ·σ 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 G F are obtained by inducing an irreducible linear character of U F which is called a regular character. We define a regular character of U F ·σ as the extension of a σ-stable regular character of U F . When σ is unipotent, σ-stable Gelfand-Graev representations of G F are obtained by inducing σ-stable regular characters of U F . In this case, we define Gelfand-Graev representations of G F ·σ as the representations obtained by inducing regular characters of U F ·σ. When σ is semi-simple, the definition of Gelfand-Graev representations is more complicated. Gelfand-Graev representations of G F ·σ have similar properties to Gelfand-Graev representations of G F . 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 G·σ is regular if the dimension of its centralizer in G is minimal among all elements of G·σ. The dual of any Gelfand-Graev representation of G F ·σ is zero outside regular unipotent elements of G F ·σ when σ is unipotent (resp. outside regular pseudo-unipotent elements of G F ·σ, i.e. conjugates under G of regular elements of U·σ, when σ is semi-simple). Moreover, Gelfand-Graev representations can be used to calculate the average value of irreducible characters of G F ·σ on the set of G F -classes of regular unipotent (resp. pseudo-unipotent) elements of G F ·σ if σ is unipotent (resp. semi-simple). When σ is semi-simple, the characteristic can be chosen good for (G σ ) 0 and we can get the exact values of irreducible characters of G F ·σ on G F -classes of regular pseudo-unipotent elements of G F ·σ.

Soient G un groupe algébrique réductif connexe défini sur 𝔽 q 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 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 à celles vérifiées par les représentations de Gelfand-Graev dans le cas connexe en particulier par rapport aux éléments réguliers.

DOI : https://doi.org/10.24033/bsmf.2463
Classification:  20C33,  20G05
Keywords: finite reductive groups, disconnected algebraic groups
     author = {Sorlin, Karine},
     title = {\'El\'ements r\'eguliers et repr\'esentations de~Gelfand-Graev des~groupes r\'eductifs non connexes},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {132},
     number = {2},
     year = {2004},
     pages = {157-199},
     doi = {10.24033/bsmf.2463},
     zbl = {1059.20017},
     mrnumber = {2075565},
     language = {fr},
     url = {http://www.numdam.org/item/BSMF_2004__132_2_157_0}
Sorlin, Karine. Éléments réguliers et représentations de Gelfand-Graev des groupes réductifs non connexes. Bulletin de la Société Mathématique de France, Volume 132 (2004) no. 2, pp. 157-199. doi : 10.24033/bsmf.2463. http://www.numdam.org/item/BSMF_2004__132_2_157_0/

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