[Regular Elements and Gelfand-Graev Representations for Disconnected Reductive Groups]
Let be a connected reductive group defined over and let be the corresponding Frobenius endomorphism. Let be a quasi-central automorphism of , which means that is quasi-semi-simple (i.e. stabilises where is a maximal torus included in a Borel subgroup of ) and for any quasi-semi-simple automorphism , where is the conjugation by for all . We suppose also that is rational. We define in this article Gelfand-Graev representations for the group when is unipotent and when it is semi-simple, which extend the -stable Gelfand-Graev representations for connected reductive groups. Let be a -stable rational maximal torus of included in a -stable rational Borel subgroup of . Let be the unipotent radical of . In the connected reductive case, Gelfand-Graev representations of are obtained by inducing an irreducible linear character of which is called a regular character. We define a regular character of as the extension of a -stable regular character of . When is unipotent, -stable Gelfand-Graev representations of are obtained by inducing -stable regular characters of . In this case, we define Gelfand-Graev representations of as the representations obtained by inducing regular characters of . When is semi-simple, the definition of Gelfand-Graev representations is more complicated. Gelfand-Graev representations of have similar properties to Gelfand-Graev representations of . They are multiplicity free and their Harish-Chandra restrictions to a rational -stable Levi subgroup included in a rational -stable parabolic subgroup still are Gelfand-Graev representations. We say that an element of is regular if the dimension of its centralizer in is minimal among all elements of . The dual of any Gelfand-Graev representation of is zero outside regular unipotent elements of when is unipotent (resp. outside regular pseudo-unipotent elements of , i.e. conjugates under of regular elements of , when is semi-simple). Moreover, Gelfand-Graev representations can be used to calculate the average value of irreducible characters of on the set of -classes of regular unipotent (resp. pseudo-unipotent) elements of if is unipotent (resp. semi-simple). When is semi-simple, the characteristic can be chosen good for and we can get the exact values of irreducible characters of on -classes of regular pseudo-unipotent elements of .
Soient un groupe algébrique réductif connexe défini sur et l’endomorphisme de Frobenius correspondant. Soit un automorphisme rationnel quasi-central de . Nous construisons ci-dessous l’équivalent des représentations de Gelfand-Graev du groupe , lorsque 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.
Mot clés : groupes réductifs finis, groupes algébriques non connexes
Keywords: finite reductive groups, disconnected algebraic groups
@article{BSMF_2004__132_2_157_0, 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}, pages = {157--199}, publisher = {Soci\'et\'e math\'ematique de France}, volume = {132}, number = {2}, year = {2004}, doi = {10.24033/bsmf.2463}, mrnumber = {2075565}, zbl = {1059.20017}, language = {fr}, url = {http://archive.numdam.org/articles/10.24033/bsmf.2463/} }
TY - JOUR AU - Sorlin, Karine TI - Éléments réguliers et représentations de Gelfand-Graev des groupes réductifs non connexes JO - Bulletin de la Société Mathématique de France PY - 2004 SP - 157 EP - 199 VL - 132 IS - 2 PB - Société mathématique de France UR - http://archive.numdam.org/articles/10.24033/bsmf.2463/ DO - 10.24033/bsmf.2463 LA - fr ID - BSMF_2004__132_2_157_0 ER -
%0 Journal Article %A Sorlin, Karine %T Éléments réguliers et représentations de Gelfand-Graev des groupes réductifs non connexes %J Bulletin de la Société Mathématique de France %D 2004 %P 157-199 %V 132 %N 2 %I Société mathématique de France %U http://archive.numdam.org/articles/10.24033/bsmf.2463/ %R 10.24033/bsmf.2463 %G fr %F 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://archive.numdam.org/articles/10.24033/bsmf.2463/
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