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

Sorlin, Karine

Bulletin de la Société Mathématique de France, Tome 132 (2004), p. 157-199 / Harvested from Numdam

Résumé

Soient $G$ un groupe algébrique réductif connexe défini sur $\mathbb {F}_q$ et $F$ l’endomorphisme de Frobenius correspondant. Soit $\sigma $ un automorphisme rationnel quasi-central de $G$. Nous construisons ci-dessous l’équivalent des représentations de Gelfand-Graev du groupe $\widetilde{G}^F=G^F\!\cdot \langle \sigma \rangle $, lorsque $\sigma $ 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.

Abstract

Let $G$ be a connected reductive group defined over $\mathbb {F}_q$ and let $F$ be the corresponding Frobenius endomorphism. Let $\sigma $ be a quasi-central automorphism of $G$, which means that $\sigma $ is quasi-semi-simple (i.e. $\sigma $ stabilises $(T\subset B)$ where $T$ is a maximal torus included in a Borel subgroup $B$ of $G$) and $\dim (G^\sigma )>\dim (G^{\sigma ^{\prime }})$ for any quasi-semi-simple automorphism $\sigma ^{\prime }= \sigma \circ {\rm ad}(g)$, where ${\rm ad}(g)$ is the conjugation by $g$ for all $g\in G$. We suppose also that $\sigma $ is rational. We define in this article Gelfand-Graev representations for the group $\widetilde{G}^F=G^F\!\cdot \langle \sigma \rangle $ when $\sigma $ is unipotent and when it is semi-simple, which extend the $\sigma $-stable Gelfand-Graev representations for connected reductive groups. Let $T$ be a $\sigma $-stable rational maximal torus of $G$ included in a $\sigma $-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\!{\cdot } \langle \sigma \rangle $ as the extension of a $\sigma $-stable regular character of $U^F\!$. When $\sigma $ is unipotent, $\sigma $-stable Gelfand-Graev representations of $G^F\!$ are obtained by inducing $\sigma $-stable regular characters of $U^F\!$. In this case, we define Gelfand-Graev representations of $G^F\!{\cdot }\langle \sigma \rangle $ as the representations obtained by inducing regular characters of $U^F\!{\cdot }\langle \sigma \rangle $. When $\sigma $ is semi-simple, the definition of Gelfand-Graev representations is more complicated. Gelfand-Graev representations of $G^F\!{\cdot }\langle \sigma \rangle $ have similar properties to Gelfand-Graev representations of $G^F\!$. They are multiplicity free and their Harish-Chandra restrictions to a rational $\sigma $-stable Levi subgroup included in a rational $\sigma $-stable parabolic subgroup still are Gelfand-Graev representations. We say that an element of $G{\cdot }\sigma $ is regular if the dimension of its centralizer in $G$ is minimal among all elements of $G{\cdot }\sigma $. The dual of any Gelfand-Graev representation of $G^F\!{\cdot }\sigma $ is zero outside regular unipotent elements of $G^F\!{\cdot }\sigma $ when $\sigma $ is unipotent (resp. outside regular pseudo-unipotent elements of $G^F\!{\cdot }\sigma $, i.e. conjugates under $G$ of regular elements of $U{\cdot }\sigma $, when $\sigma $ is semi-simple). Moreover, Gelfand-Graev representations can be used to calculate the average value of irreducible characters of $G^F\!{\cdot }\langle \sigma \rangle $ on the set of $G^F\!$-classes of regular unipotent (resp. pseudo-unipotent) elements of $G^F\!{\cdot }\sigma $ if $\sigma $ is unipotent (resp. semi-simple). When $\sigma $ is semi-simple, the characteristic can be chosen good for $(G^\sigma )^0$ and we can get the exact values of irreducible characters of $G^F\!{\cdot }\langle \sigma \rangle $ on $G^F\!$-classes of regular pseudo-unipotent elements of $G^F\!{\cdot }\sigma $.

Détail

Publié le : 2004-01-01

MR 2075565 | Zbl 1059.20017 | 1 citation dans Numdam

DOI : https://doi.org/10.24033/bsmf.2463
Classification:  20C33,  20G05
Mots clés: groupes réductifs finis, groupes algébriques non connexes

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