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

Sorlin, Karine

Bulletin de la Société Mathématique de France, Tome 132 (2004) no. 2, pp. 157-199.

Soient $G$ un groupe algébrique réductif connexe défini sur ${𝔽}_q$ 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 ${\tilde{G}}^F=G^F·\langle \sigma \rangle$, 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.

Let $G$ be a connected reductive group defined over ${𝔽}_q$ 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 $dim\left(G^{\sigma }\right)>dim\left(G^{{\sigma }^{\text{'}}}\right)$ for any quasi-semi-simple automorphism ${\sigma }^{\text{'}}=\sigma \circ \mathrm{ad}\left(g\right)$, where $\mathrm{ad}\left(g\right)$ 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 ${\tilde{G}}^F=G^F·\langle \sigma \rangle$ 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 $U^F·\langle \sigma \rangle$ 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 $G^F·\langle \sigma \rangle$ as the representations obtained by inducing regular characters of $U^F·\langle \sigma \rangle$. When $\sigma$ is semi-simple, the definition of Gelfand-Graev representations is more complicated. Gelfand-Graev representations of $G^F·\langle \sigma \rangle$ 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 $G·\sigma$ is regular if the dimension of its centralizer in $G$ is minimal among all elements of $G·\sigma$. The dual of any Gelfand-Graev representation of $G^F·\sigma$ is zero outside regular unipotent elements of $G^F·\sigma$ when $\sigma$ is unipotent (resp. outside regular pseudo-unipotent elements of $G^F·\sigma$, i.e. conjugates under $G$ of regular elements of $U·\sigma$, when $\sigma$ is semi-simple). Moreover, Gelfand-Graev representations can be used to calculate the average value of irreducible characters of $G^F·\langle \sigma \rangle$ on the set of $G^F$-classes of regular unipotent (resp. pseudo-unipotent) elements of $G^F·\sigma$ if $\sigma$ is unipotent (resp. semi-simple). When $\sigma$ is semi-simple, the characteristic can be chosen good for ${\left(G^{\sigma }\right)}^0$ and we can get the exact values of irreducible characters of $G^F·\langle \sigma \rangle$ on $G^F$-classes of regular pseudo-unipotent elements of $G^F·\sigma$.

Classification : 20C33, 20G05
Mot clés : groupes réductifs finis, groupes algébriques non connexes
Keywords: finite reductive groups, disconnected algebraic groups

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