Éléments réguliers et représentations de Gelfand-Graev des groupes réductifs non connexes
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 ${\stackrel{˜}{G}}^{F}={G}^{F}\phantom{\rule{-0.166667em}{0ex}}·〈\sigma 〉$, 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 $\left(T\subset B\right)$ 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 ${\stackrel{˜}{G}}^{F}={G}^{F}\phantom{\rule{-0.166667em}{0ex}}·〈\sigma 〉$ 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 ${G}^{F}\phantom{\rule{-0.166667em}{0ex}}$ are obtained by inducing an irreducible linear character of ${U}^{F}\phantom{\rule{-0.166667em}{0ex}}$ which is called a regular character. We define a regular character of ${U}^{F}\phantom{\rule{-0.166667em}{0ex}}·〈\sigma 〉$ as the extension of a $\sigma$-stable regular character of ${U}^{F}\phantom{\rule{-0.166667em}{0ex}}$. When $\sigma$ is unipotent, $\sigma$-stable Gelfand-Graev representations of ${G}^{F}\phantom{\rule{-0.166667em}{0ex}}$ are obtained by inducing $\sigma$-stable regular characters of ${U}^{F}\phantom{\rule{-0.166667em}{0ex}}$. In this case, we define Gelfand-Graev representations of ${G}^{F}\phantom{\rule{-0.166667em}{0ex}}·〈\sigma 〉$ as the representations obtained by inducing regular characters of ${U}^{F}\phantom{\rule{-0.166667em}{0ex}}·〈\sigma 〉$. When $\sigma$ is semi-simple, the definition of Gelfand-Graev representations is more complicated. Gelfand-Graev representations of ${G}^{F}\phantom{\rule{-0.166667em}{0ex}}·〈\sigma 〉$ have similar properties to Gelfand-Graev representations of ${G}^{F}\phantom{\rule{-0.166667em}{0ex}}$. 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 ${G}^{F}\phantom{\rule{-0.166667em}{0ex}}·\sigma$ is zero outside regular unipotent elements of ${G}^{F}\phantom{\rule{-0.166667em}{0ex}}·\sigma$ when $\sigma$ is unipotent (resp. outside regular pseudo-unipotent elements of ${G}^{F}\phantom{\rule{-0.166667em}{0ex}}·\sigma$, 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 ${G}^{F}\phantom{\rule{-0.166667em}{0ex}}·〈\sigma 〉$ on the set of ${G}^{F}\phantom{\rule{-0.166667em}{0ex}}$-classes of regular unipotent (resp. pseudo-unipotent) elements of ${G}^{F}\phantom{\rule{-0.166667em}{0ex}}·\sigma$ 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 ${G}^{F}\phantom{\rule{-0.166667em}{0ex}}·〈\sigma 〉$ on ${G}^{F}\phantom{\rule{-0.166667em}{0ex}}$-classes of regular pseudo-unipotent elements of ${G}^{F}\phantom{\rule{-0.166667em}{0ex}}·\sigma$.

DOI : https://doi.org/10.24033/bsmf.2463
Classification : 20C33,  20G05
Mots clés : groupes réductifs finis, groupes algébriques non connexes
@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},
zbl = {1059.20017},
mrnumber = {2075565},
language = {fr},
url = {http://www.numdam.org/articles/10.24033/bsmf.2463/}
}
Sorlin, Karine. Éléments réguliers et représentations de Gelfand-Graev des groupes réductifs non connexes. Bulletin de la Société Mathématique de France, Tome 132 (2004) no. 2, pp. 157-199. doi : 10.24033/bsmf.2463. http://www.numdam.org/articles/10.24033/bsmf.2463/

[1] R. Carter - Finite groups of Lie type, Wiley-Interscience, 1985. | MR 794307 | Zbl 0567.20023

[2] F. Digne, G. Lehrer & J. Michel - « The characters of the group of rational points of a reductive group with non-connected centre », J. reine angew. Math. 425 (1992), p. 155-192. | MR 1151318 | Zbl 0739.20018

[3] F. Digne & J. Michel - Representations of Finite Groups of Lie Type, London Math. Soc. Student Texts, vol. 21, Cambridge University Press, Cambridge, 1991. | MR 1118841 | Zbl 0815.20014

[4] -, « Groupes réductifs non connexes », Ann. Sci. École Normale Sup. 27 (1994), p. 345-406. | Numdam | MR 1272294 | Zbl 0846.20040

[5] -, « Points fixes des automorphismes quasi-semi-simples », C.R. Acad. Sci. Paris, Sér. I 334 (2002), p. 1055-1060. | MR 1911646 | Zbl 1001.20043

[6] G. Malle - « Generalized Deligne-Lusztig characters », J. Alg. 159 (1993), no. 1, p. 64-97. | MR 1231204 | Zbl 0812.20024

[7] N. Spaltenstein - Classes unipotentes et sous-groupes de Borel, Lectures Notes in Math., vol. 946, Springer, 1982. | MR 672610 | Zbl 0486.20025

[8] R. Steinberg - Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc., vol. 80, American Mathematical Society, Providence, RI, 1968. | MR 230728 | Zbl 0164.02902