Nagoya Mathematical Journal

Le groupe $GL_{n}$ tordu, sur un corps fini

J.-L. Waldspurger

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Let $\mathbb{F}_{q}$ be a finite field, $G = GL_{n}(\mathbb{F}_{q})$, $\theta$ be the outer automorphism of $G$, suitably normalized. Consider the non-connected group $G \rtimes \{1, \theta\}$ and its connected component $\tilde{G} = G\theta$. We know two ways to produce functions on $\tilde{G}$, with complex values and invariant by conjugation by $G$: on one hand, let $\pi$ be an irreducible representation of $G$ we can and do extend to a representation $\pi^{+}$ of $G \rtimes \{1, \theta\}$, then the restriction $\operatorname{\it trace}_{\tilde{G}}\pi^{+}$ to $\tilde{G}$ of the character of $\pi^{+}$ is such a function; on the other hand, Lusztig define character-sheaves ${\bf a}$, whose characteristic functions $\phi({\bf a})$ are such functions too. We consider only "quadratic-unipotent" representations. For all such representation $\pi$, we define a suitable extension $\pi^{+}$, a character-sheave ${\mathfrak f}(\pi)$ and we prove an identity $\operatorname{\it trace}_{\tilde{G}}\pi^{+} = \gamma(\pi)\phi(\mathfrak{f}(\pi))$ with an explicit complex number $\gamma(\pi)$.

Article information

Nagoya Math. J., Volume 182 (2006), 313-379.

First available in Project Euclid: 20 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20C33: Representations of finite groups of Lie type 20G40: Linear algebraic groups over finite fields

non-connected finite group representation character-sheave Whittaker model


Waldspurger, J.-L. Le groupe $GL_{n}$ tordu, sur un corps fini. Nagoya Math. J. 182 (2006), 313--379.

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  • T. Asai, Unipotent class functions of split special orthogonal groups $SO^+_2n$ over finite fields , Comm. Algebra, 12 (1984), 517--615.
  • A.-M. Aubert, J. Michel and R. Rouquier, Correspondance de Howe pour les groupes réductifs sur les corps finis , Duke Math. J., 83 (1996), 353--397.
  • C. Curtis and I. Reiner, Methods of representation theory, with applications to finite groups and orders, vol. 1, Wiley Interscience (1981).
  • F. Digne and J. Michel, Groupes réductifs non connexes , Ann. Scient. ENS, 27 (1994), 345--406.
  • G. Lusztig, Character sheaves V , Adv. in Math., 61 (1986), 103--165.
  • --------, Character sheaves on disconnected groups III , Representation Th., 8 (2004), 125--154.
  • --------, Character sheaves on disconnected groups IV , Representation Th., 8 (2004), 155--188.
  • --------, Character sheaves on disconnected groups V , Representation Th., 8 (2004), 346--376.
  • --------, Green functions and character sheaves , Annals of Math., 131 (1990), 355--408.
  • N. Spaltenstein, Classes unipotentes et sous-groupes de Borel, Springer L\.N. 946 (1980).
  • J.-L. Waldspurger, Une conjecture de Lusztig pour les groupes classiques \bookinfoMémoires SMF 96 (2004).
  • --------, Intégrales orbitales nilpotentes et endoscopie pour les groupes classiques non ramifiés, Astérisque 269 (2001).