Nagoya Mathematical Journal

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

J.-L. Waldspurger

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 20 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1150810011

Mathematical Reviews number (MathSciNet)
MR2235346

Zentralblatt MATH identifier
1170.20308

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

Keywords
non-connected finite group representation character-sheave Whittaker model

Citation

Waldspurger, J.-L. Le groupe $GL_{n}$ tordu, sur un corps fini. Nagoya Math. J. 182 (2006), 313--379. https://projecteuclid.org/euclid.nmj/1150810011


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