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

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 $\gamma(\pi)$.