Caract\`eres tordus des repr\'esentations admissibles

Abstract

Let $F$ be a non--Archimedean locally compact field (${\rm car}(F)\geq 0$), ${\bf G}$ be a connected reductive group defined over $F$, $\theta$ be an $F$--automorphism of ${\bf G}$, and $\omega$ be a character of ${\bf G}(F)$. We fix a Haar measure $dg$ on ${\bf G}(F)$. For a smooth irreducible $(\theta,\omega)$--stable complex representation $\pi$ of ${\bf G}(F)$, that is such that $\pi\circ \theta\simeq \pi\otimes \omega$, the choice of an isomorphism $A$ from $\pi\otimes \omega$ to $\pi\circ \theta$ defines a distribution $\Theta_\pi^A$, called the \og ($A$--)twisted character of $\pi$\fg: for a compactly supported locally constant function $f$ on ${\bf G}(F)$, we put $\Theta_\pi^A(f)={\rm trace}(\pi(fdg)\circ A)$. In this paper, we study these distributions $\Theta_\pi^A$, without any restrictive hypothesis on $F$, ${\bf G}$ or $\theta$. We prove in particular that the restriction of $\Theta_\pi^A$ on the open dense subset of ${\bf G}(F)$ formed of those elements which are $\theta$--quasi--regular is given by a locally constant function, and we describe how this function behaves with respect to parabolic induction and Jacquet restriction. This leads us to take up again the Steinberg theory of automorphisms of an algebraic group, from a rationnal point of view.