Caractères tordus des représentations admissibles

Details Caractères tordus des représentations admissibles Caractères tordus des représentations admissibles

2010-07-21

arxiv.org/abs/1007.3576v1

Mathematics

Representation Theory

100 pages

Scientific paper

Let $F$ be a non-Archimedean locally compact field (of any characteristic), ${\bf G}$ be a connected reductive group defined over $F$, $\theta$ be an $F$-automorphism of ${\bf G}$, and $\kappa$ be a character of ${\bf G}(F)$. Let us fix a Haar measure $dg$ on ${\bf G}(F)$. If $\pi$ is a smooth irreducible $(\theta,\kappa)$-stable (complex) representation of ${\bf G}(F)$, that is such that the representation $\pi^\theta=\pi\circ \theta$ of ${\bf G}(F)$ is isomorphic to $\kappa\pi=\pi\otimes \kappa$, then the choice of an isomorphism $A$ from $\kappa\pi$ to $\pi^\theta$ defines a distribution $\Theta_\pi^A$, called the ($A$-)twisted character of $\pi$: for all function $f$ on ${\bf G}(F)$ which is locally constant with compact support, 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 $\theta$ --- moreover, instead of fixing $\theta$, we work with a twisted ${\bf G}$-space defined over $F$. We prove in particular that they are locally constant functions on the open dense subset of ${\bf G}(F)$ formed of those elements which are $\theta$-quasi-regular, and we describe how these functions characters behave with respect to parabolic induction and Jacquet restriction. This leads us to take again the Steinberg theory of automorphisms of an algebraic group, from a rationnal point of view.

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