La transformée de Fourier pour les espaces tordus sur un groupe réductif $\mathfrak{p}$-adique I. Le théorème de Paley-Wiener

Abstract

Let ${\boldsymbol{G}}$ be a connected reductive group defined over a non--Archimedean local field $F$. Put $G={\boldsymbol{G}}(F)$. Let $θ$ be an $F$--automorphism of ${\boldsymbol{G}}$, and let $ω$ be a smooth character of $G$. This paper is concerned with the smooth complex representations $π$ of $G$ such that $π^θ=π\circθ$ is isomorphic to $ωπ=ω\otimesπ$. If $π$ is admissible, in particular irreducible, the choice of an isomorphism $A$ from $ωπ$ to $π^θ$ (and of a Haar measure on $G$) defines a distribution $Θ_π^A={\rm tr}(π\circ A)$ on $G$. The twisted Fourier transform associates to a compactly supported locally constant function $f$ on $G$, the function $(π,A)\mapsto Θ_π^A(f)$ on a suitable Grothendieck group. Here we describe its image (Paley--Wiener theorem), and we reduce the description of its kernel (spectral density theorem) to a result on the discrete part of the theory.