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

Abstract

Let $\mathbf{G}$ be a connected reductive group defined over a non--Archimedean local field $F$. Put $G=\mathbf{G}(F)$. Let $\theta$ be an $F$--automorphism of $\mathbf{G}$, and let $\omega$ be a smooth character of $G$. This paper is concerned with the smooth complex representations $\pi$ of $G$ such that ${\pi }^{\theta }=\pi \circ \theta$ is isomorphic to $\omega \pi =\omega \otimes \pi$. If $\pi$ is admissible, in particular irreducible, the choice of an isomorphism $A$ from $\omega \pi$ to ${\pi }^{\theta }$ (and of a Haar measure on $G$) defines a distribution ${\Theta }_{\pi }^A=\mathrm{tr}(\pi \circ A)$ on $G$. The twisted Fourier transform associates to a compactly supported locally constant function $f$ on $G$, the function $(\pi ,A)\mapsto {\Theta }_{\pi }^A(f)$ 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.

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