La transform\'ee de Fourier pour les espaces tordus sur un groupe r\'eductif $\mathfrak{p}$-adique I. Le th\'eor\`eme de Paley-Wiener

Let $\mathit{G}$ be a connected reductive group defined over a non–Archimedean local field $F$. Put $G=\mathit{G}(F)$. Let $\theta$ be an $F$–automorphism of $\mathit{G}$, 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 ${\mathrm{\Theta }}_{\pi }^A=\mathrm{t}\mathrm{r}(\pi \circ A)$ on $G$. The twisted Fourier transform associates to a compactly supported locally constant function $f$ on $G$, the function $(\pi ,A)\mapsto {\mathrm{\Theta }}_{\pi }^A(f)$ 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.