Let ${ \boldsymbol{ G}}$ be a connected reductive group defined over a non--Archimedean local field $F$. Put $G={ \boldsymbol{ G}}(F)$. Let $\theta$ be an $F$--automorphism of ${ \boldsymbol{ G}}$, 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 $\Theta_\pi^A={ \rm tr}(\pi\circ A)$ 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.
Categories: math.RT