Abstract: This talk is related to the relative Langlands' program, which aims to extend the classical Langlands' program to spherical varieties. In the classical case, a well-known \emph{trace Paley-Wiener theorem} was given by Bernstein, Deligne and Kazhdan in 1986. It gives a characterization of the functions
$$\pi \mapsto \mathrm{Tr}(\pi(f)),$$
with $G$ a reductive $p$-adic group, and where $\pi$ ranges over isomorphism classes of smooth…
