Raphael Beuzart-Plessis (IAS)
The local Gan-Gross-Prasad conjecture for tempered representations of unitary groups.

Abstract: Let $E/F$ be a quadratic extension of $p$-adic fields. Let $V_n?V_{n+1}$ be hermitian spaces of dimension $n$ and $n+1$ respectively. For $\sigma$ and $\pi$ smooth irreducible representations of $U(V_n)$ and $U(V_{n+1})$ set $m(\pi,\sigma)=dim \; Hom_{U(V_n)}(\pi,\sigma)$. This multiplicity is always less or equal to 1 and the Gan-Gross-Prasad conjecture predicts for which pairs of representations we get multiplicity 1. Their predictions are based on the conjectural Langlands correspondence. In this talk, I will explain a proof of the Gan-Gross-Prasad conjecture for the so-called tempered representations. This is in straight continuation of Waldspurger's work dealing with special orthogonal groups.