# The local Gan-Gross-Prasad conjecture for tempered representations of unitary groups

 Joint IAS/Princeton University Number Theory Seminar Topic: The local Gan-Gross-Prasad conjecture for tempered representations of unitary groups Speaker: Raphaël Beuzart-Plessis Affiliation: Member, School of Mathematics Date: Thursday, October 24 Time/Room: 4:30pm - 5:30pm/Fine 214, Princeton University

Let $E/F$ be a quadratic extension of $p$-adic fields. Let $V_n\subset V_{n+1}$ 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 $m(\pi,\sigma)=\dim \mathrm{Hom}_{U(V_n)}(\pi,\sigma)$. This multiplicity is always less than 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.