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

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Raphaël Beuzart-Plessis , IAS
Fine Hall 214

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 $m(\pi,\sigma)=dim\; Hom_{U(V_n)}(\pi,\sigma)$. 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.