Joint IAS/Princeton University Number Theory Seminar

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.