The local Gan-Gross-Prasad for tempered representations of unitary groups (1/2)

by Dr R. Beuzart (Univ. of Paris, France)

Monday 22 Apr 2013, 16:00 → 17:00 Asia/Kolkata

AG-77 (Colaba Campus)

ABSTRACT: Let $E/F$ be a quadratic extension of $p$-adic fields. Let $W\subset V$ be a pair of Hermitian spaces whose dimensions have different parities and $H=U(W)\subset G=U(V)$ the associated unitary groups. Then Gan, Gross and Prasad have defined a multiplicity $m(\pi,\sigma)$ for all smooth irreducible representations $\pi$ and $\sigma$ of $G(F)$ and $H(F)$ respectively. If $dim(W)=dim(V)-1$, it is just the dimension of $Hom_H(\pi,\sigma)$. This multiplicity is always less than 1 and the Gan-Gross-Prasad conjecture predicts for which pairs of representations we get the multiplicity one. Their predictions are based on the conjectural Langlands correspondence. In four recent papers, Waldspurger and Moeglin-Waldspurger proved the analogue of the conjecture for special orthogonal groups. In this serie of two lectures, I will try to explain a similar proof in the case of unitary groups. It is based on two parallels integral formulas: one for the multiplicity and one for certain $\epsilon$-factors. The first lecture will be more elementary and aims to give an overview of the proof, in the second lecture I will try to go more deeply into the details.