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

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.