La conjecture locale de Gross-Prasad pour les repr\'esentations temp\'er\'ees des groupes unitaires

TL;DR

This paper proves a part of the local Gross-Prasad conjecture for tempered representations of unitary groups over non-archimedean fields, providing an integral formula for multiplicities and assuming properties of tempered L-packets.

Contribution

It introduces an integral formula for multiplicities in the context of tempered representations of unitary groups and advances the proof of the local Gross-Prasad conjecture.

Findings

Established an integral formula for multiplicities m(π,σ).

Proved a part of the local Gross-Prasad conjecture for tempered unitary groups.

Connected the conjecture to properties of tempered L-packets.

Abstract

Let $E/F$ be a quadratic extension of non-archimedean local fields of characteristic 0 and let $G=U(n)$, $H=U(m)$ be unitary groups of hermitian spaces $V$ and $W$. Assume that $V$ contains $W$ and that the orthogonal complement of $W$ is a quasisplit hermitian space (i.e. whose unitary group is quasisplit over $F$). Let $\pi$ and $\sigma$ be smooth irreducible representations of $G(F)$ and $H(F)$ respectively. Then Gan, Gross and Prasad have defined a multiplicity $m(\pi,\sigma)$ which for $m=n-1$ is just the dimension of $Hom_{H(F)}(\pi,\sigma)$. For $\pi$ and $\sigma$ tempered, we state and prove an integral formula for this multiplicity. As a consequence, assuming some expected properties of tempered $L$-packets, we prove a part of the local Gross-Prasad conjecture for tempered representations of unitary groups. This article represents a straight continuation of recent papers of Waldspurger dealing with special orthogonal groups.