La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes unitaires Auteur(s): Beuzart-Plessis Raphaël (Document sans référence bibliographique) 2012-05-12 Ref HAL: hal-00696745_v2 Ref Arxiv: 1205.2987 Ref. & Cit.: NASA ADS Exporter : BibTex | endNote Résumé: 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. Commentaires: 113 p.