%0 Journal Article
%T La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes unitaires
%+ Institut de Mathématiques de Jussieu (IMJ)
%A Beuzart-Plessis, Raphaël
%Z 113 p.
%< avec comité de lecture
%@ 0249-633X
%J Mémoires de la Société Mathématique de France
%I SMF
%S Mémoire de la SMF
%V 149
%P 191p.
%8 2016-09
%D 2016
%Z 1205.2987
%R 10.24033/msmf.457
%K conjecture de Gross-Prasad
%K groupes unitaires
%Z Mathematics [math]/Representation Theory [math.RT]
%Z Mathematics [math]/Number Theory [math.NT]Journal articles
%X 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.
%G French
%2 https://hal.science/hal-00696745v2/document
%2 https://hal.science/hal-00696745v2/file/articlefinal3.pdf
%L hal-00696745
%U https://hal.science/hal-00696745
%~ UNIV-PARIS7
%~ UPMC
%~ CNRS
%~ IMJ
%~ USPC
%~ UPMC_POLE_1
%~ SORBONNE-UNIVERSITE
%~ SU-SCIENCES
%~ UNIV-PARIS
%~ UP-SCIENCES
%~ SU-TI
%~ ALLIANCE-SU