La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes unitaires
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
π
and
σ
be smooth irreducible representations of
G(F)
and
H(F)
respectively. Then Gan, Gross and Prasad have defined a multiplicity
m(π,σ)
which for
m=n−1
is just the dimension of
Ho
m
H(F)
(π,σ)
. For
π
and
σ
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.