arXiv subjects
Publications and source records attributed to Raphaël Beuzart-Plessis.
Under endoscopic assumptions about $L$-packets of unitary groups, we prove the local Gan-Gross-Prasad conjecture for tempered representations of unitary groups over $p$-adic fields. Roughly, this conjecture says that branching laws for $U(n-1)\subset U(n)$ can be computed using epsilon factors.
Let $E/F$ be a quadratic extension of $p$-adic fields and let $d$, $m$ be nonnegative integers of distinct parities. Fix admissible irreducible tempered representations $π$ and $σ$ of $GL_d(E)$ and $GL_m(E)$ respectively. We assume that $π$ and $σ$ are conjugate-dual. That is to say $π\simeq π^{\vee,c}$ and $σ\simeq σ^{\vee,c}$) where $c$ is the non trivial $F$-automorphism of $E$. This implies, we can extend $π$ to an unitary representation $\tildeπ$ of a nonconnected group $GL_d(E)\rtimes {1,θ}$. Define $\tildeσ$ the same way. We state and prove an integral formula for $ε(1/2,π\times σ,ψ_E)$ involving the characters of $\tildeπ$ and $\tildeσ$. This formula is related to the local Gan-Gross-Prasad conjecture for unitary groups.
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 $Hom_{H(F)}(π,σ)$. 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.