Une formule intégrale reliée à la conjecture locale de Gross-Prasad, 2ème partie: extension aux représentations tempérées

Abstract

Let $F$ be a non-archimedean local field, of characteristic 0. Let $V$ be a finite dimensional vector space over $F$ and $q$ be a non-degenerate quadratic form on $V$. Denote $G$ the special orthogonal group of $(V,q)$. Let $W$ a non-degenerate hyperplane of $V$, denote $H$ the special orthogonal group of $W$. Let $π$, resp. $σ$, an admissible irreducible representation of $G(F)$, resp. $H(F)$. Denote $m(σ,π)$ the dimension of the complex space $Hom_{H(F)}(π_{| H(F)},σ)$. It's know that $m(σ,π)=0$ or 1. In a first paper, we have defined another term $m_{geom}(σ,π)$. It's an explicit sum of integrals of functions that can be deduced from the characters of $σ$ and $π$. Assume that $π$ and $σ$ are tempered. Then we prove the equality $m(σ,π)=m_{geom}(σ,π)$. This generalize the result of the first paper, where $π$ was supercuspidal. As in this paper, the previous equality implies as corollary (assuming certain properties of tempered $L$-packets) a weak form of the local Gross-Prasad conjecture, now for pairs of tempered $L$-packets.