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

Jean-Loup Waldspurger (1)

1 IMJ - Institut de Mathématiques de Jussieu

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 $\pi$, resp. $\sigma$, an admissible irreducible representation of $G(F)$, resp. $H(F)$. Denote $m(\sigma,\pi)$ the dimension of the complex space $Hom_{H(F)}(\pi_{\vert H(F)},\sigma)$. It's know that $m(\sigma,\pi)=0$ or $1$. In a first paper, we have defined another term $m_{geom}(\sigma,\pi)$. It's an explicit sum of integrals of functions that can be deduced from the characters of $\sigma$ and $\pi$. Assume that $\pi$ and $\sigma$ are tempered. Then we prove the equality $m(\sigma,\pi)=m_{geom}(\sigma,\pi)$. This generalize the result of the first paper, where $\pi$ 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.

Mots clés: conjecture de Gross-Prasad; représentations tempérées

Domaines: Théorie des représentations [math.RT]

Jean-Loup Waldspurger. Une formule intégrale reliée à la conjecture locale de Gross-Prasad, 2ème partie: extension aux représentations tempérées. 2009. ⟨hal-00372663⟩