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 $\pi$, resp. $\sigma$, an admissible irreducible representation of $G(F)$, resp. $H(F)$. Denote $m(\sigma ,\pi )$ the dimension of the complex space $Ho{m}_{H(F)}({\pi }_{|H(F)},\sigma )$. 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.

References in corpus (1)

• Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups

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