Une formule int\'egrale reli\'ee \`a la conjecture locale de Gross-Prasad, 2\`eme partie: extension aux repr\'esentations temp\'er\'ees

TL;DR

This paper proves a geometric formula for the multiplicity of certain representations related to the local Gross-Prasad conjecture for tempered representations over non-archimedean fields, extending previous results to a broader class.

Contribution

It establishes the equality between multiplicity and a geometric sum for tempered representations, generalizing earlier supercuspidal cases and supporting the local Gross-Prasad conjecture.

Findings

Proves $m(\sigma,\pi)=m_{geom}(\sigma,\pi)$ for tempered representations.

Extends previous supercuspidal results to tempered representations.

Provides a geometric interpretation of multiplicities in the local Gross-Prasad setting.

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 $Hom_{H(F)}(\pi_{| H(F)},\sigma)$. 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.