Une formule int\'egrale reli\'ee \`a la conjecture locale de Gross-Prasad

TL;DR

This paper establishes an explicit integral formula relating to the local Gross-Prasad conjecture for special orthogonal groups over non-archimedean fields, proving a key equality for supercuspidal representations and implications for L-packets.

Contribution

It introduces a new explicit integral formula for the multiplicity m(\sigma,\pi) and proves its equality to a geometric term for supercuspidal representations, advancing the understanding of the conjecture.

Findings

Proves m(\sigma,\pi)=m_{geom}(\sigma,\pi) for supercuspidal \\pi.

Derives a weak form of the Gross-Prasad conjecture under certain assumptions.

Provides an explicit sum of integrals from characters of representations.

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 d the dimension of V and G=SO(d) the special orthogonal group of (V,q). Let v_{0}\in V such that q(v_{0})\not=0, denote W the subspace of V orthogonal to v_{0} and H=SO(d-1) 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). By a theorem of Aizenbud, Gourevitch, Rallis and Schiffmann, we know that m(\sigma,\pi)=0 or 1. We define 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 is supercuspidal. Then we prove the equality m(\sigma,\pi)=m_{geom}(\sigma,\pi). Now, let \Pi, resp. \Sigma, an L-packet of tempered representations of G(F), resp. H(F). We use the sophisticated notion of L-paquet due to Vogan: the representations in the packets can be representations of inner forms of G(F), resp. H(F). We assume that certain conjectural properties of tempered L-packets are true. Assume that all elements of \Pi are supercuspidal. Then our integral formula implies the weak form of the Gross-Prasad conjecture: there exist a unique pair \sigma\times \pi\in \Sigma\times \Pi such that m(\sigma,\pi)=1.