Une formule intégrale reliée à la conjecture locale de Gross-Prasad

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 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). By a theorem of Aizenbud, Gourevitch, Rallis and Schiffmann, we know that m(σ,π)=0 or 1. We define another term m_{geom}(σ,π). It's an explicit sum of integrals of functions that can be deduced from the characters of σand π. Assume that πis supercuspidal. Then we prove the equality m(σ,π)=m_{geom}(σ,π). Now, let Π, resp. Σ, 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 Πare supercuspidal. Then our integral formula implies the weak form of the Gross-Prasad conjecture: there exist a unique pair σ\times π\in Σ\times Πsuch that m(σ,π)=1.