2009/02/11 by Jean-Loup Waldspurger, Waldspurger, Jean-Loup
Mathematics · #22E35 #22E50 #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT) #math.NT #math.RT #msc:22E35 #msc:22E50
paper · pdf · doi:10.48550/arxiv.0902.1875
arxiv created 2009/02/11 · arxiv updated 2009/12/01
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 v0∈ V such that q(v0)\not=0, denote W the subspace of V orthogonal to v0 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 HomH(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 mgeom(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)=mgeom(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× π∈ Σ× Πsuch that m(σ,π)=1.