Let V be a vector space over a p-adic field F, of finite dimension, let q be a non-degenerate quadratic form over V and let D be a non-isotropic line in V. Denote by W the hyperplane orthogonal to D, and by G and H the special orthogonal groups of V and W. Let π, respectively σ, be an irreducible admissible representation of G(F) , respectively H(F) . The representation σ appears as a quotient of the restriction of π to H(F) with a certain multiplicity m(π,σ) . We know that m(π,σ)≤1 . We assume that π is supercuspidal. Then we prove a formula that computes m(π,σ) as an integral of functions deduced from the characters of π and σ. Let Π, respectively Σ, be an L-packet of tempered irreducible representations of G(F) , respectively H(F) . Here we use the sophisticated notion of L-packet due to Vogan and we assume some usual conjectural properties of those packets. A weak form of the local Gross–Prasad conjecture says that there exists a unique pair (π,σ)Π×Σ such that m(π,σ)=1 . Assuming that the elements of Π are supercuspidal, we prove this assertion.
(Received February 23 2009)
(Accepted November 14 2009)
(Online publication May 04 2010)
2000 Mathematics Subject Classification22E50 (primary)
Keywordsreprésentations des groupes spéciaux orthogonaux; conjecture locale de Gross–Prasad