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

Abstract

Abstract 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.