The local Gross-Prasad conjecture for tempered representations of unitary groups (Q2829380)

Let \(E/F\) be a quadratic extension of non-archimedean local fields of characteristic 0. Let \(V\) and \(W\) be two hermitian spaces relative to \(E/F\), such that \(V\) is the orthogonal direct sum of \(W\), a line and a hyperbolic space. Let \(G\) and \(H\) be the unitary groups of \(V\) and \(W\). Denote by \(m(\pi,\sigma)\) the multiplicity assigned by Gan, Gross and Prasad to a pair of irreducible smooth representations \(\pi\) and \(\sigma\) of \(G(F)\) resp. \(H(F)\). It is known that \(m(\pi,\sigma)\) equals 0 or 1. The conjecture of Gan-Gross-Prasad, which concerns the determination of the value of \(m(\pi,\sigma)\) in terms of \(\varepsilon\)-factors of pairs of Langlands parameters, has been proved by the author for tempered representations \(\pi\) and \(\sigma\), cf. [Compos. Math. 151, No. 7, 1309--1371 (2015; Zbl 1328.22013)]. The scheme of the proof is that of Waldspurger's proof of the Gross-Prasad conjecture for special orthogonal groups. The main result of the present article, an integral formula for \(m(\pi,\sigma)\), is part of the proof in [loc. cit.] of the conjecture.NEWLINENEWLINETo obtain the formula in question a certain integral involving the character of \(\check{\sigma}\) and a very cuspidal function \(f\) on \(G(F)\) is computed in two ways. The resulting equality is made invariant. That gives a formula which is valid for all cuspidal \(f\) and all tempered \(\sigma\) and from which the desired formula for \(m(\pi,\sigma)\) can be deduced.NEWLINENEWLINEThe details and proofs in this article are generally analogous to Waldspurger's in the case of special orthogonal groups. The author proves the estimations of some integrals (analogous to Waldspurger's estimations) in a different way. He also proves in a different way that all intertwining operators are tempered. For both points the method of proof relies on the following result. Suppose \(V\) is equal to the direct sum of \(W\) and a line. Then the author proves a decomposition of \(G(F)\) of the form \(G(F) = C_HA^+_HA^+_GC_G\). Here \(C_H\) and \(C_G\) are compact subsets of \(H(F)\) resp. \(G(F)\); \(A_H\) and \(A_G\) are maximal split tori of \(H\) and \(G\).