On a generalization of a result of Waldspurger

We consider a generalization of a trace formula identity of Jacquet, in the context of the symmetric spaces GL(2n)/GL(n) x GL(n) and G'/H'. Here G' is an inner form of GL(2n) over F with a subgroup H' isomorphic to GL(n,E) where E/F is a quadratic extension of number field attached to a quadratic idele class character eta of F. A consequence of this identity would be the following conjecture: Let eta be an automorphic cuspidal representation of GL(2n). If there exists an automorphic representation pi' of G' which is related to a by the Jacquet-Langlands correspondence, and a vector phi in the space of pi' whose integral over H' is nonzero, then both L(1/2, pi) and L(1/2, pi x eta) are nonvanishing. Moreover, we have L(1/2, pi)L(1/2, pi x eta) > 0. Here the nonvanishing part of the conjecture is a generalization of a result of Waldspurger for GL(2) and the nonnegativity of the product is predicted from the generalized Riemann Hypothesis. In this article, we study the corresponding local orbital integrals for the symmetric spaces. We prove the ''fundamental lemma for the unit Hecke functions'' which says that unit Hecke functions have ''matching'' orbital integrals. This serves as the first step toward establishing the trace formula identity and in the same time it provides strong evidence for what we proposed.