Unitary Friedberg-Jacquet periods and central values of L functions

Let G be a reductive group over a number field F and H a subgroup. Automorphic periods study the integrals
of cuspidal automorphic forms on G over H(F)\H(A_F). They are often related to special values of certain L functions. One of the most notable case is when (G,H)=(U(n+1)☓U(n), U(n)), and these periods are related to central values of Rankin-Selberg L functions
on GL(n+1)☓GL(n). In this talk, I will explain my work in progress with Wei Zhang that studies central values of standard L functions on GL(2n) using (G,H)=(U(2n), U(n)☓U(n)) and some variants. I shall explain the conjecture and a relative trace formula approach
to study it. We prove the required fundamental lemma using a limit of the Jacquet-Rallis fundamental lemma and Hironaka’s characterization of spherical functions on the space of Hermitian matrices. Also, the question admits an arithmetic analogy.