Harmonic analysis for relative trace formula

Many questions on special values of L-functions are tied to the study of period integrals of automorphic forms. A notable example is the formula of Waldspurger ([22]) that relates the toric period on GL2 or its inner form to some central critical L-value. The conjecture of Gan, Gross and Prasad ([4]) started with the Gross-Prasad conjecture ([5]), as well as the refinement of Ichino and Ikeda ([8]) of the Gross-Prasad conjecture, vastly generalizes the Waldspurger formula to higher rank groups. We briefly state their conjecture. Let E be either F or a quadratic extension of F . If E = F (resp., if E is a quadratic extension of F ), let Wn+1 be a quadratic space over E = F (resp., a Hermitian space associated to E/F ) of E-dimension n+ 1. Let Wn ⊂ Wn+1 be a non-degenerate subspace of codimension one. Let Gi be SO(Wi) or U(Wi) for i = n, n+1. The Gan-Gross-Prasad period is attached to the pair (H,G) where G = Gn ×Gn+1 and H ⊂ G is the diagonal embedding of Gn. Assume further that π is tempered. The conjecture of Gan-Gross-Prasad ([4, Conjecture 24.1]) asserts that the following two statements are equivalent: