Special periods and some non-tempered cases of the Gan-Gross-Prasad conjecture

Abstract: In this paper, we establish a relationship between special periods and special L-values of automorphic representations of classical groups, and prove the non-tempered global Gan--Gross--Prasad conjecture in several cases. Our approach consists of two main steps. First, inspired by Rallis' tower property, we study the interaction between special periods and the tower property for the genericity of global theta lifts. Second, we investigate the relationship between the analytic properties of L-functions and special periods via the Rankin--Selberg integral method. Combining these results with non-vanishing criteria for global theta lifts in terms of various L-values, we prove three explicit higher-corank families of non-tempered cases of the global Gan--Gross--Prasad conjecture.