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

TL;DR

This paper links special periods and L-values of automorphic representations, proving new non-tempered cases of the Gan-Gross-Prasad conjecture using theta lifts and Rankin-Selberg methods.

Contribution

It establishes explicit non-tempered cases of the Gan-Gross-Prasad conjecture by connecting periods, L-values, and theta lifts through novel analytic techniques.

Findings

Proved non-tempered global Gan-Gross-Prasad conjecture in several cases.

Linked special periods with special L-values via Rankin-Selberg integrals.

Identified criteria for non-vanishing of global theta lifts in relation to L-values.

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.