The tower property on the genericity of global theta lifts

TL;DR

This paper investigates the tower property of global theta lifts, showing that the first occurrence preserves genericity and applying this to prove cases of the Gan-Gross-Prasad conjecture.

Contribution

It establishes the preservation of genericity in the first occurrence of global theta lifts and applies this to prove a case of the Gan-Gross-Prasad conjecture.

Findings

First occurrence of global theta lifts preserves genericity.

Established the Gan-Gross-Prasad conjecture for specific orthogonal groups.

Linked analytic properties of L-functions with theta lift behavior.

Abstract

In this paper, we examine the tower property concerning the genericity of global theta lifts between various classical groups, drawing inspiration from Rallis' tower property. By exploring the relationship between the analytic properties of $L$-functions and special Bessel and Fourier-Jacobi periods, we demonstrate that the first occurrence of global theta lifts between dual reductive groups preserves genericity. As an application, we establish the global Gan-Gross-Prasad conjecture for $\SO_{2n+1} \times \SO_{2}$ under the assumption that $\SO_{2}$ is split and its representation is trivial.