The tower property on the genericity of global theta lifts

TL;DR

This work develops a tower-property framework for the genericity of global theta lifts between orthogonal, symplectic, and metaplectic groups, extending Rallis’ tower property to quasi-split and non-split settings. By tying the first occurrence in Witt towers to the preservation of nontrivial generic (Whittaker/Fourier–Jacobi) periods and connecting these periods to the analytic behavior of L-functions via Rankin–Selberg type integrals and the Rallis inner product formula, the authors prove that the first nonzero theta lift is generic exactly in the expected cases. They establish global equivalences between poles of completed L-functions and non-vanishing/genericity of theta lifts, using a network of special Bessel and Fourier–Jacobi periods. As an application, they prove the tempered Gan–Gross–Prasad conjecture for the pair $( ext{SO}_{2n+1}, ext{SO}_2)$ with split $ ext{SO}_2$ and trivial representation, illustrating the arithmetic significance of the tower-property–period interplay for global automorphic forms.

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.

Theorems & Definitions (51)

• Theorem 1.1: Ra84
• Theorem 1.2: Theorem \ref{['a1']}, Theorem \ref{['p6']}
• Theorem 1.3: Theorem \ref{['b10']}, Theorem \ref{['q1']}
• Remark 2.1
• Theorem 3.1: cf. GS12, MS20
• proof
• Remark 3.2
• Proposition 3.3: cf. GS12, MS20
• proof
• Proposition 3.4