The Fourier-Jacobi Periods : The case of $Mp(2n+2r) \times Sp(2n)$

TL;DR

This paper proves one direction of the Gan--Gross--Prasad conjecture for certain metaplectic-symplectic groups and residual representations, with applications to automorphic L-functions.

Contribution

It establishes new cases of the GGP conjecture for both tempered and non-tempered representations, advancing understanding of automorphic periods.

Findings

Proved one direction of the GGP conjecture for tempered metaplectic-symplectic groups.

Proved one direction of the GGP conjecture for residual non-tempered representations.

Discussed implications for the non-vanishing of quadratic twists of automorphic L-functions.

Abstract

In this paper, we prove one direction of the Gan--Gross--Prasad conjecture on metaplectic-symplectic groups for tempered cases. Furthermore, we also prove one direction of the non-tempered GGP conjecture for residual representations with relevant $A$-parameters. As an application, we discuss the non-vanishing of the central value of quadratic twists of automorphic $L$-functions of $GL_{2n}$.