Fourier-Jacobi periods and the non-tempered Gan--Gross--Prasad conjecture for $\Mp_{2n} \times \Sp_{2m}$

Abstract

In this paper, we establish one direction of the non-tempered global Gan--Gross--Prasad (GGP) conjecture for the symplectic-metaplectic pairs $\Sp_{2n} \times \Mp_{2m}$ and $\Mp_{2n} \times \Sp_{2m}$. Our study focuses on two distinct families of non-tempered global $A$-parameters spanning all coranks, where standard relative trace formula methods remain unavailable. Our approach proceeds along two distinct lines of attack. First, we treat the case where both members of the pair are non-tempered, utilizing regularized Fourier-Jacobi periods of residual Eisenstein series and establishing a reciprocal non-vanishing theorem. Second, we address the setting where only the metaplectic member is non-tempered and its central $L$-value vanishes, relying heavily on the global theta correspondence and explicit seesaw identities. Along the way, we establish one direction of the tempered GGP conjecture for these pairs in arbitrary coranks, and prove a dichotomy for the global associated $L$-packet of the metaplectic parameter $[1] \boxplus M'$. We show that this packet lies entirely in the residual spectrum or entirely in the cuspidal spectrum, a behavior determined uniformly by the non-vanishing of the central $L$-value $L(1/2, M')$.