Refined global Gan-Gross-Prasad conjecture for Fourier-Jacobi periods on symplectic groups

In this paper, we propose a conjectural identity between the Fourier-Jacobi periods on symplectic groups and the central value of certain Rankin-Selberg -functions. This identity can be viewed as a refinement to the global Gan-Gross-Prasad conjecture for . To support this conjectural identity, we show that when and , it can be deduced from the Ichino-Ikeda conjecture in some cases via theta correspondences. As a corollary, the conjectural identity holds when or when , and the automorphic representation on the bigger group is endoscopic.