Fourier–jacobi Periods and the Central Value of Rankin–selberg L-functions

1.1. The refined Gan–Gross–Prasad conjecture. In this paper, as a sequel of [Xue14], we formulate a refinement to the global Gan–Gross–Prasad conjecture for the Fourier–Jacobi periods on U(n) × U(n) and prove it under some local conditions, assuming some expected properties of the L-packets and some parts of the local Gan–Gross–Prasad conjecture. This refinement is modeled on some recent work on the refined Gan–Gross–Prasad conjecture for the Bessel periods. In a seminal paper [II10], Ichino and Ikeda formulated a refinement of the Gan–Gross– Prasad conjecture for orthogonal groups SO(n+ 1)× SO(n). The n = 2 case corresponds to Waldspurger’s formula [Wal85] and the n = 3 case corresponds to the triple product formula [Ich08]. Gan and Ichino proved some cases of n = 4 using theta correspondences. But little is known beyond this. N. Harris then formulated the refined conjecture for the unitary groups U(n+1)×U(n) in his Ph.D. thesis at the University of California, San Diego [Har12]. Harris also verified the case n = 1 and some special cases of n = 2. W. Zhang proved the refinement for the unitary groups U(n+ 1)×U(n) under some local conditions [Zha14b]. Zhang’s work is built on the relative trace formulae formulated by Jacquet–Rallis [JR11], the relevant fundamental lemma established by Yun [Yun11] and the existence of smooth transfer established by himself [Zha14a]. There is a parallel theory for the Fourier–Jacobi periods on U(n) × U(n). The non-refined conjecture for the Fourier–Jacobi periods was formulated in [GGP12]. A relative trace formulae approach to these conjectures for unitary groups was then proposed by Liu [Liu14]. The relevant fundamental lemma for U(n) × U(n) was also proved by Liu. Building on the work of Liu, we proved the non-refined Gan–Gross– Prasad conjecture for U(n)×U(n) in [Xue14] under some local conditions. The techniques in [Xue14] were largely inspired by [Zha14a]. In fact, we proved the existence of smooth transfer by reducing it to the case established in [Zha14a]. In this paper, we first formulate a refinement to the Gan–Gross–Prasad conjecture for U(n)×U(n), and then prove it under some local conditions. This is an analogue of [Zha14b] for the case of Fourier–Jacobi periods.