The global Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on unitary groups III: Proof of the main theorems
Paul Boisseau, Weixiao Lu, Hang Xue
TL;DR
This work completes the Gan--Gross--Prasad program for Fourier--Jacobi periods on unitary groups by establishing both the global GGP and the Ichino--Ikeda refinements through a careful comparison of Liu's relative trace formulae. The authors develop and compare spectral expansions on the unitary side (via Fourier--Jacobi periods and Weil representations) and on the general linear side (via Rankin--Selberg and Flicker--Rallis frameworks), then use test-function transfer and a robust regularization scheme to prove a global identity that ties central L-values to period integrals. A key technical achievement is reducing the non-corank-zero case to corank-zero via Eisenstein series and regular Hermitian Arthur parameters, enabling unfolding arguments and a complete spectral analysis that yields explicit product formulas. The results provide a full validation of GGP and Ichino--Ikeda type conjectures for Fourier--Jacobi periods, enhance the relative trace formula toolkit with corank-robust transfer and regularization, and illuminate the role of base-change and automorphic Arthur parameters in the restriction problems for unitary groups. Overall, the paper delivers a comprehensive, end-to-end treatment of Fourier--Jacobi periods, establishing precise non-vanishing criteria and L-value factorization with broad implications for automorphic representation theory and special value conjectures.
Abstract
This is the third and the last of a series of three papers where we prove the Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on unitary groups and an Ichino--Ikeda type refinement. Our strategy is based on the comparison of relative trace formulae formulated by Liu. In this paper, we compute the spectral expansions of these formulae and end the proof of the conjectures via a reduction to the corank zero case.