The global Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on unitary groups III: Proof of the main theorems

TL;DR

This paper completes the proof of the Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on unitary groups by analyzing spectral expansions and reducing the problem to a simpler case.

Contribution

It provides the final proof of the conjecture using spectral analysis and trace formula comparisons, completing a series of three papers.

Findings

Spectral expansions of relative trace formulae computed

Conjecture reduced to the corank zero case

Proof of the main theorems established

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.