The global Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on unitary groups I: Coarse expansions of the relative trace formulae

TL;DR

This work initiates a three-paper program to prove the Gan–Gross–Prasad conjecture for Fourier–Jacobi periods on unitary groups and its Ichino–Ikeda refinement. It develops the coarse expansions of Liu’s relative trace formulas by introducing Jacobi groups and D-parabolic subgroups, and by formulating a truncation framework that preserves the geometry-spectral correspondence. The paper constructs both the coarse spectral and geometric expansions for general linear groups, via modified kernels and constant-term approximations, laying the groundwork for a full comparison and the eventual global GGP results in the series. The approach combines theta correspondences, Jacobi group analysis, and Arthur–type truncations to connect Fourier–Jacobi periods with central L-values in a global automorphic setting, with the subsequent papers completing the transfer and culminating in the predicted identities and refinements.

Abstract

This is the first 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. The goal of this first paper is to introduce the relative trace formulae and establish the coarse expansions.

Theorems & Definitions (122)

• Theorem 1.1
• Proposition 3.1
• Remark 3.2
• Lemma 3.3
• proof
• Remark 3.4
• Lemma 3.5
• proof
• Lemma 3.6
• proof