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

TL;DR

This paper introduces the relative trace formulae and establishes their coarse expansions as a foundational step towards proving the Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on unitary groups, using Liu's comparison method.

Contribution

It provides the first detailed formulation and coarse expansion of the relative trace formulae crucial for the conjecture's proof.

Findings

Formulation of relative trace formulae for Fourier--Jacobi periods

Establishment of coarse expansions of these trace formulae

Foundation for subsequent proof of the Gan--Gross--Prasad conjecture

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.