Relative trace formulae toward Bessel and Fourier-Jacobi periods of unitary groups

TL;DR

This paper develops a relative trace formula framework for Bessel and Fourier-Jacobi periods of unitary groups, extending previous work and proving a fundamental lemma in positive characteristic to advance the understanding of these periods.

Contribution

It introduces a new relative trace formula approach for unitary groups and proves the fundamental lemma in positive characteristic, extending prior results to broader cases.

Findings

Established a relative trace formula for unitary groups involving Bessel and Fourier-Jacobi periods.

Proved the fundamental lemma for U_n×U_n in positive characteristic.

Extended the work of Jacquet-Rallis and Flicker to new settings.

Abstract

We propose a relative trace formula approach and state the corresponding fundamental lemma toward the global restriction problem involving Bessel or Fourier-Jacobi periods of unitary groups $\mathrm{U}_n\times\mathrm{U}_m$, extending the work of Jacquet-Rallis for $m=n-1$ (which is a Bessel period). In particular, when $m=0$, we recover a relative trace formula proposed by Flicker concerning Kloosterman/Fourier integrals on quasi-split unitary groups. As evidence for our approach, we prove the fundamental lemma for $\mathrm{U}_n\times\mathrm{U}_n$ in positive characteristics.