On the geometric side of the Jacquet-Rallis relative trace formula

TL;DR

This paper analyzes the geometric aspects of the Jacquet-Rallis relative trace formula, computing geometric terms globally and comparing orbital integrals locally between general linear and unitary groups.

Contribution

It provides explicit computations of geometric terms and establishes a comparison between orbital integrals on different groups, advancing understanding of the trace formula's geometric side.

Findings

Explicit formulas for geometric terms on GL(n)

Comparison between orbital integrals on GL(n) and unitary groups

Establishment of regular orbital integral descriptions

Abstract

We study some aspects of the geometric side of the Jacquet-Rallis relative trace formula. Globally, we compute each geometric term of the Jacquet-Rallis relative trace formula on the general linear group for regular supported test functions. We prove that it can be described by the regular orbital integral. Locally, we show that the regular orbital integral can be compared with the semisimple orbital integral on the unitary group.