On the $\theta$-split side of the local relative trace formula

TL;DR

This paper develops a new expression for one side of the local relative trace formula at the Lie algebra level, integrating Arthur and Harish-Chandra methods with reductive symmetric space theory.

Contribution

It introduces a novel approach to the local relative trace formula by combining classical methods with the structure theory of reductive symmetric spaces.

Findings

Derived an explicit expression for the local relative trace formula side

Integrated Arthur and Harish-Chandra techniques with symmetric space theory

Enhanced understanding of trace formula at the Lie algebra level

Abstract

The author derives an expression for one side of the local relative trace formula, at the level of Lie algebras, by combining methods of Arthur and Harish-Chandra with the structure theory for reductive symmetric spaces.