A Local Relative Trace Formula for F*\SL(2,F)

TL;DR

This paper explicitly derives the local relative trace formula for the symmetric space F* ext{SL}(2,F) over p-adic fields, providing a fundamental example for further research in the area.

Contribution

It presents the first explicit derivation of the local relative trace formula at the Lie algebra level for this symmetric space, serving as a key example for future work.

Findings

Explicit local relative trace formula derived

Lie algebra level formulation achieved

Serves as a foundational example for further research

Abstract

In this note, we derive explicitly the local relative trace formula for the symmetric space F*\SL(2,F) at the level of Lie algebras, where F is a p-adic field of residue characteristic greater than two and F* is the set of invertible elements in F. This is perhaps one of the simplest non-trivial analogs of the trace formula, and also a motivating example for the author's work (in preparation) on the relative trace formula.