Geometric side of a local relative trace formula

TL;DR

This paper develops a geometric expansion of a local relative trace formula for a reductive p-adic group relative to a symmetric subgroup, adapting Arthur's techniques to this setting.

Contribution

It introduces a geometric expansion of the local relative trace formula for reductive p-adic groups with symmetric subgroups, extending Arthur's methods.

Findings

Derived a geometric expansion for the local relative trace formula.

Adapted Arthur's proof techniques to the symmetric subgroup setting.

Provided a framework for analyzing harmonic analysis on p-adic symmetric spaces.

Abstract

Following a scheme suggested by B. Feigon, we investigate a local relative trace formula in the situation of a reductive $p$ -adic group $G$ relative to a symmetric subgroup $H= \underline{H}(F)$ where $\underline{H}$ is split over the local field $F$ of characteristic zero and $G = \underline{G} (F)$ is the restriction of scalars of $\underline{H} _{I E}$ relative to a quadratic unramified extension $E$ of $F$. We adapt techniques of the proof of the local trace formula by J. Arthur in order to get a geometric expansion of the integral over $H \times H$ of a truncated kernel associated to the regular representation of $G$.