A local relative trace formula for PGL(2)

TL;DR

This paper develops a local relative trace formula for PGL(2) over a quadratic extension, linking spectral data with geometric orbital integrals, and applies it to invert orbital integrals.

Contribution

It introduces a new local relative trace formula for PGL(2) relative to PGL(2) over a quadratic extension, connecting spectral and geometric sides.

Findings

Derived a spectral side using normalized periods and C-functions.

Established a geometric side based on previous work by Delorme, Harinck, and Souaifi.

Applied the formula to invert specific orbital integrals.

Abstract

Following a scheme inspired by B. Feigon, we describe the spectral side of a local relative trace formula for $G:= PGL(2,\rm E)$ relative to the symmetric subgroup $H:=PGL(2,\rm F)$ where $\rm E/\rm F$ is an unramified quadratic extension of local non archimedean fields of characteristic $0$. This spectral side is given in terms of regularized normalized periods and normalized $C$-functions of Harish-Chandra. Using the geometric side obtained in a more general setting by P. Delorme, P. Harinck and S. Souaifi , we deduce a local relative trace formula for $G$ relative to $H$. We apply our result to invert some orbital integrals.