Spherical character of a supercuspidal representation as weighted orbital integral

TL;DR

This paper characterizes the spherical characters of certain supercuspidal representations over unramified quadratic extensions using weighted orbital integrals, building on recent advances in local trace formulas.

Contribution

It provides a new description of spherical characters for distinguished supercuspidal representations in terms of weighted orbital integrals, extending the understanding of harmonic analysis on p-adic groups.

Findings

Spherical characters are expressed via weighted orbital integrals.

Utilizes recent results of Zhang and geometric trace formulas.

Connects representation theory with orbital integral techniques.

Abstract

Let $\rm E/\rm F$ be an unramified quadratic extension of local non archimedean fields of characteristic 0. Let $\underline{H}$ be an algebraic reductive group, defined and split over $\rm F$. We assume that the split connected component of the center of $\underline{H}$ is trivial. Let $(\tau,V)$ be a $\underline{H}(\rm F)$-distinguished supercuspidal representation of $\underline{H}(\rm E)$. Using the recent results of C. Zhang, and the geometric side of a local relative trace formula obtained by P. Delorme, P. Harinck and S. Souaifi, we describe spherical characters associated to $\underline{H}(\rm F)$-invariant linear forms on $V$ in terms of weighted orbital integrals of matrix coefficients of $\tau$.