Endoscopic Relative Orbital Integrals on U$_3$

TL;DR

This paper computes endoscopic orbital integrals for a specific spherical variety over nonarchimedean fields, advancing the understanding of transfer in relative harmonic analysis on spherical varieties.

Contribution

It provides the first known computation of endoscopic orbital integrals for spherical varieties with type N-spherical roots, a key step in transfer theory.

Findings

Computed endoscopic orbital integrals for (U_3/O_3)(F)

Established foundational results for transfer in this setting

First such computation for N-spherical roots

Abstract

Let $F$ be a nonarchimedean local field and consider the action of the reductive group SO$_3(F)$ on the spherical variety (U$_3$/O$_3)(F)$. We compute the endoscopic orbital integrals of the basic function in this situation. Knowing the endoscopic orbital integrals is essential for observing the existence of transfer in this relative setting. This would be the first time such a computation has appeared in the literature for spherical varieties with type $N$-spherical roots.