Relative Endoscopy and Orbital Integrals on U$_3$

Abstract

This thesis aims to study the relative theory of endoscopy. Let $F$ be a nonarchimedean local field and consider the reductive group $\mathrm{O}_3(F)$ acting on the symmetric variety $\mathcal{S}(F)=(\mathrm{U}_3/\mathrm{O}_3)(F)$ over $F$. We compute the endoscopic orbital integrals of the basic function in this situation. Knowledge on the endoscopic orbital integrals is essential for observing the existence of endoscopy groups and transfer of smooth functions in the relative setting. This would be the first time such a computation has appeared in the literature for a spherical variety with type $N$-spherical roots.