Endoscopic Relative Orbital Integrals on U$_3$

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.