Fundamental Lemma for Rank One Spherical Varieties of Classical Types

Abstract

According to the relative Langlands functoriality conjecture, an admissible morphism between the $L$-groups of spherical varieties should induce a functorial transfer of the corresponding local and global automorphic spectra. Via the relative trace formula approach, two basic problems are the local transfer and the fundamental lemma on the geometric side of the relative trace formulae. In this paper, we consider the rank one spherical variety case, where the admissible morphism between the $L$-groups is the identity morphism, in which case, Y. Sakellaridis has already established the local transfer. We formulate the statement of the fundamental lemma for the general rank one spherical variety case and prove the fundamental lemma for the rank one spherical varieties of classical types.

Extend the previous results to general Hecke elements