Fundamental Lemma for Rank One Spherical Varieties of Classical Types

TL;DR

This paper proves the fundamental lemma for rank one spherical varieties of classical types, advancing the understanding of functorial transfers in the Langlands program using the relative trace formula.

Contribution

It formulates and proves the fundamental lemma for rank one spherical varieties of classical types, filling a key gap in the relative trace formula approach.

Findings

Established the fundamental lemma for classical types

Confirmed the local transfer in rank one cases

Enhanced the framework for functoriality in spherical varieties

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.