On the stabilization of relative trace formulae: descent and the fundamental lemma

TL;DR

This paper develops descent theory and proves a fundamental lemma for symmetric spaces over p-adic fields, advancing the understanding of orbital integrals and automorphic periods in the Langlands program.

Contribution

It introduces a notion of topological Jordan decomposition for symmetric spaces and establishes a relative Kazhdan lemma, crucial for fundamental lemma proofs.

Findings

Proved the existence of topological Jordan decomposition for symmetric spaces.

Established a relative version of Kazhdan's lemma for orbital integrals.

Proved the endoscopic fundamental lemma for certain symmetric spaces.

Abstract

Motivated by the study of periods of automorphic forms and relative trace formulae, we develop the theory of descent necessary to study orbital integrals arising in the fundamental lemma for a general class of symmetric spaces over a $p$-adic field $F$. More precisely, we prove that a connected symmetric space over $F$ enjoys a notion of topological Jordan decomposition, which may be of independent interest, and establish a relative version of a lemma of Kazhdan that played a crucial role in the proof of the Langlands-Shelstad fundamental lemma. As our main application, we use these results to prove the endoscopic fundamental lemma for the unit element of the Hecke algebra for the symmetric space associated to unitary Friedberg-Jacquet periods.