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

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