  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.
