  We prove the endoscopic fundamental lemma for the Lie algebra of the
symmetric space $U(2n)/U(n)\times U(n)$, where $U(n)$ denotes a unitary group
of rank $n$. This is the first major step in the stabilization of the relative
trace formula associated to the $U(n)\times U(n)$-periods of automorphic forms
on $U(2n)$.
