Endoscopy for unitary symmetric spaces

Motivated by global applications, we propose a theory of relative endoscopic data and transfer factors for the symmetric pair $(U(2n),U(n)\times U(n))$ over a local field. We then formulate the smooth transfer conjecture and fundamental lemma, establish the existence of smooth transfer for many test functions, and prove the fundamental lemma for the symmetric pair $(U(4),U(2)\times U(2))$.