Endoscopy for unitary symmetric spaces

Abstract

Motivated by global applications, we propose a theory of relative endoscopic data and transfer factors for the symmetric pair (U (2n), U (n) × U (n)) over a local field. We then formulate the smooth transfer and fundamental lemma conjectures, establish the existence of smooth transfer for many test functions, and prove the fundamental lemma for the symmetric pair (U (4), U (2) × U (2)). Contents 1. Introduction 1 2. Endoscopy 6 3. The Lie algebra of the symmetric space 7 4. Relative endoscopy for (U (W ), U (W 1 ) × U (W 2 )) 10 5. The relative fundamental lemma for (U (V 4 ), U (V 2 ) × U (V 2 )) 13 Appendix A. Comment on Torsors 21 References 22

The endoscopic fundamental lemma for unitary Friedberg-Jacquet periods

“…The characteristic functions 1 End(Λn) and 1 End(Λa) ⊗ 1 End(Λ b ) are smooth transfers of each other. This is the endoscopic fundamental lemma referred to in the title and was conjectured in [Les19a] where we proved the special case n = 2 and a = b = 1 via explicit computation.…”

“…In this paper, we prove the endoscopic fundamental lemma for the Lie algebra of the symmetric space U (2n)/U (n) × U (n), stated below as Theorem 1.2. Conjectured in [Les19a], this is the first example of such a fundamental lemma and is the first major step in the stabilization of the relative trace formula associated to the U (n) × U (n)-periods of automorphic forms on U (2n). Let us now explain the context and motivation.…”

“…Theorem 1.5 of [PWZ19] proves one direction of this conjecture under the assumption that π is discrete series at a split place of F . In [Les19a], we outlined a comparison of relative trace formulas conjectured by Wei Zhang (first suggested in a less precise form in [GW14]) which would prove this conjecture if established.…”

“…Now suppose that E/F is a quadratic extension of non-archimedean local fields of characteristic zero and set W = W 1 ⊕W 2 . In [Les19a], we initiated a program to stabilize the relative trace formula associated to the symmetric pair (U (W ), U (W 1 )×U (W 2 )) by developing the local theory of endoscopy for the "Lie algebra" of the symmetric space…”

On the p-adic interpolation of unitary Friedberg--Jacquet periods

“…• The "relative trace formula approach" in forthcoming work of Jingwei Xiao and Wei Zhang, and the work of Spencer Leslie [Les19a], [Les19b]. • Applications of the residue method in the work of Pollack-Wan-Zydor [PWZ19].…”