Exposing Relative Endoscopy in Unitary Symmetric Spaces

TL;DR

This paper introduces a new class of symmetric space orbital integrals relevant to automorphic representations and verifies a fundamental lemma for a specific unitary group case, suggesting a broader endoscopy theory.

Contribution

It presents a novel class of orbital integrals for symmetric spaces and provides explicit verification of a fundamental lemma in a specific unitary group embedding, indicating a potential general endoscopy framework.

Findings

Verified fundamental lemma for U_2 x U_2 in U_4

Introduced new symmetric space orbital integrals

Evidence for a general endoscopy theory

Abstract

We introduce a new class of symmetric space orbital integrals important for applications in certain relative trace formulas appearing in the theory of automorphic representations. We verify a fundamental lemma for $U_2\times U_2\hookrightarrow U_4$ via an explicit calculation, showing strong evidence that there is a general theory of endoscopy lurking in this situation.