Unitary Friedberg-Jacquet periods and their twists: Fundamental lemmas

Abstract

We formulate a global conjecture for the automorphic period integral associated to the symmetric pairs defined by unitary groups over number fields, generalizing a theorem of Waldspurger's toric period for $\mathrm{GL}(2)$. We introduce a new relative trace formula to prove our global conjecture under some local hypotheses. A new feature is the presence of the relative endoscopy. In this paper we prove the main local theorem: a new relative fundamental lemma comparing certain orbital integrals of functions matched in terms of Hironaka and Satake transforms.