Twisted linear periods and a new relative trace formula

TL;DR

This paper develops a new relative trace formula for twisted linear periods on $GL_{2n}$, proving fundamental lemmas and transferring orbital integrals, leading to generalizations of Waldspurger's theorem.

Contribution

It introduces a novel relative trace formula for twisted linear periods, establishing fundamental lemmas and orbital transfer, enabling new comparisons and generalizations.

Findings

Established the relative fundamental lemma.

Transferred orbital integrals between groups.

Generalized Waldspurger's theorem for $n>1$.

Abstract

We study the linear periods on $GL_{2n}$ twisted by a character using a new relative trace formula. We establish the relative fundamental lemma and the transfer of orbital integrals. Together with the spectral isolation technique of Beuzart-Plessis--Liu--Zhang--Zhu, we are able to compare the elliptic part of the relative trace formulae and to obtain new results generalizing Waldspurger's theorem in the $n=1$ case.