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.
Relative trace formulas and subconvexity estimates of L-functions for Hilbert modular forms
Bessel Periods on $U(2,1) \times U(1,1)$, Relative Trace Formula and Non-Vanishing of Central L-values
Beyond endoscopy for the relative trace formula I: local theory
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