A unitary analogy of Friedberg-Jacquet and Guo-Jacquet periods and central values of standard L functions on GL(2n)

Let GG be a reductive group over a number field FF and HH a subgroup. Automorphic periods study the integrals of cuspidal automorphic forms on GG over H(F)∖H(AF)H(F)∖H(AF). They are often related to special values of certain L functions. One of the most notable case is when (G,H)=(U(n+1)×U(n),U(n))(G,H)=(U(n+1)×U(n),U(n)), and these periods are related to central values of Rankin-Selberg L functions on GL(n+1)×GL(n)GL(n+1)×GL(n). In this talk, I will explain my work in progress with Wei Zhang that studies central values of standard L functions on GL(2n)GL(2n) using (G,H)=(U(2n),U(n)×U(n))(G,H)=(U(2n),U(n)×U(n)) and some variants. I shall explain the conjecture and a relative trace formula approach to study it. We prove the required fundamental lemma using a limit of the Jacquet-Rallis fundamental lemma and Hironaka’s characterization of spherical functions on the space of non degenerate Hermitian matrices. Also, the question admits an arithmetic analogy.



Institute for Advanced Study and Princeton University; Veblen Research Instructor, School of Mathematics