Joint IAS/Princeton University Number Theory Seminar

Let $G$ be a reductive group over a number field $F$ and $H$ a subgroup. Automorphic periods study the integrals of cuspidal automorphic forms on $G$ over $H(F)\backslash H(A_F)$. They are often related to special values of certain L functions. One of the most notable case is when $(G,H)=(U(n+1) \times U(n), U(n))$, and these periods are related to central values of Rankin-Selberg L functions on $GL(n+1) \times GL(n)$. In this talk, I will explain my work in progress with Wei Zhang that studies central values of standard L functions on $GL(2n)$ using $(G,H)=(U(2n), U(n)\times U(n))$ 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.