Speaker: Jialiang Zou (MIT)

Title: Godement Jacquet L-function and homological theta lift.

Abstract: Let F be a local field of characteristic zero. The general linear groups GL_n(F) and GL_m(F) act naturally on the space M_m,n(F) of m x n matrices, with associated representation ω = C∞_c(M_m,n(F)), the space of compactly supported functions on M_m,n(F). Given an irreducible representation π of GL_n(F), a basic problem is to describe the structure of the big theta lift Θ(π), the maximal π-isotypic quotient of ω, as a representation of GL_m(F). This may be viewed as a kind of transcendental invariant theory, and the case m = n = 1 already appears in Tate’s thesis. The problem has been studied extensively (by Howe, Mínguez, Fang-Sun-Xue, among others), yet it is still not fully understood.

Following ideas of Adams–Prasad–Savin, one can enrich the picture by considering derived theta lifting. In this talk I will discuss some recent progress in this direction, and highlight the phenomenon relating the vanishing of higher theta lifts and the holomorphic of the Godement–Jacquet L-function of π and π^v at certain critical points. This is based on joint work with Rui Chen, Yufeng Li, Xiaohuan Long, and Chenhao Tang.