A trace Paley-Wiener theorem for GL_n(F)\GL_n(E)

BIMSA-YMSC Tsinghua Number Theory Seminar

Organizers：

Hansheng Diao, Heng Du, Yueke Hu, Bin Xu, Yihang Zhu

Speaker：

Juliette Coutens (Institute of Mathematics of Marseille)

Time：

Thur., 10:00-11:00 am, May 29, 2025

Venue：

C654, Shuangqing Complex Building A

Title：

A trace Paley-Wiener theorem for GL_n(F)\GL_n(E)

Abstract:

This talk is related to the relative Langlands program, which aims to extend the classical Langlands program to spherical varieties. In the classical case, a well-known trace Paley-Wiener theorem was given by Bernstein, Deligne and Kazhdan in 1986. It gives a characterization of the functions \pi \mapsto Tr(\pi(f)) with G a reductive p-adic group, \pi ranges over isomorphism classes of smooth irreducible representations of G and f in C_c^\infty(G).

We will explain how to extend this to the relative case. That is when E/F is a quadratic extension of p-adic fields, the theorem is a scalar Paley-Wiener theorem for relative Bessel distributions on GL_n(F)\GL_n(E). These distributions are relative character of the form \pi\mapsto I_\pi (f) for f in C_c^\infty(GL_n(E)) as \pi ranges over GL_n(F)-distinguished irreducible tempered representations, and are constructed from a GL_n(F)-invariant functional and a Whittaker functional. We will explain how by using the local Langlands correspondence, and the base-change from a unitary group, the relative characters can be described as elements of the "generic" Bernstein center of the unitary group U(n).