On Discrete Spectrum of Quasi-split Classical Groups and the Generalized Ramanujan Problem

One of the fundamental problems in the modern theory of automorphic forms is to consider the spectral decomposition of the space of all square-integrable automorphic forms on a reductive algebraic group G defined over a number field F. This can be viewed as a vast generalization of the classical theory of Fourier analysis on $\mathbb{R}/\mathbb{Z}$ and $\mathbb{R}$. In this talk, we start with classical examples and then review the recent work of Arthur. The objective is to show our recent study on various refined properties of the spectral decomposition for quasi-split classical groups, with connection to the generalized Ramanujan problem: how bad the local components of the cuspidal spectrum could be?

The speaker is a candidate for a position in the Department.