Unitary Representations of Lie Groups

Abstract

This course will introduce unitary representations of Lie groups, discuss spectral gap in general, and discuss concrete unitary representations of SL(2,R).

Objective

The goal is to acquire familiarity with the basic formalism and results concerning Lie groups and their unitary representations. In the second part we will consider concrete representations of SL(2,R) and decompose these into irreducible representations or at least understand whether spectral gap is present.

Content

The course will start with the general framework of unitary representations of locally compact groups, which is in some sense a general theory of Fourier analysis related to groups. For this some functional analysis (in particular spectral theory of bounded selfadjoint operators, Krein-Milman and Choquet) will be important. In the interest of time we will only summarise the case abelian groups and use the abelian theory to understand some metabelian groups. After this we will discuss some more general theory. Some of the general phenomena will be discussed for the concrete group of SL(2,R). Moreover, we will understand the unitary dual of SL(2,R), discuss the notion of spectral gap for SL(2,R), and decompose the unitary representation of SL(2,R) arising from the hyperbolic plane into irreducible representation.

Resources