Topics in Representation Theory: Cultural Background

This semester we will be covering various topics in representation theory, see the separate syllabus for a detailed list of topics, including some that we may or may not get to. Much of the material will be rather standard for a course of this kind, but an attempt will be made to explain some of the relations between representation theory and quantum mechanics. The notion of “quantization” has motivated several different approaches to the construction of group representations and some of the most important applications of representation theory are to the solution of quantum mechanical problems. Some important things we won’t be covering include the representation theory of non-compact Lie groups such as SL(2,R), and of arithmetically interesting groups such as GL(n,Qp). Our approach to representation theory will be quite geometric, but in the context of differential geometry over the real numbers, not that of algebraic geometry over more general fields, i.e. the theory of “algebraic groups”.