Automorphic Forms

Abstract

The purpose of these lectures is to give a relatively easily accessible introduction to the theory of automorphic forms. For most of the lectures we will confine our attention to the least complicated (but still important in its own right) case of SL(2, R). The lectures begin with a discussion of the relationships between the η—function, the θ-function and classical Eisenstein series. The emphasis here is on their q-expansions which we will see have interesting group and representation theoretic interpretations. Here we give some implications of a combinatorial nature to the explicit calculations of the expansions. These functions were singled out for study for two reasons. The first is that they are the simplest automorphic forms that can be explicitly written. The second is that they are intimately related to the representation theory of loop groups and the Virasoro algebra (precisely through their q-expansions).