Basic Representation Theory of the Jacobi Group

Abstract

Depending on whether we look at the archimedean, a p-adic or the adelic case, the methods for studying representations are sometimes very different. In this chapter we will collect some general material, mainly going back to Mackey, which will be useful in all three cases. We start by explaining the induction procedure, and apply it to describe the representations of the Heisenberg group. We treat the representations of the Jacobi group G J with trivial central character and set the way for all further discussions of the cases with non-trivial central character by introducing a certain projective representation of G J , the Schrodinger-Weil representation (others would perhaps call it the oscillator representation). This fundamental representation will later on be elaborated thoroughly in the different cases, and will allow to reduce, in a sense to be made precise later, the G J -theory to the metaplectic theory.