Department Colloquium
Department of Mathematics
Princeton University
February 23rd, 2000
The Metaplectic Group, Harmonic Oscillators, Transformation of Theta Functions, Representation Theory, Orthogonal Polynomials, and Multivariate Statistics
For some time I have been interested in the connections between the classical mechanics of harmonic oscillators, the corresponding quantum systems, the Heisenberg group, the inhomogeneous metaplectic group, the Schwarz class of functions and tempered distributions, and the transformation law for theta functions. All of this material is well-known, but there doesn't seems to a be a single source that puts all the connections together. My original interest in these matters came from a problem in singular perturbation theory in quantum mechanics. Recently, through some work of Bert Kostant, I realized that there are further connections with the representation theory of the universal cover of the metaplectic group and with multivariate statistics. There should also be a connection with some known families of symmetric orthogonal polynomials. Siddharta Sahi and I are currently trying to understand that connection. That sounds like rather a lot, and I may have to skip a few of the more interesting digressions, but I hope to get through the essentials of all of it.