On A Notion of Rank for Unitary Representations of the Classical Groups

Abstract

1. Review of the Heisenberg group and the oscillator representation.

The goal of these lectures is to introduce some general concepts concerning unitary representations of locally compact groups, and to apply these concepts to the study of representations of semisimple Lie groups, especially the symplectic group. Historically, what we may call “general representation theory”, on one hand, and the representation theory of semisimple Lie groups on the other, have, with some notable exceptions, tended to go their separate ways; general concepts have not proven very powerful in semisimple harmonic analysis, which has needed its own special methods. The key phenomenon permitting at least a partial merger here is the oscillator representation of the symplectic group. Although this representation has received increasing attention in recent years, it may be still unfamiliar to some. Because of that, and because it will play such a pervasive role in the present study, I will begin by reviewing the basic definitions and salient properties of the oscillator representation. General references for the following material are [Cr], [H1], [Wil].

Let F be a local field - ℝ, ℂ, or non-Archimedean. We will assume F is not of characteristic 2. We let W be a vector space of dimension 2m over F, on which is defined a symplectic form < , >. We may choose a basis $\left\{ {{\text{e}}_{\text{i}} ,\,{\text{f}}_{\text{i}} } \right\}_{{\text{i = 1}}}^{\text{m}}$ for W, such that

$$ \left\langle {{\text{e}}_{\text{i}} ,{\text{e}}_{\text{j}} } \right\rangle = 0 = \left\langle {{\text{f}}_{\text{i}} ,{\text{f}}_{\text{j}} } \right\rangle \,\,\,\,\,\,\,\,\,\,\,\left\langle {{\text{e}}_{\text{i}} ,{\text{f}}_{\text{j}} } \right\rangle = {{\delta }}_{{\text{ij}}} $$

(1.1)