Principal Series Representations of Infinite Dimensional Lie Groups, I: Minimal Parabolic Subgroups

TL;DR

This paper investigates the structure of minimal parabolic subgroups in classical infinite dimensional real simple Lie groups and explores their role in constructing principal series representations, extending finite-dimensional theories.

Contribution

It introduces a new structure theory for parabolic subgroups not necessarily arising as direct limits, and applies it to develop principal series representations for infinite dimensional Lie groups.

Findings

Characterization of minimal parabolic subgroups in infinite dimensions

Construction of principal series representations for classical lim--compact groups

Analysis of analytic aspects in the representation construction

Abstract

We study the structure of minimal parabolic subgroups of the classical infinite dimensional real simple Lie groups, corresponding to the classical simple direct limit Lie algebras. This depends on the recently developed structure of parabolic subgroups and subalgebras that are not necessarily direct limits of finite dimensional parabolics. We then discuss the use of that structure theory for the infinite dimensional analog of the classical principal series representations. We look at the unitary representation theory of the classical lim--compact groups $U(\infty)$, $SO(\infty)$ and $Sp(\infty)$ in order to construct the inducing representations, and we indicate some of the analytic considerations in the actual construction of the induced representations.