Principal series representations of infinite dimensional Lie groups, II: Construction of induced representations

TL;DR

This paper develops a framework for constructing and analyzing principal series representations of infinite dimensional classical Lie groups, extending finite dimensional theories and exploring properties of parabolic subgroups.

Contribution

It extends the structure theory of parabolic subgroups in infinite dimensional Lie groups and constructs induced representations using amenability and geometric conditions.

Findings

Maximal lim-compact subgroup acts transitively on G/P when P is flag-closed.

Minimal parabolics are shown to be amenable, enabling induced representation construction.

Decomposition P=MAN provides K-spectrum information similar to finite dimensional cases.

Abstract

We study representations of the classical infinite dimensional real simple Lie groups $G$ induced from factor representations of minimal parabolic subgroups $P$. This makes strong use of the recently developed structure theory for those parabolic subgroups and subalgebras. In general parabolics in the infinite dimensional classical Lie groups are are somewhat more complicated than in the finite dimensional case, and are not direct limits of finite dimensional parabolics. We extend their structure theory and use it for the infinite dimensional analog of the classical principal series representations. In order to do this we examine two types of conditions on $P$: the flag-closed condition and minimality. We use some riemannian symmetric space theory to prove that if $P$ is flag-closed then any maximal lim-compact subgroup $K$ of $G$ is transitive on $G /P$\,. When $P$ is minimal we prove that it is amenable, and we use properties of amenable groups to induce unitary representations $\tau$ of $P$ up to continuous representations $\Ind_P^G(\tau)$ of $G$ on complete locally convex topological vector spaces. When $P$ is both minimal and flag-closed we have a decomposition $P = MAN$ similar to that of the finite dimensional case, and we show how this gives $K$--spectrum information $\Ind_P^G(\tau)|_K = \Ind_M^K(\tau|_M)$.