Semi-bounded unitary representations of infinite-dimensional Lie groups

TL;DR

This paper introduces semi-bounded unitary representations for infinite-dimensional Lie groups, characterizing their properties via momentum sets and spectral measures, and providing methods for computing these sets in specific contexts.

Contribution

It defines semi-bounded representations for infinite-dimensional Lie groups and characterizes them using momentum sets and spectral measures, extending the understanding of such representations.

Findings

Characterization of semi-bounded representations via momentum sets.

Analysis of semi-equicontinuous subsets in dual spaces.

Method for computing momentum sets in reproducing kernel Hilbert spaces.

Abstract

In this note we introduce the concept of a semi-bounded unitary representations of an infinite-dimensional Lie group $G$. Semi-boundedness is defined in terms of the corresponding momentum set in the dual $\g'$ of the Lie algebra $\g$ of $G$. After dealing with some functional analytic issues concerning certain weak-$*$-locally compact subsets of dual spaces, called semi-equicontinuous, we characterize unitary representations which are bounded in the sense that their momentum set is equicontinuous, we characterize semi-bounded representations of locally convex spaces in terms of spectral measures, and we also describe a method to compute momentum sets of unitary representations of reproducing kernel Hilbert spaces of holomorphic functions.