Holomorphic Realization of Unitary Representations of Banach--Lie Groups

TL;DR

This paper develops a holomorphic induction framework for classifying and analyzing unitary representations of Banach--Lie groups, extending finite-dimensional techniques to infinite-dimensional settings.

Contribution

It introduces a holomorphic induction method for Banach--Lie groups, preserving irreducibility and providing criteria for identifying positive energy representations.

Findings

Classification of complex bundle structures similar to finite-dimensional case

Holomorphic induction preserves irreducibility of representations

Criteria for identifying holomorphically induced representations

Abstract

In this paper we explore the method of holomorphic induction for unitary representations of Banach--Lie groups. First we show that the classification of complex bundle structures on homogeneous Banach bundles over complex homogeneous spaces of real Banach--Lie groups formally looks as in the finite dimensional case. We then turn to a suitable concept of holomorphic unitary induction and show that this process preserves commutants. In particular, holomorphic induction from irreducible representations leads to irreducible ones. Finally we develop criteria to identify representations as holomorphically induced ones and apply these to the class of so-called positive energy representations. All this is based on extensions of Arveson's concept of spectral subspaces to representations on Fr\'echet spaces, in particular on spaces of smooth vectors.