Multiplicity freeness of unitary representations in sections of holomorphic Hilbert bundles

TL;DR

This paper establishes conditions under which the action of Banach-Lie groups on sections of holomorphic Hilbert bundles is multiplicity free, extending visible action concepts and providing examples in operator theory.

Contribution

It introduces new criteria for multiplicity freeness of group actions on holomorphic sections, generalizing visible actions and applying to larger groups and operator algebra contexts.

Findings

Proves multiplicity freeness under compatible antiholomorphic bundle maps.

Introduces $(S,\sigma)$-weakly visible actions for broader applicability.

Provides examples involving operator groups and von Neumann algebras.

Abstract

We prove several results asserting that the action of a Banach-Lie group on Hilbert spaces of holomorphic sections of a holomorphic Hilbert space bundle over a complex Banach manifold is multiplicity free. These results require the existence of compatible antiholomorphic bundle maps and certain multiplicity freeness assumptions for stabilizer groups. For the group action on the base, the notion of an $(S,\sigma)$-weakly visible action (generalizing T.Koboyashi's visible actions) provides an effective way to express the assumptions in an economical fashion. In particular, we derive a version for group actions on homogeneous bundles for larger groups. We illustrate these general results by several examples related to operator groups and von Neumann algebras.