Complex Semigroups for Oscillator Groups

TL;DR

This paper constructs complex semigroups for infinite-dimensional oscillator groups and establishes a correspondence between semibounded representations of these groups and holomorphic representations of the semigroups, aiding classification and analysis.

Contribution

It introduces Olshanski semigroups for infinite-dimensional oscillator groups and links semibounded group representations to holomorphic semigroup representations.

Findings

Every semibounded representation extends to a holomorphic semigroup representation.

Holomorphic semigroup representations correspond to semibounded group representations.

Results facilitate classification of CCR representations with positive Hamiltonian.

Abstract

An oscillator group $G$ is a semidirect product of a Heisenberg group with a one-parameter group. In this article we construct Olshanski semigroups for infinite-dimensional oscillator groups. These are complex involutive semigroups which have a polar decomposition. The main application will be for representations $\pi$ of $G$ which are semibounded, i.e., there exists a non-empty open subset $U$ of the Lie algebra $\mathfrak{g}$ such that the operators $id\pi(x)$ from the derived representation are uniformly bounded from above for $x\in U$. More precisely we show that every semibounded representation of an oscillator group $G$ extends to a non-degenerate holomorphic representation of such a semigroup and conversely each non-degenerate holomorphic representation of such a semigroup gives rise to a semibounded representation of $G$. The main application of this result is a classification of representations of the canonical commutation relations with a positive Hamiltonian, which will be obtained in a subsequent paper. Moreover it yields direct integral decomposition into irreducible ones and implies the existence of a dense subspace of analytic vectors for semibounded representations of $G$.