Oscillator algebras with semi-equicontinuous coadjoint orbits

TL;DR

This paper characterizes when generalized oscillator groups admit non-trivial semi-bounded unitary representations by linking this to the existence of semi-equicontinuous coadjoint orbits, a geometric condition related to the Hamiltonian.

Contribution

It provides a complete characterization of semi-bounded representations for generalized oscillator groups without spectral restrictions on the generator D.

Findings

Existence of semi-bounded representations is equivalent to semi-equicontinuous coadjoint orbits.

The geometric condition can be expressed via a positivity condition on the Hamiltonian.

No spectral assumptions are made on the operator D.

Abstract

A unitary representation of a, possibly infinite dimensional, Lie group G is called semi-bounded if the corresponding operators id\pi(x) from the derived representations are uniformly bounded from above on some non-empty open subset of the Lie algebra g. Not every Lie group has non-trivial semibounded unitary representations, so that it becomes an important issue to decide when this is the case. In the present paper we describe a complete solution of this problem for the class of generalized oscillator groups, which are semidirect products of Heisenberg groups with a one-parameter group \gamma. For these groups it turns out that the existence of non-trivial semibounded representations is equivalent to the existence of so-called semi-equicontinuous non-trivial coadjoint orbits, a purely geometric condition on the coadjoint action. This in turn can be expressed by a positivity condition on the Hamiltonian function corresponding to the infinitesimal generator D of \gamma. A central point of our investigations is that we make no assumption on the structure of the spectrum of D. In particular, D can be any skew-adjoint operator on a Hilbert space.