Neeb KH, Zellner C (2013)

**Publication Type:** Journal article, Original article

**Publication year:** 2013

**Publisher:** Elsevier

**Book Volume:** 31

**Pages Range:** 268-283

**Journal Issue:** 2

**DOI:** 10.1016/j.difgeo.2012.10.010

A unitary representation of a, possibly infinite dimensional, Lie group *G * is called semibounded if the corresponding operators idπ(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 *γ*. For these groups it turns out that the existence of non-trivial semibounded representations is equivalent to the existence of the 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 *γ*. 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.

Karl Hermann Neeb
Lehrstuhl für Mathematik (Lie-Gruppen und Darstellungstheorie)
Christoph Zellner
Department Mathematik

**APA:**

Neeb, K.H., & Zellner, C. (2013). Oscillator algebras with semi-equicontinuous coadjoint orbits. *Differential Geometry and its Applications*, *31*(2), 268-283. https://doi.org/10.1016/j.difgeo.2012.10.010

**MLA:**

Neeb, Karl Hermann, and Christoph Zellner. "Oscillator algebras with semi-equicontinuous coadjoint orbits." *Differential Geometry and its Applications* 31.2 (2013): 268-283.

**BibTeX:** Download