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@article{faucris.116126824,
abstract = {A unitary representation of a, possibly infinite dimensional, Lie group *G * is called semibounded if the corresponding operators idπ(x)$i\mathtt{d}\pi (x)$ from the derived representations are uniformly bounded from above on some non-empty open subset of the Lie algebra g$\mathfrak{g}$. 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.},
author = {Neeb, Karl-Hermann and Zellner, Christoph},
doi = {10.1016/j.difgeo.2012.10.010},
faupublication = {yes},
journal = {Differential Geometry and its Applications},
keywords = {Infinite dimensional Lie group; Semibounded representation; Oscillator group; Coadjoint orbit; Complex structure},
pages = {268-283},
peerreviewed = {Yes},
title = {{Oscillator} algebras with semi-equicontinuous coadjoint orbits},
volume = {31},
year = {2013}
}