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Communications in Mathematical Physics

, Volume 210, Issue 2, pp 309–321 | Cite as

The Oscillator Representation and¶Groups of Heisenberg Type

  • E. Galina
  • A. Kaplan
  • F. Levstein
  • 75 Downloads

Abstract:

We obtain the explicit reduction of the Oscillator representation of the symplectic group, on the subgroups of automorphisms of certain vector-valued skew forms Φ of “Clifford type”-equivalently, of automorphisms of Lie algebras of Heisenberg type. These subgroups are of the form G⋅ \Spin(k), with G a real reductive matrix group, in general not compact, commuting with Spin(k) with finite intersection. The reduction turns out to be free of multiplicity in all the cases studied here, which include some where the factors do not form a Howe pair. If G is maximal compact in G, the restriction to K⋅ \Spin(k) is essentially the action on the symmetric algebra on a space of spinors. The cases when this is multiplicity-free are listed in [R]; our examples show that replacing K by G does make a difference. Our question is motivated to a large extent by the geometric object that comes with such a Φ: a Fock-space bundle over a sphere, with G acting fiberwise via the oscillator representation. It carries a Dirac operator invariant under G and determines special derivations of the corresponding gauge algebra.

Keywords

Dirac Operator Geometric Object Symplectic Group Oscillator Representation Finite Intersection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • E. Galina
    • 1
  • A. Kaplan
    • 2
  • F. Levstein
    • 1
  1. 1.FaMAF, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina.¶E-mail: galina@mate.uncor.edu; levstein@mate.uncor.eduAR
  2. 2.University of Massachusetts, Amherst, MA 01003, USA. E-mail: kaplan@math.umass.eduUS

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