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SUPERORBITS

Part of: Special varieties Analytic spaces General theory of differentiable manifolds Symplectic geometry, contact geometry Algebraic groups

Published online by Cambridge University Press:  20 July 2016

Alexander Alldridge ,
Joachim Hilgert  and
Tilmann Wurzbacher
Alexander Alldridge
Affiliation:
Universität zu Köln, Mathematisches Institut, Weyertal 86-90, 50931 Köln, Germany (alldridg@math.uni-koeln.de)
Joachim Hilgert
Affiliation:
Universität Paderborn, Institut für Mathematik, 33095 Paderborn, Germany (hilgert@math.upb.de)
Tilmann Wurzbacher
Affiliation:
Institut É. Cartan (IECL), Université de Lorraine et C.N.R.S., 57045 Metz, France (tilmann.wurzbacher@univ-lorraine.fr)
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Abstract

We study actions of Lie supergroups, in particular, the hitherto elusive notion of orbits through odd (or more general) points. Following categorical principles, we derive a conceptual framework for their treatment and therein prove general existence theorems for the isotropy (or stabiliser) supergroups and orbits through general points. In this setting, we show that the coadjoint orbits always admit a (relative) supersymplectic structure of Kirillov–Kostant–Souriau type. Applying a family version of Kirillov’s orbit method, we decompose the regular representation of an odd Abelian supergroup into an odd direct integral of characters and construct universal families of representations, parametrised by a supermanifold, for two different super variants of the Heisenberg group.

Keywords

categorical quotientLie supergroupcoadjoint actiongeneralised pointKirillov’s orbit methodsupermanifold

MSC classification

Primary: 14L30: Group actions on varieties or schemes (quotients) 58A50: Supermanifolds and graded manifolds
Secondary: 14M30: Supervarieties 32C11: Complex supergeometry 53D50: Geometric quantization 57S20: Noncompact Lie groups of transformations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 17 , Issue 5 , November 2018 , pp. 1065 - 1120
Copyright
© Cambridge University Press 2016 

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Footnotes

Research supported by Deutsche Forschungsgemeinschaft (DFG), grant nos. SFB/TR 12 (all authors), the Heisenberg grant AL 698/3-1 (A.A.), the Leibniz prize to M. Zirnbauer ZI 513/2-1 (A.A.), SFB TRR 183 (A.A.), and the Institutional Strategy of the University of Cologne within the German Excellence Initiative (all authors).

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