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SUPERORBITS

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.

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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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