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.

47 pages; v3: final version, accepted for publication in J. Math. Inst. Jussieu

References in corpus (2)

• Fréchet globalisations of Harish-Chandra supermodules
• $Q$-representations and unitary representations of the super-Heisenberg group and harmonic superanalysis

Cited by in corpus (2)

• A minimal representation of the orthosymplectic Lie supergroup
• Non-Euclidean Fourier inversion on super-hyperbolic space