Superorbits

TL;DR

This paper develops a categorical framework for Lie supergroup actions, establishing existence theorems for orbits and isotropy supergroups, and explores supersymplectic structures and representation decompositions in supergeometry.

Contribution

It introduces a conceptual framework for orbits through odd points and proves existence theorems for supergroup stabilizers and orbits, advancing supergeometry theory.

Findings

Coadjoint orbits admit supersymplectic structures.

Decomposition of regular representations into characters.

Construction of universal families of supergroup representations.

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.