Geometric Quantization of Superorbits: a Case Study

TL;DR

This paper investigates the geometric quantization of superorbits in a Heisenberg-like Lie supergroup, revealing limitations of traditional methods and proposing an extended approach involving non-homogeneous points and forms.

Contribution

It demonstrates that not all irreducible representations can be obtained via standard geometric quantization of coadjoint orbits, and introduces a broader method involving non-homogeneous orbits and forms.

Findings

Standard geometric quantization does not produce all representations.

Non-homogeneous orbits and forms can generate all irreducible representations.

The choice of polarization significantly affects the resulting representation.

Abstract

By decomposing the regular representation of a particular (Heisenberg-like) Lie supergroup into irreducible subspaces, we show that not all of them can be obtained by applying geometric quantization to coadjoint orbits with an even symplectic form. However, all of them can be obtained by introducing coadjoint orbits through non-homogeneous points and with non-homogeneous symplectic forms as described in \cite{Tu1}. In this approach it turns out that the choice of a polarization can change (dramatically) the representation associated to an orbit. On the other hand, the procedure is not completely mechanical (meaning that some parts have to be done "by hand"), hence work remains to be done in order to understand all details of what is happening.