Geometric quantization and unitary highest weight Harish-Chandra supermodules

TL;DR

This paper extends geometric quantization to the super setting, constructing unitary highest weight Harish-Chandra supermodules for real forms of Lie supergroups acting on pseudo-K"ahler supermanifolds, and demonstrates that quantization commutes with reduction.

Contribution

It introduces a super geometric quantization framework for Lie supergroups, constructing new unitary representations and establishing the commutation of quantization and reduction.

Findings

Constructed unitary representations from sections of line bundles.

Identified conditions for highest weight Harish-Chandra supermodules.

Proved that quantization commutes with symplectic reduction.

Abstract

Geometric quantization transforms a symplectic manifold with Lie group action to a unitary representation. In this article, we extend geometric quantization to the super setting. We consider real forms of contragredient Lie supergroups with compact Cartan subgroups, and study their actions on some pseudo-K\"ahler supermanifolds. We construct their unitary representations in terms of sections of some line bundles. These unitary representations contain highest weight Harish-Chandra supermodules, whose occurrences depend on the image of the moment map. As a result, we construct a Gelfand model of highest weight Harish-Chandra supermodules. We also perform symplectic reduction, and show that quantization commutes with reduction.