Geometric quantization and unitary highest weight Harish-Chandra supermodules

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.