Super K\"ahler structures on the complex Abelian Lie supergroups

Abstract

We consider a real Abelian Lie supergroup $G$ acting on its complexification $M$, equipped with a $G$-invariant super K\"ahler form. We extend the scheme of classical geometric quantization to this setting and construct a unitary $G$-representation. We show that the occurrences of its irreducible subrepresentations are governed by the image of the moment map of the super K\"ahler form. As an application, we construct a Gelfand model of $G$, namely a unitary $G$-representation in which every unitary irreducible representation occurs exactly once.