Super Kähler structures on the complex Abelian Lie supergroups

We consider a real Abelian Lie supergroup $G$ acting on its complexification $M$, equipped with a $G$-invariant super Kähler 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 Kähler 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.