Super Kähler structures on the complex Abelian Lie supergroups

Abstract

Let $G$ be a real Abelian Lie supergroup, let $M$ be its complexification. We classify the $G$-invariant super Kähler forms on $M$. For the super Kähler forms with Hamiltonian actions, we extend the scheme of geometric quantization to the super setting and construct unitary $G$-representations. We show that the irreducible representations that occur in these unitary representations are governed by the image of the moment maps of super Kähler forms. As an application, we construct a Gelfand model of $G$, namely a unitary $G$-representation in which every unitary irreducible representation occurs exactly once.

24 pages - Submitted. W.r.t. the previous version, the entire paper has been deeply revised: the main results (that had some flaws) have been amended, and the presentation of the torus case and the Euclidean case have been fused together, which makes the presentation shorter