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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 ofclassical geometric quantization to this setting and construct a unitaryG-representation. We show that the occurrences of its irreduciblesubrepresentations are governed by the image of the moment map of the superK\"ahler form. As an application, we construct a Gelfand model of G, namely aunitary G-representation in which every unitary irreducible representationoccurs exactly once.

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