Superunitary Representations of Heisenberg Supergroups

TL;DR

This paper broadens the concept of superunitary representations for Heisenberg supergroups using a new Hilbert superspace framework, enabling a generalized representation theory and Fourier analysis applicable across various signatures.

Contribution

It introduces a new definition of superunitary representations based on Hilbert superspaces, extending the classical theory to all signatures and including metaplectic supergroups.

Findings

Established a generalized Stone-von Neumann theorem for supergroups.

Constructed superunitary dual and Fourier transform satisfying Parseval's theorem.

Extended superunitary representations to metaplectic supergroups.

Abstract

Numerous Lie supergroups do not admit superunitary representations except the trivial one, e.g., Heisenberg and orthosymplectic supergroups in mixed signature. To avoid this situation, we introduce in this paper a broader definition of superunitary representation, relying on a new definition of Hilbert superspace. The latter is inspired by the notion of Krein space and was developed initially for noncommutative supergeometry. For Heisenberg supergroups, this new approach yields a smooth generalization, whatever the signature, of the unitary representation theory of the classical Heisenberg group. First, we obtain Schrodinger-like representations by quantizing generic coadjoint orbits. They satisfy the new definition of irreducible superunitary representations and serve as ground to the main result of this paper: a generalized Stone-von Neumann theorem. Then, we obtain the superunitary dual and build a group Fourier transformation, satisfying Parseval theorem. We eventually show that metaplectic representations, which extend Schrodinger-like representations to metaplectic supergroups, also fit into this definition of superunitary representations.