  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.
