Superunitary Representations of Heisenberg 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 Schrödinger-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 Schrödingerlike representations to metaplectic supergroups, also fit into this definition of superunitary representations.

A Fock Model and the Segal-Bargmann Transform for the Minimal Representation of the Orthosymplectic Lie Superalgebra osp(m,2|2n)

“…Note that it is only an inner product in the bosonic case. However, in [13] a new definition of Hilbert superspaces was introduced where the preserved form is no longer an inner product, but rather a non-degenerate, sesquilinear, superhermitian form. We will prove that the Bessel-Fischer product is such a form when restricted to F with M − 2 ∈ −2N.…”

“…However, a large class of Lie superalgebras, including osp(p, q|2n), do not allow for any super unitary representation in this sense [25,Theorem 6.2.1]. This highly unsatisfactory situation has inspired the search for a new or extended definition of a unitary representation [13,27]. At the moment it is still unclear what the right definition should be, but we believe that the construction of explicit examples which ought to be 'unitary' could be useful for this endeavour.…”

A Schrödinger model, Fock model and intertwining Segal-Bargmann transform for the exceptional Lie superalgebra $D(2,1;α)$

“…The Bessel-Fischer product is no longer an inner product in this case. However, under the right conditions it is a non-degenerate, sesquilinear, superhermitian form, which is consistent with the definition of Hilbert superspaces given in [22]. We will use the same approach in this section to construct the Fock space associated with D(2, 1; α).…”

“…The existing accepted definition [21,Definition 2 ] has the drawback that a lot of Lie superalgebras do not admit unitary representations at all, [12,Theorem 6.2.1]. This is a highly unsatisfying situation, which has inspired many to look for alternative definitions [22,23]. We strongly believe that the construction of explicit models of irreducible representations that 'ought' to be unitary such as the ones constructed in this paper should help in this undertaking.…”

A Fock Model and the Segal-Bargmann Transform for the Minimal Representation of the Orthosymplectic Lie Superalgebra $\mathfrak{osp}(m,2|2n)$

“…However, a large class of Lie superalgebras, including osp(p, q|2n), do not allow for any super unitary representation in this sense [11,Theorem 6.2.1]. This highly unsatisfactory situation has inspired the search for a new or extended definition of a unitary representation [16,17]. At the moment it is still unclear what the right definition should be, but we believe that the construction of explicit examples which ought to be 'unitary' could be useful for this endeavour.…”