1. Superunitary Representations of Heisenberg Supergroups
- Author
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Axel Marcillaud de Goursac and Jean-Philippe Michel
- Subjects
Pure mathematics ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Superspace ,01 natural sciences ,Noncommutative geometry ,Functional Analysis (math.FA) ,Parseval's theorem ,Mathematics - Functional Analysis ,Unitary representation ,FOS: Mathematics ,Supergeometry ,Heisenberg group ,Representation Theory (math.RT) ,0101 mathematics ,22E27, 58C50, 43A32 ,Mathematics::Representation Theory ,Signature (topology) ,Mathematics - Representation Theory ,Mathematical Physics ,Mathematics - 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., Comment: 59 pages. v2: section 6 (Proof of Stone-von Neumann theorem) and section 7 (super unitary dual) were corrected and rewritten
- Published
- 2018