Harmonic analysis on Heisenberg--Clifford Lie supergroups

We define a Fourier transform and a convolution product for functions and distributions on Heisenberg--Clifford Lie supergroups. The Fourier transform exchanges the convolution and a pointwise product, and is an intertwining operator for the left regular representation. We generalize various classical theorems, including the Paley--Wiener--Schwartz theorem, and define a convolution Banach algebra.Comment: 28 page

Superunitary Representations of Heisenberg Supergroups

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 rewritte

Fr\'echet Quantum Supergroups

In this paper, we introduce Fr\'echet quantum supergroups and their representations. By using the universal deformation formula of the abelian supergroups R^{m|n} we construct various classes of Fr\'echet quantum supergroups that are deformation of classical ones. For such quantum supergroups, we find an analog of Kac-Takesaki operators that are superunitary and satisfy the pentagonal relation.Comment: 21 pages, published versio

Representations of Lie Groups and Supergroups

The workshop focussed on recent developments in the representation theory of group objects in several categories, mostly finite and infinite dimensional smooth manifolds and supermanifolds. The talks covered a broad range of topics, with a certain emphasis on benchmark problems and examples such as branching, limit behavior, and dual pairs. In many talks the relation to physics played an important role