Fr\'echet globalisations of Harish-Chandra supermodules

For any Lie supergroup whose underlying Lie group is reductive, we prove an extension of the Casselman-Wallach globalisation theorem: There is an equivalence between the category of Harish-Chandra modules and the category of SF-representations (smooth Fr\'echet representations of moderate growth) whose module of finite vectors is Harish-Chandra. As an application, we extend to Lie supergroups a general general form of the Gel'fand-Kazhdan criterion due to Sun-Zhu.Comment: 33 pages; final version accepted for publication in Int. Math. Res. Not. IMR

Non-Euclidean Fourier inversion on super-hyperbolic space

For the super-hyperbolic space in any dimension, we introduce the non-Euclidean Helgason--Fourier transform. We prove an inversion formula exhibiting residue contributions at the poles of the Harish-Chandra c-function, signalling discrete parts in the spectrum. The proof is based on a detailed study of the spherical superfunctions, using recursion relations and localization techniques to normalize them precisely, careful estimates of their derivatives, and a rigorous analysis of the boundary terms appearing in the polar coordinate expression of the invariant integralComment: 30 pages; final version accepted for publication in Comm. Math. Phy

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

A convenient category of supermanifolds

With a view towards applications in the theory of infinite-dimensional representations of finite-dimensional Lie supergroups, we introduce a new category of supermanifolds. In this category, supermanifolds of `maps' and `fields' (fibre bundle sections) exist. In particular, loop supergroups can be realised globally in this framework. It also provides a convenient setting for induced representations of supergroups, allowing for a version of Frobenius reciprocity. Finally, convolution algebras of finite-dimensional Lie supergroups are introduced and applied to a prove a supergroup Dixmier-Malliavin Theorem: The space of smooth vectors of a continuous representation of a supergroup pair equals the Garding space given by the convolution with compactly supported smooth supergroup densities.Comment: 32 page

Superorbits

We study actions of Lie supergroups, in particular, the hitherto elusive notion of orbits through odd (or more general) points. Following categorical principles, we derive a conceptual framework for their treatment and therein prove general existence theorems for the isotropy (or stabiliser) supergroups and orbits through general points. In this setting, we show that the coadjoint orbits always admit a (relative) supersymplectic structure of Kirillov-Kostant-Souriau type. Applying a family version of Kirillov's orbit method, we decompose the regular representation of an odd Abelian supergroup into an odd direct integral of characters and construct universal families of representations, parametrised by a supermanifold, for two different super variants of the Heisenberg group.Comment: 47 pages; v3: final version, accepted for publication in J. Math. Inst. Jussie

Spherical representations of Lie supergroups

The classical Cartan-Helgason theorem characterises finite-dimensional spherical representations of reductive Lie groups in terms of their highest weights. We generalise the theorem to the case of a reductive symmetric supergroup pair $(G,K)$ of even type. Along the way, we compute the Harish-Chandra $c$-function of the symmetric superspace $G/K$. By way of an application, we show that all spherical representations are self-dual in type AIII|AIII.Comment: 37 pages; title changed; substantially revised version; accepted for publication, J. Func. Anal. (2014