286 research outputs found
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
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
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
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 of even type. Along the way, we compute the
Harish-Chandra -function of the symmetric superspace . 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
Asymptotics of spherical superfunctions on rank one Riemannian symmetric superspaces
We compute the Harish-Chandra -function for a generic class of rank-one
purely non-compact Riemannian symmetric superspaces in terms of Euler
functions, proving that it is meromorphic. Compared to the even case,
the poles of the -function are shifted into the right half-space. We derive
the full asymptotic Harish-Chandra series expansion of the spherical
superfunctions on . In the case where the multiplicity of the simple root is
an even negative number, they have a closed expression as Jacobi polynomials
for an unusual choice of parameters.Comment: 41 pages; slightly expanded version; accepted for publication in Doc.
Math (2014
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
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