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Integrability of unitary representations on reproducing kernel spaces


Authors: Stéphane Merigon, Karl-Hermann Neeb and Gestur Ólafsson
Journal: Represent. Theory 19 (2015), 24-55
MSC (2010): Primary 17B15, 22E30, 22E70
DOI: https://doi.org/10.1090/S1088-4165-2015-00461-3
Published electronically: March 10, 2015
MathSciNet review: 3319397
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Abstract: Let $ \mathfrak{g}$ be a Banach-Lie algebra and $ \tau : \mathfrak{g} \to \mathfrak{g}$ an involution. Write $ \mathfrak{g}=\mathfrak{h}\oplus \mathfrak{q}$ for the eigenspace decomposition of $ \mathfrak{g}$ with respect to $ \tau $ and $ \mathfrak{g}^c := \mathfrak{h}\oplus i\mathfrak{q}$ for the dual Lie algebra. In this article we show the integrability of two types of infinitesimally unitary representations of $ \mathfrak{g}^c$. The first class of representation is determined by a smooth positive definite kernel $ K$ on a locally convex manifold $ M$. The kernel is assumed to satisfy a natural invariance condition with respect to an infinitesimal action $ \beta \colon \mathfrak{g} \to \mathcal {V}(M)$ by locally integrable vector fields that is compatible with a smooth action of a connected Lie group $ H$ with Lie algebra $ \mathfrak{h}$. The second class is constructed from a positive definite kernel corresponding to a positive definite distribution $ K \in C^{-\infty }(M \times M)$ on a finite dimensional smooth manifold $ M$ which satisfies a similar invariance condition with respect to a homomorphism $ \beta \colon \mathfrak{g} \to \mathcal {V}(M)$. As a consequence, we get a generalization of the Lüscher-Mack Theorem which applies to a class of semigroups that need not have a polar decomposition. Our integrability results also apply naturally to local representations and representations arising in the context of reflection positivity.


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Additional Information

Stéphane Merigon
Affiliation: Department of Mathematics, Friedrich-Alexander University, Erlangen-Nuremberg, Cauerstrasse 11, 91058 Erlangen, Germany
Email: merigon@math.fau.de

Karl-Hermann Neeb
Affiliation: Department of Mathematics, Friedrich-Alexander University, Erlangen-Nuremberg, Cauerstrasse 11, 91058 Erlangen, Germany
Email: neeb@math.fau.de

Gestur Ólafsson
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
Email: olafsson@math.lsu.edu

DOI: https://doi.org/10.1090/S1088-4165-2015-00461-3
Keywords: Unitary representations, finite and infinite dimensional Lie groups, manifolds, reproducing kernel
Received by editor(s): June 10, 2014
Received by editor(s) in revised form: June 11, 2014, July 11, 2014, and October 1, 2014
Published electronically: March 10, 2015
Additional Notes: The first author would like to thank Louisiana State University for their hospitality during his visit in 2014 when most of the work on this article was done
The research of the second author was supported by DFG-grant NE 413/7-2, SPP “Representation Theory”.
The research of the third author was supported by NSF grant DMS-1101337, “Representation Theory and Harmonic Analysis on Homogeneous Spaces”
Article copyright: © Copyright 2015 American Mathematical Society

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