## Integrability of unitary representations on reproducing kernel spaces

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- by Stéphane Merigon, Karl-Hermann Neeb and Gestur Ólafsson
- Represent. Theory
**19**(2015), 24-55 - DOI: https://doi.org/10.1090/S1088-4165-2015-00461-3
- Published electronically: March 10, 2015
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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.## References

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## Bibliographic 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
- MR Author ID: 288679
- Email: neeb@math.fau.de
**Gestur Ólafsson**- Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
- MR Author ID: 133515
- Email: olafsson@math.lsu.edu
- 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” - © Copyright 2015 American Mathematical Society
- Journal: Represent. Theory
**19**(2015), 24-55 - MSC (2010): Primary 17B15, 22E30, 22E70
- DOI: https://doi.org/10.1090/S1088-4165-2015-00461-3
- MathSciNet review: 3515153