Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs

We study Gelfand pairs for locally compact quantum groups. We give an operator algebraic interpretation and show that the quantum Plancherel transformation restricts to a spherical Plancherel transformation. As an example, we turn the quantum group analogue of the normaliser of SU(1,1) in SL(2,C) to...

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Published in:Symmetry, Integrability and Geometry: Methods and Applications
Date:2011
Main Author: Caspers, M.
Format: Article
Language:English
Published: Інститут математики НАН України 2011
Online Access:https://nasplib.isofts.kiev.ua/handle/123456789/147385
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Journal Title:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs / M. Caspers // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 44 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
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author Caspers, M.
author_facet Caspers, M.
citation_txt Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs / M. Caspers // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 44 назв. — англ.
collection DSpace DC
container_title Symmetry, Integrability and Geometry: Methods and Applications
description We study Gelfand pairs for locally compact quantum groups. We give an operator algebraic interpretation and show that the quantum Plancherel transformation restricts to a spherical Plancherel transformation. As an example, we turn the quantum group analogue of the normaliser of SU(1,1) in SL(2,C) together with its diagonal subgroup into a pair for which every irreducible corepresentation admits at most two vectors that are invariant with respect to the quantum subgroup. Using a Z₂-grading, we obtain product formulae for little q-Jacobi functions.
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institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn 1815-0659
language English
last_indexed 2025-12-07T13:15:48Z
publishDate 2011
publisher Інститут математики НАН України
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spelling Caspers, M.
2019-02-14T16:55:15Z
2019-02-14T16:55:15Z
2011
Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs / M. Caspers // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 44 назв. — англ.
1815-0659
2010 Mathematics Subject Classification: 16T99; 43A90
https://nasplib.isofts.kiev.ua/handle/123456789/147385
We study Gelfand pairs for locally compact quantum groups. We give an operator algebraic interpretation and show that the quantum Plancherel transformation restricts to a spherical Plancherel transformation. As an example, we turn the quantum group analogue of the normaliser of SU(1,1) in SL(2,C) together with its diagonal subgroup into a pair for which every irreducible corepresentation admits at most two vectors that are invariant with respect to the quantum subgroup. Using a Z₂-grading, we obtain product formulae for little q-Jacobi functions.
This paper is a contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”. The full collection is available at http://www.emis.de/journals/SIGMA/OPSF.html.
 The author likes to thank Erik Koelink for the useful discussions and Noud Aldenhoven for
 providing Fig. 1. Also, the author benefits from a detailed referee report.
en
Інститут математики НАН України
Symmetry, Integrability and Geometry: Methods and Applications
Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs
Article
published earlier
spellingShingle Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs
Caspers, M.
title Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs
title_full Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs
title_fullStr Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs
title_full_unstemmed Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs
title_short Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs
title_sort spherical fourier transforms on locally compact quantum gelfand pairs
url https://nasplib.isofts.kiev.ua/handle/123456789/147385
work_keys_str_mv AT caspersm sphericalfouriertransformsonlocallycompactquantumgelfandpairs