Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One
We present a method to obtain infinitely many examples of pairs (W,D) consisting of a matrix weight W in one variable and a symmetric second-order differential operator D. The method is based on a uniform construction of matrix valued polynomials starting from compact Gelfand pairs (G,K) of rank one...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2014 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2014
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/146404 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One / Maarten van Pruijssen , P. Román // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 40 назв. — англ. |
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Maarten van Pruijssen Román, P. 2019-02-09T11:22:55Z 2019-02-09T11:22:55Z 2014 Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One / Maarten van Pruijssen , P. Román // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 40 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 22E46; 33C47 DOI: http://dx.doi.org/10.3842/SIGMA.2014.113 https://nasplib.isofts.kiev.ua/handle/123456789/146404 We present a method to obtain infinitely many examples of pairs (W,D) consisting of a matrix weight W in one variable and a symmetric second-order differential operator D. The method is based on a uniform construction of matrix valued polynomials starting from compact Gelfand pairs (G,K) of rank one and a suitable irreducible K-representation. The heart of the construction is the existence of a suitable base change Ψ₀. We analyze the base change and derive several properties. The most important one is that Ψ₀ satisfies a first-order differential equation which enables us to compute the radial part of the Casimir operator of the group G as soon as we have an explicit expression for Ψ0. The weight W is also determined by Ψ₀. We provide an algorithm to calculate Ψ₀ explicitly. For the pair (USp(2n),USp(2n−2)×USp(2)) we have implemented the algorithm in GAP so that individual pairs (W,D) can be calculated explicitly. Finally we classify the Gelfand pairs (G,K) and the K-representations that yield pairs (W,D) of size 2×2 and we provide explicit expressions for most of these cases. The research for this paper was partly conducted when the first author visited the University of Cordoba in August and September 2012. We would like to thank the Mathematics Departments of the Universities of Nijmegen and Cordoba for their generous supports that made this visit possible. Finally we would like to thank to the anonymous referees, whose comments and suggestions have helped us to improve the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One |
| spellingShingle |
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One Maarten van Pruijssen Román, P. |
| title_short |
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One |
| title_full |
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One |
| title_fullStr |
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One |
| title_full_unstemmed |
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One |
| title_sort |
matrix valued classical pairs related to compact gelfand pairs of rank one |
| author |
Maarten van Pruijssen Román, P. |
| author_facet |
Maarten van Pruijssen Román, P. |
| publishDate |
2014 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We present a method to obtain infinitely many examples of pairs (W,D) consisting of a matrix weight W in one variable and a symmetric second-order differential operator D. The method is based on a uniform construction of matrix valued polynomials starting from compact Gelfand pairs (G,K) of rank one and a suitable irreducible K-representation. The heart of the construction is the existence of a suitable base change Ψ₀. We analyze the base change and derive several properties. The most important one is that Ψ₀ satisfies a first-order differential equation which enables us to compute the radial part of the Casimir operator of the group G as soon as we have an explicit expression for Ψ0. The weight W is also determined by Ψ₀. We provide an algorithm to calculate Ψ₀ explicitly. For the pair (USp(2n),USp(2n−2)×USp(2)) we have implemented the algorithm in GAP so that individual pairs (W,D) can be calculated explicitly. Finally we classify the Gelfand pairs (G,K) and the K-representations that yield pairs (W,D) of size 2×2 and we provide explicit expressions for most of these cases.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/146404 |
| citation_txt |
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One / Maarten van Pruijssen , P. Román // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 40 назв. — англ. |
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2025-12-07T15:39:33Z |
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2025-12-07T15:39:33Z |
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1850864538624196608 |