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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| Date: | 2014 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2014
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/146404 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | 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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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | 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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