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This article is cited in 9 scientific papers (total in 9 papers)
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One
Maarten Van Pruijssena, Pablo Románb a Universität Paderborn, Institut für Mathematik, Warburger Str. 100, 33098 Paderborn, Germany
b CIEM, FaMAF, Universidad Nacional de Córdoba, Medina Allende s/n Ciudad Universitaria, Córdoba, Argentina
Abstract:
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 $\Psi_{0}$. We analyze the base change and derive several properties. The most important one is that $\Psi_{0}$ 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 $\Psi_{0}$. The weight $W$ is also determined by $\Psi_{0}$. We provide an algorithm to calculate $\Psi_{0}$ explicitly. For the pair $(\mathrm{USp}(2n),\mathrm{USp}(2n-2)\times\mathrm{USp}(2))$ 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 $2\times2$ and we provide explicit expressions for most of these cases.
Keywords:
matrix valued classical pairs; multiplicity free branching.
Received: April 30, 2014; in final form December 12, 2014; Published online December 20, 2014
Citation:
Maarten Van Pruijssen, Pablo Román, “Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One”, SIGMA, 10 (2014), 113, 28 pp.
Linking options:
https://www.mathnet.ru/eng/sigma978 https://www.mathnet.ru/eng/sigma/v10/p113
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