### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 113, 28 pages      arXiv:1312.6577      https://doi.org/10.3842/SIGMA.2014.113

### Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One

Maarten van Pruijssen a and Pablo Román b
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

Received April 30, 2014, in final form December 12, 2014; Published online December 20, 2014

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.

Key words: matrix valued classical pairs; multiplicity free branching.

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