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

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 $2\times 2$ and we provide explicit expressions for most of these cases.

References in corpus (2)

• Infinite J-matrices and a matrix moment problem
• Spherical Functions of Fundamental $K$-Types Associated with the $n$-Dimensional Sphere

Cited by in corpus (6)

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• The Algebra of Differential Operators for a Gegenbauer Weight Matrix
• Ladder relations for a class of matrix valued orthogonal polynomials
• Deformation of matrix-valued orthogonal polynomials related to Gelfand pairs
• Pre-sequences of matrix orthogonal polynomials
• Non-symmetric Jacobi polynomials of type $B{C}_1$ as vector-valued polynomials Part 1: spherical functions