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 Ψ0. We analyze the base change and derive several properties. The most important one is that Ψ0satisfies 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 Ψ0explicitly. 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.