MATRIX VALUED ORTHOGONAL POLYNOMIALS FOR GELFAND PAIRS OF RANK ONE

Abstract

In this paper we study matrix valued orthogonal polynomials of one variable associated with a compact connected Gelfand pair $(G,K)$ of rank one, as a generalization of earlier work by Koornwinder [30] and subsequently by Koelink, van Pruijssen and Roman [28], [29] for the pair (SU (2)$\times$SU (2), SU (2)), and by Grünbaum, Pacharoni and Tirao [13] for the pair (SU (3), U (2)). Our method is based on representation theory using an explicit determination of the relevant branching rules. Our matrix valued orthogonal polynomials have the Sturm–Liouville property of being eigenfunctions of a second order matrix valued linear differential operator coming from the Casimir operator, and in fact are eigenfunctions of a commutative algebra of matrix valued linear differential operators coming from the $K$-invariant elements in the universal enveloping algebra of the Lie algebra of $G$.