## Tohoku Mathematical Journal 2016 September SECOND SERIES VOL. 68, NO. 3

 Tohoku Math. J. 68 (2016), 407-437

Title MATRIX VALUED ORTHOGONAL POLYNOMIALS FOR GELFAND PAIRS OF RANK ONE

Author Gert Heckman and Maarten van Pruijssen

(Received March 17, 2014, revised November 18, 2014)
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 $G$.

Mathematics Subject Classification. Primary 22E46; Secondary 33C47.

Key words and phrases. Spherical varieties of rank one, multiplicity free branching, matrix valued orthogonal polynomials.