In a previous paper we have introduced matrix-valued analogues of the Chebyshev polynomials by studying matrix-valued spherical functions on SU(2) × SU(2). In particular the matrix-size of the polynomials is arbitrarily large. The matrix-valued orthogonal polynomials and the corresponding weight function are studied. In particular, we calculate the LDU-decomposition of the weight where the matrix entries of L are given in terms of Gegenbauer polynomials. The monic matrix-valued orthogonal polynomials P n are expressed in terms of Tirao's matrix-valued hypergeometric function using the matrix-valued differential operator of first and second order to which the P n 's are eigenfunctions. From this result we obtain an explicit formula for coefficients in the three-term recurrence relation satisfied by the polynomials P n . These differential operators are also crucial in expressing the matrix entries of P n L as a product of a Racah and a Gegenbauer polynomial. We also present a group theoretic derivation of the matrix-valued differential operators by considering the Casimir operators corresponding to SU(2) × SU(2).

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 Ψ 0 . We analyze the base change and derive several properties. The most important one is that Ψ 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 Ψ 0 . The weight W is also determined by Ψ 0 . We provide an algorithm to calculate Ψ 0 explicitly. 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.

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Abstract. Let (G, H) be a reductive spherical pair and P ⊂ H a parabolic subgroup such that (G, P ) is spherical. The triples (G, H, P ) with this property are called multiplicity free systems and they are classified in this paper. Denote by π H µ = ind H P µ the Borel-Weil realization of the irreducible H-representation of highest weight µ ∈ P + H and consider the induced representation indµ , a multiplicity free induced representation. Some properties of the spectrum of the multiplicity free induced representations are discussed. For three multiplicity free systems the spectra are calculated explicitly. The spectra give rise to families of multi-variable orthogonal polynomials which generalize families of Jacobi polynomials: they are simultaneous eigenfunctions of a commutative algebra of differential operators, they satisfy recurrence relations and they are orthogonal with respect to integrating against a matrix weight on a compact subset. We discuss some difficulties in describing the theory for these families of polynomials in the generality of the classification. Contents

In Part 1 we study the spherical functions on compact symmetric pairs of arbitrary rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules. The spherical functions are taking values in the spaces of linear operators of a finite dimensional representation of the subgroup, so the spherical functions are matrix-valued. Under these assumptions these functions can be described in terms of matrix-valued orthogonal polynomials in several variables, where the number of variables is the rank of the compact symmetric pair. Moreover, these polynomials are uniquely determined as simultaneous eigenfunctions of a commutative algebra of differential operators.In Part 2 we verify that the group case SU(n + 1) meets all the conditions that we impose in Part 1. For any k ∈ N 0 we obtain families of orthogonal polynomials in n variables with values in the N × N -matrices, where N = n+k k . The case k = 0 leads to the classical Heckman-Opdam polynomials of type A n with geometric parameter. For k = 1 we obtain the most complete results. In this case we give an explicit expression of the matrix weight, which we show to be irreducible whenever n ≥ 2. We also give explicit expressions of the spherical functions that determine the matrix weight for k = 1. These expressions are used to calculate the spherical functions that determine the matrix weight for general k up to invertible upper-triangular matrices. This generalizes and gives a new proof of a formula originally obtained by Koornwinder for the case n = 1. The commuting family of differential operators that have the matrix-valued polynomials as simultaneous eigenfunctions contains an element of order one. We give explicit formulas for differential operators of order one and two for (n, k) equal to (2, 1) and (3, 1). 4 5