Matrix elements of irreducible representations of SU(n + 1)×SU(n + 1) and multivariable matrix-valued orthogonal polynomials

Abstract

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∈N0 we obtain families of orthogonal polynomials in n variables with values in the N×N-matrices, where N=(n+kk). The case k=0 leads to the classical Heckman-Opdam polynomials of type An 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).