Matrix Elements of Irreducible Representations of SU(n+1) x 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 $k \in \mathbb{N}_0$ we obtain families of orthogonal polynomials in n variables with values in the $N \times N$-matrices, where $N = \binom{n+k}{k}$

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 $(3, 1)$.