Matrix elements of irreducible representations of $\mathrm{SU}(n+1)\times\mathrm{SU}(n+1)$ and multivariable matrix-valued orthogonal polynomials

TL;DR

This paper develops a framework for matrix-valued orthogonal polynomials associated with compact symmetric pairs, providing explicit formulas and extending classical results to multivariable and higher-rank cases.

Contribution

It introduces a new approach to matrix-valued orthogonal polynomials on symmetric pairs, explicitly constructs these polynomials for $ ext{SU}(n+1)$, and generalizes known formulas to higher dimensions.

Findings

Explicit matrix weight functions for $k=1$ case.

Generalization of Koornwinder's formula to higher dimensions.

Construction of differential operators with these polynomials as eigenfunctions.

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 $\mathrm{SU}(n+1)$ 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 $n\ge2$. 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 $(3,1)$.