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In Part 1 we study the spherical functions on compact symmetric pairs ofarbitrary rank under a suitable multiplicity freeness assumption and additionalconditions on the branching rules. The spherical functions are taking values inthe spaces of linear operators of a finite dimensional representation of thesubgroup, so the spherical functions are matrix-valued. Under these assumptionsthese functions can be described in terms of matrix-valued orthogonalpolynomials in several variables, where the number of variables is the rank ofthe compact symmetric pair. Moreover, these polynomials are uniquely determinedas simultaneous eigenfunctions of a commutative algebra of differentialoperators.  In Part 2 we verify that the group case $\mathrm{SU}(n+1)$ meets all theconditions that we impose in Part 1. For any $k\in\mathbb{N}_{0}$ we obtainfamilies of orthogonal polynomials in $n$ variables with values in the $N\timesN$-matrices, where $N=\binom{n+k}{k}$. The case $k=0$ leads to the classicalHeckman-Opdam polynomials of type $A_{n}$ with geometric parameter. For $k=1$we obtain the most complete results. In this case we give an explicitexpression of the matrix weight, which we show to be irreducible whenever$n\ge2$. We also give explicit expressions of the spherical functions thatdetermine the matrix weight for $k=1$. These expressions are used to calculatethe spherical functions that determine the matrix weight for general $k$ up toinvertible upper-triangular matrices. This generalizes and gives a new proof ofa formula originally obtained by Koornwinder for the case $n=1$. The commutingfamily of differential operators that have the matrix-valued polynomials assimultaneous eigenfunctions contains an element of order one. We give explicitformulas for differential operators of order one and two for $(n,k)$ equal to$(2,1)$ and $(3,1)$.

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