  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 $(3,1)$.
