For a specific case of a symmetric pair $(G,K)$ corresponding to the group case $SU(n+1)$, we discussmatrix-valued spherical functions as a generalization of the spherical functions, which arise as bi-K-invariant functions.
The main input is an irreducible representation of $K$, such that the induced representation splits multiplicity free. To such a fixed representation one can associate matrix valued spherical functions on $G$. The restriction to the corresponding abelian subgroup gives a column in the matrix-valued orthogonal polynomial upon choosing suitable coordinates depending on $n$ variables.
The polynomials satisfy orthogonality relations, and can also be described in terms of eigenfunctions of matrix-valued (partial) differential operators.
The case $n=1$ gives matrix valued analogs of Chebyshev polynomials. The case $n=2$ gives matrix valued $2$-variable orthogonal polynomials on Steiner’s hypocycloid, generalizing the scalar case by Koornwinder (1974).

Joint work with Maarten van Pruijssen (U. Paderborn, Germany) and Pablo Rom\’an (UN Córdoba).