Matrix Spherical Functions for $(\mathrm{S}\mathrm{U}(n+m),\mathrm{S}(\mathrm{U}(n)\times \mathrm{U}(m)))$: Two Specific Classes

We consider the matrix spherical function related to the compact symmetric pair $(G,K)=(\mathrm{S}\mathrm{U}(n+m),\mathrm{S}(\mathrm{U}(n)\times \mathrm{U}(m)))$. The irreducible $K$ representations $(\pi ,V)$ in the $\mathrm{U}(n)$ part are considered and the induced representation ${\mathrm{I}\mathrm{n}\mathrm{d}}_K^G\pi$ splits multiplicity free. In this case, the irreducible $K$ representations in the $\mathrm{U}(n)$ part are studied. The corresponding spherical functions can be approximated in terms of the simpler matrix-valued functions. We can determine the explicit spherical functions using the action of a differential operator. We consider several cases of irreducible $K$ representations and the orthogonality relations are also described.