We consider the matrix spherical function related to the compact symmetric pair $(G,K)=(\mathrm{ SU}(n+m),\mathrm{ S}(\mathrm{ U}(n)\times\mathrm{ U}(m)))$. The irreducible $K$ representations $(\pi,V)$ in the ${ \rm U}(n)$ part are considered and the induced representation $\mathrm{ Ind}_K^G\pi$ splits multiplicity free. In this case, the irreducible $K$ representations in the ${ \rm U}(n)$ 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.
Categories: math.RT