Matrix Spherical Functions for (SU(𝑛 + 𝑚), S(U(𝑛) × U(𝑚))) : Two Specific Classes

Abstract

We consider the matrix spherical function related to the compact symmetric pair (G, K) = (SU(𝑛 + 𝑚), S(U(𝑛)×U(𝑚))). The irreducible 𝛫 representations (π, V) in the U(𝑛) part are considered, and the induced representation Indᴳ𝛫π splits multiplicity-free. In this case, the irreducible 𝛫 representations in the 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.