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

TL;DR

This paper studies matrix spherical functions associated with a specific symmetric pair, providing explicit formulas and orthogonality relations for certain irreducible representations, advancing understanding of harmonic analysis on these structures.

Contribution

The paper explicitly determines matrix spherical functions for specific irreducible representations of the symmetric pair $( ext{SU}(n+m), ext{S}( ext{U}(n) imes ext{U}(m)))$, including their orthogonality relations.

Findings

Explicit formulas for spherical functions derived

Orthogonality relations established for these functions

Approximation methods for matrix-valued functions discussed

Abstract

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.