Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 055, 33 pages      arXiv:2210.03041      https://doi.org/10.3842/SIGMA.2023.055

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

Jie Liu
Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands

Received October 18, 2022, in final form July 13, 2023; Published online August 04, 2023

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.

Key words: representation theory; Lie group; special functions.

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