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Symmetry, Integrability and Geometry: Methods and Applications, 2023, Volume 19, 055, 33 pp.
DOI: https://doi.org/10.3842/SIGMA.2023.055
(Mi sigma1950)
 

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
References:
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.
Keywords: representation theory, Lie group, special functions.
Received: October 18, 2022; in final form July 13, 2023; Published online August 4, 2023
Document Type: Article
MSC: 17B10, 22E46, 33C50
Language: English
Citation: Jie Liu, “Matrix Spherical Functions for $(\mathrm{SU}(n+m),\mathrm{S}(\mathrm{U}(n)\times\mathrm{U}(m)))$: Two Specific Classes”, SIGMA, 19 (2023), 055, 33 pp.
Citation in format AMSBIB
\Bibitem{Liu23}
\by Jie~Liu
\paper Matrix Spherical Functions for $(\mathrm{SU}(n+m),\mathrm{S}(\mathrm{U}(n)\times\mathrm{U}(m)))$: Two Specific Classes
\jour SIGMA
\yr 2023
\vol 19
\papernumber 055
\totalpages 33
\mathnet{http://mi.mathnet.ru/sigma1950}
\crossref{https://doi.org/10.3842/SIGMA.2023.055}
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