\(\mathrm{BC}_2\)-type multivariable matrix functions and matrix spherical functions. (English) Zbl 07926394
The authors study matrix spherical functions for the compact symmetric pair \((G,K)=(\mathrm{SU}(m+2), \mathrm{S}(\mathrm{U}(2)\times \mathrm{U}(m))\).
The authors show that “instead of studying the more complicated matrix spherical functions, one can study the simpler leading terms of matrix spherical functions. The leading terms turn out to be homogeneous polynomials, and homogeneity considerations allow us to prove some results, e.g. on the indecomposability of the corresponding matrix weight and the explicit derivation of the second-order matrix partial differential equation.” The leading terms are related to some hypergeometric functions. The investigation of the leading terms is done explicitly using the action of the radial part of the Casimir operator on the considered functions and their leading terms. Some new orthogonal polynomials associated with the leading terms are investigated.
The authors show that “instead of studying the more complicated matrix spherical functions, one can study the simpler leading terms of matrix spherical functions. The leading terms turn out to be homogeneous polynomials, and homogeneity considerations allow us to prove some results, e.g. on the indecomposability of the corresponding matrix weight and the explicit derivation of the second-order matrix partial differential equation.” The leading terms are related to some hypergeometric functions. The investigation of the leading terms is done explicitly using the action of the radial part of the Casimir operator on the considered functions and their leading terms. Some new orthogonal polynomials associated with the leading terms are investigated.
Reviewer: Dmitry Artamonov (Moskva)
MSC:
| 33C80 | Connections of hypergeometric functions with groups and algebras, and related topics |
| 33C52 | Orthogonal polynomials and functions associated with root systems |
| 33C47 | Other special orthogonal polynomials and functions |
| 22E30 | Analysis on real and complex Lie groups |
| 15A16 | Matrix exponential and similar functions of matrices |
Keywords:
matrix spherical functions; Lie groups; multiplicity-free triples; matrix orthogonal polynomials; multivariable functionsReferences:
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