\(\mathrm{BC}_2\)-type multivariable matrix functions and matrix spherical functions (Q6618961)

The authors study matrix spherical functions for the compact symmetric pair \((G,K)=(\mathrm{SU}(m+2), \mathrm{S}(\mathrm{U}(2)\times \mathrm{U}(m))\).\NThey 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.