2021/10/05 by Erik Koelink, Jie Liu, Koelink, Erik +1
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.2110.02287
openalex publication_date 2021/10/05 · openalex created_date 2021/10/11 · openalex updated_date 2026/07/28
Matrix spherical functions associated to the compact symmetric pair (SU(m+2), S(U(2)× U(m)), having reduced root system of type BC2, are studied. We consider an irreducible K-representation (π,V) arising from the U(2)-part of K, and the induced representation IndKG π splits multiplicity free. The corresponding spherical functions, i.e. Φ\colon G → End(V) satisfying Φ(k1gk2)=π(k1)Φ(g)π(k2) for all g∈ G, k1,k2∈ K, are studied by studying certain leading coefficients which involve hypergeometric functions. This is done explicitly using the action of the radial part of the Casimir operator on these functions and their leading coefficients. To suitably grouped matrix spherical functions we associate two-variable matrix orthogonal polynomials giving a matrix analogue of Koornwinder's 1970s two-variable orthogonal polynomials, which are Heckman-Opdam polynomials for BC2. In particular, we find explicit orthogonality relations and the matrix polynomials being eigenfunctions to an explicit second order matrix partial differential operator. The scalar part of the matrix weight is less general than Koornwinder's weight.