Matrix-valued orthogonal polynomials of several variables

M. VAN PRUIJSSEN; KOELINK, ERIK; PABLO ROMÁN

Oaxaca

Conferencia; Workshop: Special Functions, Orthogonal Polynomials and Applications, PRIMA third congress; 2017

We introduce matrix-valued orthogonal polynomials of several variables by studying matrix-valued spherical functions for higher rank symmetric spaces. For the pair G=SU(n+1)×SU(n+1) and K=SU(n+1) diagonally embedded, we can describe explicitly the properties of the family of matrix-valued polynomials. These polynomials are orthogonal with respect to a matrix weight which is given explicitly. We prove that the weight is irreducible, i.e. it does not have non-trivial invariants subspaces and we obtain two commuting matrix-valued differential operators having the matrix-valued orthogonal polynomials as eigenfunctions. Remarkably one of these differential operators is of order one. In the case n=2, we obtain matrix-valued analogues of the orthogonal polynomials on the interior of Steiner´s hypocycloid, introduced by T. Koornwinder in 1974. This is a joint work with E. Koelink (Radboud University) and M. van Pruijssen (University of Paderborn).