Matrix-valued orthogonal polynomials in several variables related to SU(n + 1) × SU(n + 1)

PABLO ROMÁN

Barcelona

Workshop; workshop Special Functions and Orthogonal Polynomials in the conference on the Foundations of Computational Mathematics (FoCM 2017); 2017

We study matrix-valued spherical functions for the symmetric pair G=SU(n+1)×SU(n+1) and K=SU(n+1) diagonally embedded. Under certain assumptions these functions give rise to a family of matrix-valued polynomials in several variables. These polynomials are orthogonal with respect to a matrix weight which is described explicitly and is irreducible, i.e. it does not have non-trivial invariants subspaces. From the group theoretic interpretation, 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 polynomials in two-variables. The weight matrix in this case is supported on the interior of the Steiner hypocycloid.