Conference abstracts
Session C7 - Special Functions and Orthogonal Polynomials
July 17, 15:00 ~ 15:30 - Room B5
Matrix-valued orthogonal polynomials in several variables related to $\mathrm{SU}(n+1)\times \mathrm{SU}(n+1)$
Pablo Román
Universidad Nacional de Córdoba, Argentina - roman@famaf.unc.edu.ar
We study matrix-valued spherical functions for the symmetric pair $G = \mathrm{SU}(n+1)\times \mathrm{SU}(n+1)$ and $K = \mathrm{SU}(n+1)$ 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.
Joint work with Erik Koelink (Radboud Universiteit) and Maarten van Pruijssen (Universität Paderborn).