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).

View abstract PDF



FoCM 2017, based on a nodethirtythree design.