Matrix-valued orthogonal polynomials related to compact Gelfand pairs and quantum groups

PABLO ROMÁN

Oaxaca

Workshop; Orthogonal an multiple orthogonal polynomials; 2015

Banff International Research Station

In this talk, we will be concerned with matrix-valued orthogonal polynomials. In order to derive new examples and study their properties,a suitable group theoretic interpretation has shown to be fruitful. This has been carried out for Gelfand pairs of rank one, leading tofamilies of matrix-valued orthogonal polynomials that can be considered as analogues of Gegenbauer and Jacobi polynomials. We will briefly discuss the construction of these families.We will focus on the analogue construction for quantum groups, in particular for the quantum analogue of the pair $(G,K)=(\operatorname{SU}(2)\times\operatorname{SU}(2),\operatorname{diag})$. We will introduce matrix-valued orthogonal polynomials in one variable $\{P_n\}_{n\geq0}$ by studying the spherical functions of any type on the quantum group and show that they are orthogonal with respect to a positive definite weight matrix $W$. We will derive $q$-difference operators that have the polynomials as eigenfunctions from the study of certain central elements in the quantum group.We will also discuss how to extend this construction to other quantum groups. In particular we are interested in thecase of the quantized universal enveloping algebra of $sl_3$.