Matrix-valued Orthogonal Polynomials Related to Quantum Groups

PABLO ROMÁN

Congreso; First Joint Meeting Brazil-Spain in Mathematics RSME-SBM-SBMAC-UFC; 2015

Recently, matrix-valued orthogonal polynomials have driven much attention. 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 to families of matrix-valued orthogonal polynomials that can be considered as analogues of Gegenbauer and Jacobi polynomials. In this work we investigate an analogue construction for quantum groups. We introduce matrix-valued orthogonal on the quantum analogue of the pair $(G,K)=(operatorname{SU}(2)imesoperatorname{SU}(2),operatorname{diag})$.The spherical functions are eigenfunctions of the Casimir operators on the quantum group and this leadsto two $q$-difference operators that have the polynomials $P_n$ as eigenfunctions. We construct explicitly a positive definite weight matrix $W$ such that$$langle P_n,P_m angle = int_{-1}^1 P_n(x)W(x)(P_m(x))^* dx =delta_{n,m} H_n, qquad n,minmathbb{N}_0,$$where $H_n$ is a constant diagonal matrix. We calculate the $LDU$ decomposition of the weight and we show that the matrix entries of $L$ are given in terms of continuous $q$-Ultraspherical polynomials.By conjugating the $q$-difference operators with $L^t$, it is possible to find an expressionfor the the matrix entries of $P_n$ in terms of $q$-Racah and continuous $q$-Ultraspherical polynomials.We will also discuss how to extend this construction to other quantum groups. In particular we will study in the detail the case of the quantized universal enveloping algebra of $sl_3$.