The algebra of differential operators for a Gegenbauer weight matrix

Abstract

In this work we study in detail the algebra of differential operators D(W) associated with a Gegenbauer matrix weight. We prove that two second order operators generate the algebra, indeed D(W) is isomorphic to the free algebra generated by two elements subject to certain relations. Also, the center is isomorphic to the affine algebra of a singular rational curve. The algebra D(W) is a finitely-generated torsion-free module over its center, but it is not at and therefore neither projective. After [Tir11], this is the second detailed study of an algebra D(W) and the first one coming from spherical functions and group representation theory.