The Algebra of Differential Operators for a Gegenbauer Weight Matrix

IGNACIO N. ZURRIÁN

INTERNATIONAL MATHEMATICS RESEARCH NOTICES

OXFORD UNIV PRESS

Lugar: Oxford; Año: 2017

1073-7928

In this article we study in detail algebraic properties of the algebra D(W ) of differentialoperators associated to a matrix weight of Gegenbauer type. We prove that two second-order operators generate the algebra, indeed D(W ) is isomorphic to the free algebragenerated by two elements subject to certain relations. Also, the center is isomorphic tothe affine algebra of a singular rational curve. The algebra D(W ) is a finitely generatedtorsion-free module over its center, but it is not flat and therefore it is not projective.This is the second detailed study of an algebra D(W ) and the first one coming fromspherical functions and group representations. We prove that the algebras for differentGegenbauer weights and the algebras studied previously, related to Hermite weights,are isomorphic to each other. We give some general results that allow us to regard thealgebra D(W ) as the centralizer of its center in the Weyl algebra. We do believe that thisshould hold for any irreducible weight and the case considered in this article representsa good step in this direction.