The Algebra of Differential Operators for a Matrix Weight: An Ultraspherical Example

Abstract

Abstract. In this paper we study in detail algebraic properties of the algebra D(W ) of differential operators 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 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 flat and therefore it is not projective. This is the second detailed study of an algebra D(W ) and the first one coming from spherical functions and group representations. We prove that the algebras for different Gegenbauer 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 the algebra D(W ) as the centralizer of its center in the Weyl algebra. We do believe that this should hold for any irreducible weight and the case considered in this paper represents a good step in this direction.

Matrix Gegenbauer Polynomials: The $$2\times 2$$ 2 × 2 Fundamental Cases

“…We refer the reader to [13] for basic definitions and main results concerning the algebra D(W ). The present paper leads to understand completely a second and more promising example of D(W ) in a forthcoming paper, [28]. There are very few examples of non-commutative algebras that arise in a natural setup at the intersection of harmonic analysis and algebras.…”

Elementary examples of solutions to Bochner’s problem for matrix differential operators

“…However then η m+1 = 0 and therefore by the same argument η (m+1)/2 = 0. Since m > 1 we have that and Pablo Roman for pointing out several errors in an earlier version of this paper, and for suggesting the application of the theory developed therein to the 2 × 2 Gegenbauer weight matrix studied by Ignacio Zurrián in [30] [31]. The author would like to thank F. Alberto Grünbaum for his encouragement and remarks on an earlier draft.…”

“…More recent papers have explored the structure of the algebra D(w) of all matrix differential operators for which the p(x, n) are eigenfunctions. In particular, [28] [3] [18] provide examples of generators and relations for D(w) for various values of w. Even more examples are [12][17] [22][30] [31]. These examples demonstrate that the structure of D(w) can be nuanced and interesting, unlike in the scalar case.…”

“…For N = 2, S 2 = p, and T 2 = r − p, this weight matrix appears in [30] [31], where explicit generators and relations of the associated algebra D( w) are computed, though the effort involved is again substantial. In this example, we verify the calculation of the algebra D( w) calculated in the paper, using the framework established above.…”

Reducibility of matrix weights

“…Given a matrix weight W and a sequence of matrix orthogonal polynomials {Q n } n∈N0 , it is natural to consider the matrix differential operators D such that Q n is an eigenfunction of D for every n ∈ N 0 . The set of these operators is a noncommutative * -algebra D(W ) (see [5,18,29,32]). In Section 3 we study the structure of the algebra D(W ) for a reducible weight, see Proposition 3.3.…”