THE MATRIX BOCHNER PROBLEM

A long standing question in the theory of orthogonal matrix polynomials is the matrix Bochner problem, the classification of N x N weight matrices W(x) whose associated orthogonal polynomials are eigenfunctions of a second order differential operator. Based on techniques from noncommutative algebra (semiprime PI algebras of Gelfand-Kirillov dimension one), we construct a framework for the systematic study of the structure of the algebra D(W) of matrix differential operators for which the orthogonal polynomials of the weight matrix W(x) are eigenfunctions. The ingredients for this algebraic setting are derived from the analytic properties of the orthogonal matrix polynomials. We use the representation theory of the algebras D(W) to resolve the matrix Bochner problem under the two natural assumptions that the sum of the sizes of the matrix algebras in the central localization of D(W) equals N (fullness of D(W)) and the leading coefficient of the second order differential operator multiplied by the weight W(x) is positive definite. In the case of 2x2 weights, it is proved that fullness is satisfied as long as D(W) is noncommutative. The two conditions are natural in that without them the problem is equivalent to much more general ones by artificially increasing the size of the matrix W(x).