Mad Subalgebras of Rings of Differential Operators on Curves

Abstract

We study the maximal abelian ad-nilpotent (mad) subalgebras of the domains D Morita equivalent to the first Weyl algebra. We give a complete description both of the individual mad subalgebras and of the space of all such. A surprising consequence is that this last space is independent of D. Our results generalize some classic theorems of Dixmier about the Weyl algebra.

Revised (and slightly extended) version, 25 pages; to appear in Advances in Math. (2007)