Automorphisms and ideals of the Weyl algebra

Abstract

Let $A_1$ be the (first) Weyl algebra, and let G be its automorphism group. We study the natural action of G on the space of isomorphism classes of right ideals of $A_1$ (equivalently, of finitely generated rank 1 torsion-free right $A_1$ -modules). We show that this space breaks up into a countable number of orbits each of which is a finite dimensional algebraic variety. Our results are strikingly similar to those for the commutative algebra of polynomials in two variables; however, we do not know of any general principle that would allow us to predict this in advance. As a key step in the proof, we obtain a new description of the bispectral involution of [W1]. We also make some comments on the group G from the viewpoint of Shafarevich's theory of infinite dimensional algebraic groups.