The Weyl algebra as a fixed ring

Abstract

We prove that the Weyl algebra over C cannot be a fixed ring of any domain under a nontrivial action of a finite group by algebra automorphisms, thus settling a 30-year old problem. In fact, we prove the following much more general result. Let X be a smooth affine variety over C, let D(X) denote the ring of algebraic differential operators on X, and let Γ be a finite group. If D(X) is isomorphic to the ring of Γ-invariants of a C-domain R on which Γ acts faithfully by C-algebra automorphisms, then R is isomorphic to the ring of differential operators on a Γ-Galois covering of X.