The Weyl algebra as a fixed ring

TL;DR

This paper proves that the Weyl algebra over complex numbers cannot be realized as a fixed ring under a nontrivial finite group action, resolving a long-standing open problem and generalizing to differential operator rings on Galois coverings.

Contribution

It establishes that the Weyl algebra cannot be a fixed ring under finite group automorphisms and characterizes rings of invariants as differential operators on Galois coverings.

Findings

Weyl algebra cannot be a fixed ring of a nontrivial finite group action.

Invariant rings of differential operators correspond to Galois coverings.

Generalization to rings of differential operators on affine varieties.

Abstract

We prove that the Weyl algebra over $\mathbb{C}$ 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 $\mathbb{C}$, let $D(X)$ denote the ring of algebraic differential operators on $X,$ and let $\Gamma$ be a finite group. If $D(X)$ is isomorphic to the ring of $\Gamma$-invariants of a $\mathbb{C}$-domain $R$ on which $\Gamma$ acts faithfully by $\mathbb{C}$-algebra automorphisms, then $R$ is isomorphic to the ring of differential operators on a $\Gamma$-Galois covering of $X.$