  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 $X.$
