Noncommutative Noether's problem is almost equivalent to the classical Noether's problem

Abstract

Motivated by the classical Noether's problem, J. Alev and F. Dumas proposed the following question, commonly referred to as the noncommutative Noether's problem: Let a finite group $G$ act linearly on $\mathbb{C}^n,$ inducing the action on $\text{Frac}(A_n(\mathbb{C}))$-the skew field of fractions of the $n$-th Weyl algebra $A_n(\mathbb{C}),$ then is $\text{Frac}(A_n(\mathbb{C}))^G$ isomorphic to $\text{Frac}(A_n(\mathbb{C}))?$ In this note we show that if $\text{Frac}(A_n(\mathbb{C}))^{G}\cong \text{Frac}(A_n(\mathbb{C})),$ then for any algebraically closed field $k$ of large enough characteristic, field $k(x_1,\cdots, x_n)^G$ is stably rational. This result allows us to produce counterexamples to the noncommutative Noether's problem based on well-known counterexamples to the Noether's problem for algebraically closed fields.