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.