The noncommutative Noether's problem is almost equivalent to the classical Noether's problem (Q2071636)

For any given domain \(A\), we denote by \(\mathrm{Frac}(A)\) the skew field of fractions of \(A\). In the paper under review, the author concentrates on a noncommutative analogue of the classical Noether's problem formulated in [\textit{J. Alev} and \textit{F. Dumas}, in: Studies in Lie theory. Dedicated to A.\ Joseph on his sixtieth birthday. Basel: Birkhauser. 21--50 (2006; Zbl 1102.16015)]: Let \(G\subseteq\mathrm{GL}_n(\mathbb{C})\) be a finite subgroup. Assuming \(G\) acts on the \(n\)-th Weyl algebra \(A_n(\mathbb{C})\), is it true that \(\mathrm{Frac}(A_n(\mathbb{C}))^G\cong\mathrm{Frac}(A_n(\mathbb{C}))\)? The author shows that counterexamples to the classical Noether's problem constructed by \textit{D. J. Saltman} in [Invent. Math. 77, 71--84 (1984; Zbl 0546.14014)] also gives counterexamples to its noncommutative analogue. Namely, let \(G\subseteq\mathrm{GL}_n(\mathbb{Z})\) be a finite subgroup. If \(\mathrm{Frac}(A_n(\mathbb{C}))^G\cong\mathrm{Frac}(A_n(\mathbb{C}))\), then for every sufficiently large prime \(p\) the field \(\overline{\mathbb{F}_p}(x_1,\ldots,x_n)^G\) is stably rational. In particular, if \(G\) is any group constructed by Saltman, the viewing \(A_n(\mathbb{C})\) as the ring of differential operators on \(\mathbb{C}[x(g)\mid g\in G]\) with the usual action of \(G\), we have that \(\mathrm{Frac}(A_n(\mathbb{C}))^G\not\cong\mathrm{Frac}(A_n(\mathbb{C}))\).