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

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 C inducing the action on Frac(An(C))-the skew field of fractions of the n-th Weyl algebra An(C), is Frac(An(C)) G isomorphic to Frac(An(C))? In this note we show that if Frac(An(C) G ∼= Frac(An(C)) then for any algebraically closed field k of large enough characterisitic, field k(x1, · · · , xn) G is stably rational. This result allows us to produce counterexamples to the noncommutativeNoether’s problem based on well-known counterexamples to the Noether’s problem for algebraically closed fields. Let F be a field. Let a finite group G act on variables x1, · · · , xn by permutations. Then the classical Noether’s problem asks whether the fixed field F (x1, · · · , xn) G is a purely transcendental extension of F, or equivalently if F (x1, · · · , xn) G ∼= F (x1, · · · , xn). The first instances when the Noether’s problem (over Q) has a negative answer was demonstrated by R. Swan [Sw]. Subsequently Saltman constructed counterexamples over algebraically closed fields [S]. More specifically, Saltman constructed infinite family of p-groups G, such that for any algebraically closed field of characteristic not dividing |G|, the fixed field F (x(g), g ∈ G) is not stably rational: no purely transcendental extension of F (x(g), g ∈ G) is purely transcendental over F. In fact to the best of our knowledge all known counterexamples to Noether’s problem have the property that the fixed field is not stably rational. Motivated by the Noether’s problem, Alev and Dumas [AD] considered its noncommutative analogue, which is now commonly referred to as the noncommutative Noether’s problem. Throughout given a noetherian domain A, by Frac(A) we denote its skew field of fractions. Problem 0.1 (Noncommutative Noether’s problem). Let G ⊂ GLn(C) be a finite subgroup, then G acts on the n-th Weyl algebra An(C), hence acting on its sckew field of fractions Frac(An(C)). Is Frac(An(C)) G isomorphic to Frac(An(C))? There is a clear analogy between the noncommutative Noetherian problem and the Gelfand-Kirillov conjecture about skew fields of enveloping algebras. Noncommutative Noether’s problem has been of interest to ring theorists for some