Noncommutative Noether’s problem vs classic Noether’s problem

Abstract

We prove that if two complex affine irreducible varieties are birational (that is their coordinate rings have isomorphic fields of fractions) then their rings of differential operators are birationally equivalent. It allows to address the Noncommutative Noether’s Problem on the invariants of Weyl fields for linear actions of finite groups. In fact, we show for any field $${\mathsf {k}}$$ of characteristic 0 that rationality of the quotient variety $${\mathbb {A}}^n({\mathsf {k}})/G$$ implies that the Noncommutative Noether’s Problem is positively solved for G. In particular, this gives affirmative answer for all pseudo-reflections groups, for the alternating groups ( $$n=3, 4, 5$$ ) and for any finite group when $$n=3$$ and $${\mathsf {k}}$$ is algebraically closed (covering and generalizing all previously known cases). Alternative proofs are given for the complex field and for all pseudo-reflections groups. In the later case an effective algorithm of finding the Weyl generators is described.