Noncommutative Noether's Problem vs Classical Noether's Problem

Abstract: We address the Noncommutative Noether's Problem on the invariants of Weyl fields for linear actions of finite groups. We prove that if the variety An(k)/G is rational then the Noncommutative Noether's Problem is positively solved for G and any field k of characteristic zero. In particular, this gives positive solution for all pseudo-reflections groups, for the alternating groups (n = 3,4,5) and for any finite group when n = 3. 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. We also study birational equivalence for the rings of invariant differential operators on complex affine irreducible varieties.