NONCOMMUTATIVE NOETHER'S PROBLEM FOR COMPLEX REFLECTION GROUPS

We solve some noncommutative analogue of the Noether's problem for the reflection groups by showing that the skew field of fractions of the invariant subalgebra of the Weyl algebra under the action of any finite complex reflection group is a Weyl field, that is, isomorphic to the skew field of fractions of some Weyl algebra. We also extend this result to the invariants of the ring of differential operators on any finite dimensional torus. The results are applied to obtain analogs of the Gelfand-Kirillov conjecture for Cherednik algebras and Galois algebras. (AU)