Noncommutative Noether's problem for complex reflection groups

TL;DR

This paper proves that the skew field of invariants under complex reflection groups is isomorphic to a Weyl field, extending to differential operators on tori, with applications to Cherednik and Galois algebras.

Contribution

It establishes the noncommutative Noether's problem for complex reflection groups and extends the result to differential operators on tori, providing new insights into invariants of Weyl algebras.

Findings

Skew field of invariants is a Weyl field for any reflection group.

Extension of results to invariants of differential operators on tori.

Applications to Gelfand-Kirillov Conjecture analogs for Cherednik and Galois algebras.

Abstract

We solve the noncommutative Noether's problem for the reflection groups by showing that the skew field of the invariants of the Weyl algebra under the action of any reection group is a Weyl field, that is isomorphic to a skew field of some Weyl algebra over a transcendental extension of the ground field. We also extend this result to the invariants of the ring of differential operators on any dimensional torus.The results are applied to obtain analogs of the Gelfand-Kirillov Conjecture for Cherednik algebras and Galois algebras.