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.

We prove that an invariant subalgebra A W n of the Weyl algebra An is a Galois order over an adequate commutative subalgebra Γ when W is a two-parameters irreducible unitary reflection group G(m, 1, n), m ≥ 1, n ≥ 1, including the Weyl group of type Bn, or alternating group An, or the product of n copies of a cyclic group of fixed finite order. Earlier this was established for the symmetric group in [15]. In each of the cases above, except for the alternating groups, we show that A W n is free as a right (left) Γ-module. Similar results are established for the algebra of W -invariant differential operators on the n-dimensional torus where W is a symmetric group Sn or orthogonal group of type Bn or Dn. As an application of our technique we prove the quantum Gelfand-Kirillov conjecture for Uq(sl 2 ), the first Witten deformation and the Woronowicz deformation.

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 A n (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 and k is algebraically closed. Alternative proofs are given for the complex field and for all pseudoreflections 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.

We define a class of quantum linear Galois algebras which include the universal enveloping algebra Uq(gln), the quantum Heisenberg Lie algebra and other quantum orthogonal Gelfand-Zetlin algebras of type A, the subalgebras of G-invariants of the quantum affine space, quantum torus for G = G(m, p, n), and of the quantum Weyl algebra for G = Sn. We show that all quantum linear Galois algebras satisfy the quantum Gelfand-Kirillov conjecture. Moreover, it is shown that the the subalgebras of invariants of the quantum affine space and of quantum torus for the reflection groups and of the quantum Weyl algebra for symmetric groups are, in fact, Galois orders over an adequate commutative subalgebras and free as right (left) modules over these subalgebras. In the rank 1 cases the results hold for an arbitrary finite group of automorphisms when the field is C.

This survey discusses the classical Noether's problem and presents some important cases where it has a positive solution. With the Weyl algebras taking place of the polynomial algebras, some aspects of noncommutative invariant theory are introduced: in particular, the noncommutative version of the Noether's problem. Some known cases are discussed and some new results are presented. As an application one obtains a new proof of the Gelfand-Kirillov conjecture for gl n . There is a striking similarity between the cases with positive solution of the Noether's problem for both commutative and noncommutative versions, and this leads to many conjectures.