Algebras of invariant differential operators

Abstract

We prove that an invariant subalgebra A_n^W of the Weyl algebra A_n is a Galois order over an adequate commutative subalgebra \Gamma when W is a two-parameters irreducible unitary reflection group G(m,1,n), m\geq 1, n\geq 1, including the Weyl group of type B_n, or alternating group, or the product of n copies of a cyclic group of fixed finite order. Earlier this was established for the symmetric group by the authors. In each of the cases above, except for the alternating groups, we show that A_n^W is free as a right (left) \Gamma-module. Similar results are established for the algebra of W-invariant differential operators on the n-dimensional torus where W is a symmetric group S_n or orthogonal group of type B_n or D_n. As an application of our technique we prove the quantum Gelfand-Kirillov conjecture for U_q(sl_2), the first Witten deformation and the Woronowicz deformation.