Cherednik algebras and differential operators on quasi-invariants

We develop representation theory of the rational Cherednik algebra Hc associated to a finite Coxeter group W in a vector space (eta), and a parameter "c." We use it to show that, for integral values of "c," the algebra Hc is simple and Morita equivalent to D(eta)#W, the cross product of W with the algebra of polynomial differential operators on (eta).O. Chalykh, M. Feigin, and A. Veselov [CV1], [FV] introduced an algebra, Qc, of quasi-invariant polynomials on (eta), such that C (eta)^W(subset) Qc (subset) C(eta). We prove that the algebra D(Qc) of differential operators on quasi-invariants is a simple algebra, Morita equivalent to D(eta). The subalgebra D(Qc)^W (subset) D(Qc) of W-invariant operators turns out to be isomorphic to the spherical subalgebra eHce (subset) Hc. We show that D(Qc) is generated, as an algebra, by Qc and its "Fourier dual" Qc^b, and that Qc is a rank-one projective (Qc * Qc^b)-module (via multiplication-action on D(Qc) on opposite sides).