Cherednik algebras and differential operators on quasi-invariants

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