  We develop representation theory of the rational Cherednik algebra H
associated to a finite Coxeter group W in a vector space h. It is applied to
show that, for integral values of parameter `c', the algebra H is simple and
Morita equivalent to D(h)#W, the cross product of W with the algebra of
polynomial differential operators on h.
  We further study an algebra Q of quasi-invariant polynomials on h introduced
by Chalykh, Feigin, and Veselov [CV], [FV], such that C[h]^W \subset Q \subset
C[h]. We prove that the algebra D(Q) of differential operators on
quasi-invariants is a simple algebra, Morita equivalent to D(h). The subalgebra
D(Q)^W of W-invariant operators turns out to be isomorphic to the spherical
subalgebra eHe \subset H. We also show that D(Q) is generated, as an algebra,
by Q and its `Fourier dual Q*, and that D(Q) is a rank one projective
(Q-Q*)-module (via multiplication-action on D(Q) on opposite sides).
