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We introduce parabolic induction and restriction functors for rationalCherednik algebras, and study their basic properties. Then we discussapplications of these functors to representation theory of rational Cherednikalgebras. In particular, we prove the Gordon-Stafford theorem about Moritaequivalence of the rational Cherednik algebra for type A and its sphericalsubalgebra, without the assumption that c is not a half-integer, which wasrequired up to now. Also, we classify representations from category O over therational Cherednik algebras of type A which do not contain an S_n-invariantvector, and confirm a conjecture of Okounkov and the first author on the numberof such representations. In the second version we have added a result on thesimplicity of the spherical Cherednik algebra of type A for -1<c<0, and astrengthened version of the main result of arXiv:math/0312474, as well as anappendix by the second author containing arXiv:0706.4308, on the reducibilityof the polynomial representation of the trigonometric Cherednik algebra.

Parabolic induction and restriction functors for rational Cherednik algebras