On category O for cyclotomic rational Cherednik algebras

We study equivalences for category Op of the rational Cherednik algebras Hp of type G_l(n)=(μ_l)n⋊Sn: a highest weight equivalence between O(p) and O(σ(p)) for σ∈S_l and an action of S_l on an explicit non-empty Zariski open set of parameters p; a derived equivalence between O(p) and O(p′) whenever p and p′ have integral difference; a highest weight equivalence between O(p) and a parabolic category O for the general linear group, under a non-rationality assumption on the parameter p. As a consequence, we confirm special cases of conjectures of Etingof and of Rouquier.