$q$-Schur algebras and complex reflection groups

We show that the category $\mathbb O$ for a rational Cherednik algebra of type $A$ is equivalent to modules over a $q$-Schur algebra (parameter $\notin\frac12+\mathbb Z$), providing thus character formulas for simple modules. We give some generalization to $B_n(d)$. We prove an “abstract” translation principle. These results follow from the unicity of certain highest weight categories covering Hecke algebras. We also provide a semi-simplicity criterion for Hecke algebras of complex reflection groups and show the isomorphism type of Hecke algebras is invariant under field automorphisms acting on parameters.