q-Schur algebras for complex reflection groups

In a joint work with Ginzburg, Guay and Opdam, we show that a category O of modules over the rational Cherednik algebra of a complex reflection group is the module category over a quasi-hereditary algebra which can be realized as the endomorphism ring of a Hecke algebra module. I will also discuss the conjecture that this algebra coincides with the usual q-Schur algebra in type A.