Categorifications and cyclotomic rational double affine Hecke algebras

Varagnolo and Vasserot conjectured an equivalence between the category for CRDAHA's and a subcategory of an affine parabolic category O of type A. We prove this conjecture. As applications, we prove a conjecture of Rouquier on the dimension of simple modules of CRDAHA's and a conjecture of Chuang-Miyachi on the Koszul duality for the category of CRDAHA's.