Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras

We define the i-restriction and i-induction functors on the category O of the cyclotomic rational double affine Hecke algebras. This yields a crystal on the set of isomorphism classes of simple modules, which is isomorphic to the crystal of a Fock space.Comment: 31 page

Canonical bases and affine Hecke algebras of type D

We prove a conjecture of Kashiwara and Miemietz on canonical bases and branching rules of affine Hecke algebras of type D. The proof is similar to the proof of the type B case.Comment: 24 page

Categorifications and cyclotomic rational double affine Hecke algebras

Varagnolo and Vasserot conjectured an equivalence between the category O 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 O of CRDAHA's.Comment: 85 page