In this paper, we study the relationship between the cocenter and the representation theory of affine Hecke algebras. The approach is based on the interaction between the rigid cocenter, an important subspace of the cocenter, and the dual object in representation theory, the rigid quotient of the Grothendieck group of finite dimensional representations.
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.
In this paper we study representation theory of the category FI^m introduced by Gadish which is a product of copies of the category FI, and show that quite a few interesting representational and homological properties of FI can be generalized to FI^m in a natural way. In particular, we prove the representation stability property of finitely generated FI^m -modules over fields of characteristic 0.