We prove a recent conjecture of Khovanov-Lauda concerning the categorification of one-half of the quantum group associated with a simply laced Cartan datum.
We prove the decomposition conjecture for the Schur algebra stated in [LT]. We also give a new approach to the Lusztig conjecture via canonical bases of the Hall algebra.
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.
Abstract. We give a proof of the parabolic/singular Koszul duality for the category O of affine Kac-Moody algebras. The main new tool is a relation between moment graphs and finite codimensional affine Schubert varieties. We apply this duality to q-Schur algebras and to cyclotomic rational double affine Hecke algebras. This yields a proof of a conjecture of Chuang-Miyachi relating the level-rank duality with the Ringel-Koszul duality of cyclotomic rational double affine Hecke algebras.
The goal of the present paper is to prove with simple algebraic methods a
Schur duality between Cherednik's double affine Hecke algebra of type GL(l) and
the toroidal quantum group of type SL(n+1) introduced by V. Ginzburg, M.
Kapranov and E. Vasserot in their "Langlands duality on algebraic surfaces".Comment: Plain TeX, 14 page
We classify finite dimensional simple spherical representations of rational double affine Hecke algebras, and we study a remarkable family of finite dimensional simple spherical representations of double affine Hecke algebras.
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