We prove that certain parabolic Kazhdan-Lusztig polynomials calculate the graded decomposition matrices of v-Schur algebras given by the Jantzen ﬁltration of Weyl modules, conﬁrming a conjecture of Leclerc and Thibon.
We categorify a coideal subalgebra of the quantum group of S[2r+ 1 by introducing
a 2-category analogous to the one defined by Khovanov一Lauda-Rouquier， and show
that self-dual indecomposable l-morphisms categorify the canonical basis of this algebra.
This allows us to define a categorical action of this coideal algebra on the categories of
modules over cohomology rings of partial flag varieties and on the category ð of type
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.
We prove a conjecture of Miemietz and Kashiwara 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 in Varagnolo and Vasserot
(in press) .
We give a proof of the parabolic/singular Koszul duality for the category O of affine Kac–Moodyalgebras. 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 Heckealgebras.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.