Duncan, JohnCase Western Reserve UniversityEtingof, PavelMITIvan Chi-Ho IpUniversity of TokyoKhovanov, MikhailColumbia UniversityLibine, MatveiIndiana UniversityLicata, AnthonyAustralian National UniversitySavage, AlistairUniversity of OttawaSchlosser, MichaelUniversity of Vienna
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.
In this paper, we propose a conjectural multiplicity formula for general spherical varieties. For all the cases where a multiplicity
formula has been proved, including Whittaker model, Gan-Gross-Prasad model, Ginzburg-Rallis model, Galois model and Shalika model, we show that the multiplicity formula in our conjecture matches the multiplicity formula that has been proved. We also give a proof of this multiplicity formula in two new cases.