Proof of Varagnolo-Vasserot conjecture on cyclotomic categories O

Abstract

We prove an asymptotic version of a conjecture by Varagnolo and Vasserot on an equivalence between the category O for a cyclotomic Rational Cherednik algebra and a suitable truncation of an affine parabolic category O. We prove an asymptotic version of a conjecture by Varagnolo and Vasserot on an equivalence between the category O for a cyclotomic Rational Cherednik algebra and a suitable truncation of an affine parabolic category O that, in particular, implies Rouquier's conjecture on the decomposition numbers in the former. Our proof uses two ingredients: an extension of Rouquier's deformation approach as well as categorical actions on highest weight categories and related combinatorics. This text replaces arXiv:1207.1299.

50 pages. arXiv admin note: substantial text overlap with arXiv:1207.1299, v2 minor changes, v2 has a serious gap in 10.4, see Remark 1.2 in v3. v3 has a simpler proof, 33 pages, v4 accepted version, some changes