Proof of Varagnolo-Vasserot conjecture on cyclotomic categories O

TL;DR

This paper proves an asymptotic version of a conjecture linking cyclotomic Rational Cherednik algebra category O to affine parabolic category O, confirming Rouquier's conjecture on decomposition numbers.

Contribution

It establishes the conjectured equivalence between categories O for cyclotomic Rational Cherednik algebras and affine parabolic categories, extending Rouquier's deformation approach.

Findings

Proves the asymptotic equivalence of categories O

Confirms Rouquier's conjecture on decomposition numbers

Uses categorical actions and combinatorics in the proof

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.