Categorifications and cyclotomic rational double affine Hecke algebras

TL;DR

This paper proves a conjectured equivalence between categories related to cyclotomic rational double affine Hecke algebras and affine parabolic category O, leading to new results on module dimensions and Koszul duality.

Contribution

It establishes the conjectured equivalence, confirming predictions about the structure and dualities of categories associated with CRDAHA.

Findings

Proved the equivalence between category O for CRDAHA and affine parabolic category O.

Confirmed Rouquier's conjecture on simple module dimensions.

Validated Chuang-Miyachi's conjecture on Koszul duality for CRDAHA category O.

Abstract

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.