Extensions of the rational Cherednik algebra and generalized KZ functors

TL;DR

This paper generalizes the category equivalence between rational Cherednik algebra modules and Hecke algebra modules to include extensions involving normalizers of reflection subgroups and lattice extensions.

Contribution

It introduces two new generalizations of the existing category equivalence, expanding the framework to more complex algebraic structures.

Findings

Established equivalences for extended Hecke algebras

Generalized the category $ ext{O}$ correspondence

Extended the applicability of KZ functors

Abstract

Ginzburg, Guay, Opdam and Rouquier established an equivalence of categories between a quotient category of the category $\mathcal{O}$ for the rational Cherednik algebra and the category of finite dimension modules of the Hecke algebra of a complex reflection group $W$. We establish two generalizations of this result. On the one hand to the extension of the Hecke algebra associated to the normaliser of a reflection subgroup and on the other hand to the extension of the Hecke algebra by a lattice.