On category O for cyclotomic rational Cherednik algebras

TL;DR

This paper explores equivalences between different categories O of rational Cherednik algebras of type G_l(n), establishing various highest weight and derived equivalences under specific parameter conditions, and confirming related conjectures.

Contribution

It introduces new highest weight and derived equivalences for category O of cyclotomic rational Cherednik algebras, linking them to parabolic categories and confirming conjectures.

Findings

Established a highest weight equivalence between O_p and O_{\sigma(p)} for permutations \sigma.

Proved a derived equivalence between O_p and O_{p'} for parameters with integral differences.

Confirmed special cases of conjectures by Etingof and Rouquier.

Abstract

We study equivalences for category O_p of the rational Cherednik algebras H_p of type G_l(n) = \mu_l^n\rtimes S_n: a highest weight equivalence between O_p and O_{\sigma(p)} for \sigma\in S_l and an action of S_l on a non-empty Zariski open set of parameters p; a derived equivalence between O_p and O_{p'} whenever p and p' have integral difference; a highest weight equivalence between O_p and a parabolic category O for the general linear group, under a non-rationality assumption on the parameter p. As a consequence, we confirm special cases of conjectures of Etingof and of Rouquier.