On category $\mathcal{O}$ for cyclotomic rational Cherednik algebras

Abstract

We study equivalences for category $\mathcal{O}_p$ of the rational Cherednik algebras ${\bf H}_p$ of type $G_{\ell}(n) = (\mu_{\ell})^n\rtimes \mathfrak{S}_n$: a highest weight equivalence between $\mathcal{O}_p$ and $\mathcal{O}_{\sigma(p)}$ for $\sigma\in \mathfrak{S}_{\ell}$ and an action of $\mathfrak{S}_{\ell}$ on an explicit non-empty Zariski open set of parameters $p$; a derived equivalence between $\mathcal{O}_p$ and $\mathcal{O}_{p'}$ whenever $p$ and $p'$ have integral difference; a highest weight equivalence between $\mathcal{O}_p$ and a parabolic category $\mathcal{O}$ 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.