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

Abstract

We study equivalences for category ${\mathcal{O}}_p$ of the rational Cherednik algebras ${\mathbf{H}}_p$ of type $G_{\ell }(n)=({\mu }_{\ell }{)}^n⋊{\mathfrak{S}}_n$: 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 ${\mathcal{O}}_{p^{\prime }}$ whenever $p$ and $p^{\prime }$ 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.