On category $\mathcal{O}$ for the rational Cherednik algebra of $G(m,1,n)$: the almost semisimple case

Abstract

We determine the structure of category $\mathcal{O}$ for the rational Cherednik algebra of the wreath product complex reflection group $G(m,1,n)$ in the case where the $\mathsf{KZ}$ functor satisfies a condition called separating simples. As a consequence, we show that the property of having exactly $N-1$ simple modules, where $N$ is the number of simple modules of $G(m,1,n)$, determines the Ariki-Koike algebra up to isomorphism.