Let $H_{c}$, for simplicity, be the rational Cherednik algebra with parameter $t=1$, with triangular strucuture $\mathbb{C}[h] \otimes \mathbb{C} W \otimes \mathbb{C}[h^*]$, and $(W,h)$ the defining complex representation.
Consider the full subcategory category $O_c$ of finitely generated $H_c$-modules $M$ such that $h \subset \mathbb{C}[h^*]$ acts locally finitely on $M$. (cf. Gordon, section 5.3 in https://arxiv.org/pdf/0712.1568.pdf)
We have $O_c$=$\bigoplus_{\lambda \in h^*/W} \mathcal{O}_c(\lambda)$, where $\mathcal{O}_c(\lambda)$ is the full subcategory of $O_c$, consisting of modules $M$ where each $P \in \mathbb{C}[h^*]^W$ is such that $P-P(\lambda)$ acts locally nilpotently. $\mathcal{O}_c(0)$ is the usual category $\mathcal{O}_c$ considered by a number of people.
$\mathcal{O}_c$ is quite well understood: it is a highest weight category in the sense of Cline-Parshall-Scott; we have analogues of Verma modules of Lie theory, BGG reciprocity, KZ-functor, etc (cf. Guay Projective modules in the category O for the Cherednik algebra; Ginzburg et al. On the category O for rational Cherednik algebras).
Question What is the current understanding of the bigger category $O_c$?
Works that discuss this bigger category that I know of are the aforementioned work of Ginzburg et al., and also to a certain extent Ginzburg, On primitive ideals, and Bezrukavnikov and Etingof, Parabolic induction and restriction functors for rational Cherednik algebras.
References or any discussion you might have on this topic would be greatly appreciated.
I finish with:
Small question If $\mathcal{O}_c(0)$ is a semisimple abelian category, is also $\mathcal{O}_c(\lambda)$ a semisimple category for every $\lambda$ as well?
