Category O for a rational Cherednik algebra

Fix $W,\mathfrak{h},c$ as in The rational Cherednik algebra H_{t,c}(W,h) and consider $H_{1,c}(W,\mathfrak{h})$. By The PBW theorem for rational Cherednik algebras, the subalgebra generated by ${\mathfrak{h}}^{\ast }$ is the polynomial algebra $\mathbb{C}[{\mathfrak{h}}^{\ast }]=S({\mathfrak{h}}^{\ast })$, and the subalgebra generated by $\mathfrak{h}$ is $S(\mathfrak{h})$.

The category O, denoted ${\mathcal{O}}_c(W,\mathfrak{h})$, is the full subcategory of left $H_{1,c}(W,\mathfrak{h})$-modules $M$ satisfying:

1. $M$ is finitely generated as an $S({\mathfrak{h}}^{\ast })$-module.
2. The augmentation ideal $S(\mathfrak{h}{)}_{>0}$ acts locally nilpotently: for every $m\in M$, there is $N\ge 1$ such that every homogeneous polynomial in $S(\mathfrak{h})$ of degree at least $N$ kills $m$.

Since $S(\mathfrak{h})$ is generated by a finite basis of $\mathfrak{h}$ and its elements commute, the second condition is equivalent to each $y\in \mathfrak{h}$ acting locally nilpotently. Morphisms in ${\mathcal{O}}_c(W,\mathfrak{h})$ are the $H_{1,c}(W,\mathfrak{h})$-module homomorphisms.

The category depends on $c$ and uses $t=1$. No semisimplicity of $H_{1,c}$, $W$-module decompositions beyond finite-dimensional $W$-representations, or regularity of KZ monodromy is built into this definition.