Standard modules lie in rational Cherednik category O

Since $E$ is finite dimensional, a finite basis of $E$ generates $\mathbb{C}[{\mathfrak{h}}^{\ast }]\otimes E$ as a $\mathbb{C}[{\mathfrak{h}}^{\ast }]$-module. By [F2] this is ${\mathrm{\Delta }}_c(E)$, so the finite-generation condition in [F1] holds.

Give $\mathbb{C}[{\mathfrak{h}}^{\ast }]\otimes E$ its polynomial-degree filtration. For $y\in \mathfrak{h}$, commute $y$ past one factor $x\in {\mathfrak{h}}^{\ast }$ by $yx=xy+[y,x]$. The first term preserves the number of polynomial factors after applying $y$ to the remaining factors; the commutator term replaces that $x$ by an element of $\mathbb{C}W$, which preserves degree and acts on the remaining polynomial. Since $y$ kills $E$, induction on polynomial degree shows that $y$ maps degree at most $d$ into degree at most $d-1$.

A product of $N$ elements of $\mathfrak{h}$ therefore maps every polynomial of degree at most $d$ to zero when $N>d$. Each element of ${\mathrm{\Delta }}_c(E)$ is a finite sum of such polynomial-degree vectors, so a sufficiently high-degree element of the augmentation ideal $S(\mathfrak{h}{)}_{>0}$ kills it. This is the local-nilpotence condition in [F1]. Both conditions hold, hence ${\mathrm{\Delta }}_c(E)\in {\mathcal{O}}_c(W,\mathfrak{h})$.