arXiv2026
Let $W$ be a finite reflection group acting on a finite-dimensional complex vector space $\h$, let $H_c(W,\h)$ be the associated rational Cherednik algebra, and let $\Dc = eH_ce$ be its spherical subalgebra. Building on the framework of BÞgvad and Källström for invariant differential operators, subsequently applied to symmetric groups, we show that, for generic parameter $c$, a Lie subalgebra $\fa\subset\Dc$ generated by basic invariants always generates $\Dc$ itself (via the Levasseur--Stafford theorem), and that, granted a mild compatibility hypothesis between the grading element and $\fa$, verified here in the rank-one case, the category of $\Dc$-submodules of the polynomial ring $B=\mathbb C[\h]$ is equivalent to the category of modules over a Cartan-type ring $R_\fa$ built from $\fa$. The ring $R_\fa$ surjects onto the more naively defined $ \displaystyle R_c = \Dc^0 \big/ \big(\Dc^0 \cap \Dc \Dc^-\big), $ and the two coincide whenever this surjection is injective, which we verify in the rank-one case; whether this holds in general, and in particular for the families treated below, is left open. As an application, we determine explicit generators of $\fa$ for the imprimitive complex reflection groups $G(m,1,n)$ ($e=1$), recovering the classical Coxeter group $B_n$ as a special case. For $D_n=G(2,2,n)$, where an extra generator obstruction arises for $e>1$, we resolve the generation problem for $n=3$ completely by exploiting the exceptional isomorphism $D_3 \cong A_3$, leaving the general case $n\geq4$ open.