  In this note we explicitly construct an action of the rational Cherednik
algebra $H_{1,m/n}(S_n,\mathbb{C}^n)$ corresponding to the permutation
representation of $S_n$ on the $\mathbb{C}^{*}$-equivariant homology of
parabolic Hilbert schemes of points on the plane curve singularity $\{x^{m} =
y^{n}\}$ for coprime $m$ and $n$. We use this to construct actions of quantized
Gieseker algebras on parabolic Hilbert schemes on the same plane curve
singularity, and actions of the Cherednik algebra at $t = 0$ on the equivariant
homology of parabolic Hilbert schemes on the non-reduced curve $\{y^{n} = 0\}.$
Our main tool is the study of the combinatorial representation theory of the
rational Cherednik algebra via the subalgebra generated by Dunkl-Opdam
elements.
