From representations of the rational Cherednik algebra to parabolic Hilbert schemes via the Dunkl-Opdam subalgebra

In this note we explicitly construct an action of the rational Cherednik algebra $H_{1,m\mathrm{/}n}(S_n,{\mathbb{C}}^n)$ corresponding to the permutation representation of $S_n$ on the ${\mathbb{C}}^{\ast }$-equivariant homology of parabolic Hilbert schemes of points on the plane curve singularity ${x^{m} = y^{n}}$forcoprime$m$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 $\{y^n=0\}\mathrm{.}$ Our main tool is the study of the combinatorial representation theory of the rational Cherednik algebra via the subalgebra generated by Dunkl-Opdam elements.