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,H-m/n (S-n, C-n) corresponding to the permutation representation of S-n on the 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