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

Abstract

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 ${\mathbb{C}}^{\ast }$-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.

63 pages; This version fixes a couple of mistakes in Propositions 7.19 and 8.9 from the published version, concerning the use of dual lattices. The article title in v3 has also changed

References in corpus (2)

• Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal{N}=4$ gauge theories, II
• Representations with minimal support for quantized Gieseker varieties