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

Eugene Gorsky, José Simental, Monica Vazirani

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

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 corresponding to the permutation representation of on the -equivariant homology of parabolic Hilbert schemes of points on the plane curve singularity for coprime and . 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 on the equivariant homology of parabolic Hilbert schemes on the non-reduced curve Our main tool is the study of the combinatorial representation theory of the rational Cherednik algebra via the subalgebra generated by Dunkl-Opdam elements.

Paper Structure

This paper contains 42 sections, 88 theorems, 191 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Hilbert schemes on singular curves
  3. Quantized Gieseker varieties
  4. Coulomb branches and generalized affine Springer fibers
  5. Rational Cherednik algebras
  6. Relation to other work
  7. Structure of the paper
  8. Affine permutations
  9. The extended affine symmetric group
  10. m-stable and m-restricted permutations
  11. Lexicographic ordering and combinatorics of integer sequences
  12. Rational Cherednik algebras
  13. Definition and basic properties
  14. An alternative presentation of Hc
  15. A Mackey formula for Ht,c
  16. ...and 27 more sections

Key Result

Theorem 1.1

There is a geometric action of the rational Cherednik algebra $H_{1,m/n}(\mathcal{S}_{n}, \mathbb{C}^n)$ on the localized $\mathbb{C}^*$-equivariant homology of $\mathrm{PHilb}^{x}(C)$. Moreover, with this action $H_{*}^{\mathbb{C}^{*}}(\mathrm{PHilb}^{x}(C))$ gets identified with the simple highest

Figures (4)

  • Figure 1: The partial order on $\mathbf{a} \in \mathcal{P}_k(3)$ for degrees $k \le 3$. So that one may compare $\prec$ to $>_{\mathtt lex}$ and to Bruhat order, above each $\mathbf{a}$ is the corresponding $\omega_{\mathbf{a}}$ both in window notation and it expression from Lemma \ref{['claim:coset']}.
  • Figure 2: An ideal on $C=\{x^4=y^3\}$ generated by $x^3y^2$ and $x^5y$. Note that $y\cdot x^3y^2=x^7$. The codimension of the ideal is $15=7+5+3 = c_1+c_2+c_3$ which is also the number of boxes under the staircase.
  • Figure 3: A flag of monomial ideals in $\mathrm{PHilb}_{15,15+3}(x^4=y^3)$: $I_{15}=\langle x^3y^2, x^5y \rangle, I_{16}=\langle x^4y^2,x^5y,x^7\rangle, I_{17}=\langle x^4y^2,x^5y\rangle.$ Here $y^{\alpha_1}x^{c_1}=x^3y^2,\ y^{\alpha_2}x^{c_2}=x^7,\ y^{\alpha_3}x^{c_3}=x^5y$.
  • Figure 4: An element of $\mathrm{CPHilb}^{6, y}(\{x^{3} = y^{4}\})$. Here, $J^{0} = J^{1} = \langle x^{2}y^{3}, xy^{5}\rangle$, $J^{2} = J^{3} = \langle x^{2}y^{3}, xy^{6}\rangle$ and $J^{4} = J^{5} = J^{6} = yJ^{0} = \langle x^{2}y^{4}, xy^{6}\rangle$. Also $\gamma = (0,1,0,2,0,0) \in \mathcal{C}_6(3)$ which corresponds to $2^1 4^2$. Note that the roles of $m$ and $n$, as well as those of $x$ and $y$ are different from those in Figures \ref{['fig: ideal']} and \ref{['fig: PHilb']}.

Theorems & Definitions (209)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Example 1.9
  • Example 1.10
  • ...and 199 more