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An explicit derived McKay correspondence for some complex reflection groups of rank two

Anirban Bhaduri, Yael Davidov, Eleonore Faber, Katrina Honigs, Peter McDonald, C. Eric Overton-Walker, Dylan Spence

Abstract

In this paper, we explore the derived McKay correspondence for several reflection groups, namely reflection groups of rank two generated by reflections of order two. We prove that for each of the reflection groups $G=G(2m,m,2)$, $G_{12}$, $G_{13}$, or $G_{22}$, there is a semiorthogonal decomposition of the following form, where $B_1,\ldots,B_r$ are the normalizations of the irreducible components of the branch divisor $\mathbb{C}^2\to \mathbb{C}^2/G$ and $E_1,\ldots,E_n$ are exceptional objects: $$D^G(\mathbb{C}^2)\cong \langle E_1,\ldots,E_n,D(B_1),\ldots, D(B_r), D(\mathbb{C}^2/G)\rangle.$$ We verify that the pieces of this decomposition correspond to the irreducible representations of $G$, verifying the Orbifold Semiorthogonal Decomposition Conjecture of Polishchuk and Van den Bergh. Due to work of Potter on the group $G(m,m,2)$, this conjecture is now proven for all finite groups $G\leq \mathrm{GL}(2,\mathbb{C})$ that are generated by order $2$ reflections. Each of these groups contains, as a subgroup of index $2$, a distinct finite group $H\leq \mathrm{SL}(2,\mathbb{C})$. A key part of our work is an explicit computation of the action of $G/H$ on the $H$-Hilbert scheme $\textrm{$H$-Hilb}(\mathbb{C}^2)$.

An explicit derived McKay correspondence for some complex reflection groups of rank two

Abstract

In this paper, we explore the derived McKay correspondence for several reflection groups, namely reflection groups of rank two generated by reflections of order two. We prove that for each of the reflection groups , , , or , there is a semiorthogonal decomposition of the following form, where are the normalizations of the irreducible components of the branch divisor and are exceptional objects: We verify that the pieces of this decomposition correspond to the irreducible representations of , verifying the Orbifold Semiorthogonal Decomposition Conjecture of Polishchuk and Van den Bergh. Due to work of Potter on the group , this conjecture is now proven for all finite groups that are generated by order reflections. Each of these groups contains, as a subgroup of index , a distinct finite group . A key part of our work is an explicit computation of the action of on the -Hilbert scheme H.

Paper Structure

This paper contains 52 sections, 14 theorems, 109 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Conventions and notation
  3. Historical Overview of the McKay correspondence
  4. Finite reflection subgroups of GL2C containing -1
  5. H-HilbC
  6. Structure of H-HilbC
  7. Exceptional locus of $Y$
  8. Outside of the exceptional locus of $Y$
  9. Structure of H-HilbC/A
  10. $G(2m,m,2)$
  11. $G_{12}$
  12. $G_{13}$
  13. $G_{22}$
  14. Smoothness of $Y/A$
  15. Semiorthogonal Decompositions of Equivariant Derived Categories
  16. ...and 37 more sections

Key Result

Lemma 4.4

Each irreducible factor of $p(u,v)$ corresponds to a component of the ramification locus of $\mathbb{C}^2/H\to \mathbb{C}^2/G$, which in turn corresponds to a component of the fixed locus of $A$ acting on $Y$ that has nontrivial intersection with $N$.

Figures (12)

  • Figure 1: $E_6$ and $E_6'$ Dynkin diagrams
  • Figure 2: Table of invariant rings, branch divisors, and number of components in branch divisors
  • Figure 3: $G(2m,m,2)$ cases
  • Figure 4: Exceptional locus in the $G_{12}$ case.
  • Figure 5: Exceptional locus in the $G_{13}$ case.
  • ...and 7 more figures

Theorems & Definitions (32)

  • Conjecture 1.1: PVdB19
  • Remark 3.1
  • Definition 4.1: Ito-Nakamura:1999
  • Remark 4.2
  • Lemma 4.4
  • proof
  • Remark 4.5
  • Proposition 4.6
  • proof
  • Corollary 4.7
  • ...and 22 more