Derived McKay correspondence for real reflection groups of rank three

TL;DR

This work extends the derived McKay correspondence to real reflection groups of rank three by constructing a maximal (logarithmic) resolution of the pair $(X, \frac{1}{2}D)$, where $X=\mathbb{A}^3/G$ and $D$ is the discriminant. It proves an equivalence between the derived category of the associated DM stack $\mathcal{Y}$ and the $G$-equivariant derived category $D^G(\mathbb{A}^3)$, and derives explicit semiorthogonal decompositions into derived categories of affine spaces whose counts mirror the dimensions of fixed-point loci. The approach uses maximal $\mathbb{Q}$-factorial terminalizations, $G$-Hilbert schemes and their variants, iterated Hilbert schemes, and Yamagishi-type results to treat each real rank-3 case (dihedral, tetrahedral, octahedral, and icosahedral) case-by-case. This verifies Polishchuk–Van den Bergh’s conjecture for real rank-3 groups and provides concrete SODs tying conjugacy-class data to components, with potential implications for stringy/crepant geometry and representation theory of finite groups acting on threefolds.

Abstract

We describe the derived McKay correspondence for real reflection groups of rank $3$ in terms of a maximal resolution of the logarithmic pair consisting of the quotient variety and the discriminant divisor with coefficient $\frac{1}{2}$. As an application, we verify a conjecture by Polishchuk and Van den Bergh on the existence of a certain semiorthgonal decomposition of the equivariant derived category into the derived categories of affine spaces for any real reflection group of rank $3$.

Figures (3)

• Figure 4.1: Toric picture for $Y_H$
• Figure 4.2: Toric picture for $Y_K$
• Figure : Toric picture for $H\text{-Hilb}(\mathbb{A}^3)$

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1.3: PVdB
• Lemma 2.1
• proof
• Remark 2.2
• Proposition 2.3
• proof
• Remark 2.4
• Theorem 3.1: kawamata_toric_III Theorem1.4