The McKay Correspondence for Dihedral Groups: The Moduli Space and the Tautological Bundles

TL;DR

This paper proves a conjecture relating resolutions of quotient singularities to moduli spaces of G-constellations specifically for dihedral groups, extending the McKay correspondence in this context.

Contribution

It confirms the conjecture for dihedral groups, establishing a link between resolutions and moduli spaces, and extends the McKay correspondence for complex reflection groups.

Findings

Affirms the conjecture for dihedral groups.

Establishes isomorphism between certain resolutions and moduli spaces.

Extends McKay correspondence to dihedral groups.

Abstract

A conjecture in [Ish20] states that for a finite subgroup $G$ of $GL(2; \mathbb{C})$, a resolution $Y$ of $\mathbb{C}^2/G$ is isomorphic to a moduli space $\mathcal{M}_{\theta}$ of $G$-constellations for some generic stability parameter $\theta$ if and only if $Y$ is dominated by the maximal resolution. This paper affirms the conjecture in the case of dihedral groups as a class of complex reflection groups, and offers an extension of McKay correspondence (via [IN1], [IN2], and [Ish02]). To appear in Hiroshima Mathematical Journal.