The fixed point locus of the smooth Jordan quiver variety under the action of the finite subgroups of $\mathrm{SL}_2(\mathbb{C})$

TL;DR

This paper investigates the structure of fixed point loci in smooth Nakajima quiver varieties related to the Jordan quiver, connecting geometric components with affine Lie algebra root lattices and resolutions of wreath product singularities.

Contribution

It describes the irreducible components of fixed point loci using McKay quivers and provides a combinatorial model linking these components to affine Lie algebra root lattices.

Findings

Irreducible components are described via McKay quivers.

A combinatorial model indexes the components using root lattices.

Existence of components is linked to resolutions of wreath product singularities.

Abstract

In this article, we study the decomposition into irreducible components of the fixed point locus under the action of $\Gamma$ a finite subgroup of $\mathrm{SL}_2(\mathbb{C})$ of the smooth Nakajima quiver variety of the Jordan quiver. The quiver variety associated with the Jordan quiver is either isomorphic to the punctual Hilbert scheme in $\mathbb{C}^2$ or to the Calogero-Moser space. We describe the irreducible components using quiver varieties of McKay's quiver associated with the finite subgroup $\Gamma$ and we have given a general combinatorial model of the indexing set of these irreducible components in terms of certain elements of the root lattice of the affine Lie algebra associated with $\Gamma$. Finally, we prove that for every projective, symplectic resolution of a wreath product singularity, there exists an irreducible component of the fixed point locus of the punctual Hilbert scheme in the plane that is isomorphic to the resolution.