Equivariant Hirzebruch classes and Molien series of quotient singularities

TL;DR

This paper explores the relationship between Hirzebruch classes of quotient singularities and Molien series of finite groups, revealing a McKay correspondence-like connection and providing explicit examples.

Contribution

It establishes that the Hirzebruch class of quotient singularities equals the Molien series of the group with variable substitution, linking geometric and algebraic invariants.

Findings

Hirzebruch class coincides with Molien series under certain substitutions

The class of crepant resolutions relates to Molien series of centralizers

Examples include 4-dimensional symplectic quotient singularities

Abstract

We study properties of the Hirzebruch class of quotient singularities $\mathbb{C}^n/G$, where $G$ is a finite matrix group. The main result states that the Hirzebruch class coincides with the Molien series of $G$ under suitable substitution of variables. The Hirzebruch class of a crepant resolution can be described specializing the orbifold elliptic genus constructed by Borisov and Libgober. It is equal to the combination of Molien series of centralizers of elements of $G$. This is an incarnation of the McKay correspondence. The results are illustrated with several examples, in particular of 4-dimensional symplectic quotient singularities.