Euler characteristics of Hilbert schemes of points on surfaces with simple singularities

TL;DR

This paper explores the generating series of Euler characteristics of Hilbert schemes on surfaces with simple singularities, proposing a Lie-theoretic formula, proving it for type D, and discussing implications for modularity and representation theory.

Contribution

It introduces a conjectural formula for the generating series on singular surfaces, proves it for type D singularities, and links the series to affine Lie algebra representations.

Findings

Conjectured formula for generating series in terms of Lie data

Proof of the conjecture for type D singularities

Demonstration of modularity for surfaces with simple singularities

Abstract

This is an announcement of conjectures and results concerning the generating series of Euler characteristics of Hilbert schemes of points on surfaces with simple (Kleinian) singularities. For a quotient surface C^2/G with G a finite subgroup of SL(2, C), we conjecture a formula for this generating series in terms of Lie-theoretic data, which is compatible with existing results for type A singularities. We announce a proof of our conjecture for singularities of type D. The generating series in our conjecture can be seen as a specialized character of the basic representation of the corresponding (extended) affine Lie algebra; we discuss possible representation-theoretic consequences of this fact. Our results, respectively conjectures, imply the modularity of the generating function for surfaces with type A and type D, respectively arbitrary, simple singularities, confirming predictions of S-duality.