Euler numbers of Hilbert schemes of points on simple surface singularities and quantum dimensions of standard modules of quantum affine algebras

Abstract

We prove a recent conjecture by Gyenge, Némethi, and Szendrői giving a formula of the generating function of Euler numbers of Hilbert schemes of points HilbN(C2∕Γ) on a simple singularity C2∕Γ, where Γ is a finite subgroup of SL(2). We deduce it from the claim that quantum dimensions of standard modules for the quantum affine algebra associated with Γ at ζ=exp(2π −1 2(h∨+1)) are always 1, which is a special case of an earlier conjecture by Kuniba. Here h∨ is the dual Coxeter number. We also prove the claim, which was not known for E7, E8 before.