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 ${\mathrm{Hilb}}^N({\mathbb{C}}^2∕\mathrm{\Gamma })$ on a simple singularity ${\mathbb{C}}^2∕\mathrm{\Gamma }$, where Γ is a finite subgroup of $\mathrm{SL}(2)$. We deduce it from the claim that quantum dimensions of standard modules for the quantum affine algebra associated with Γ at $\mathit{\zeta }=exp(\frac{2\pi \sqrt{-1}}{2(h^{\vee }+1)})$ are always 1, which is a special case of an earlier conjecture by Kuniba. Here $h^{\vee }$ is the dual Coxeter number. We also prove the claim, which was not known for $E_7$, $E_8$ before.