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

Abstract

We prove the conjecture by Gyenge, N\'emethi and Szendr\H{o}i in arXiv:1512.06844, arXiv:1512.06848 giving a formula of the generating function of Euler numbers of Hilbert schemes of points $\operatorname{Hilb}^n(\mathbb C^2/\Gamma)$ on a simple singularity $\mathbb C^2/\Gamma$, where $\Gamma$ 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 $\Gamma$ at $\zeta = \exp(\frac{2\pi i}{2(h^\vee+1)})$ are always $1$, which is a special case of a conjecture by Kuniba [Kun93]. Here $h^\vee$ is the dual Coxeter number. We also prove the claim, which was not known for $E_7$, $E_8$ before.