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

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.