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 ${\mathrm{Hilb}}^n({\mathbb{C}}^2\mathrm{/}\mathrm{\Gamma })$ on a simple singularity ${\mathbb{C}}^2\mathrm{/}\mathrm{\Gamma }$, where $\mathrm{\Gamma }$ is a finite subgroup of $\mathrm{S}\mathrm{L}(2)$. We deduce it from the claim that quantum dimensions of standard modules for the quantum affine algebra associated with $\mathrm{\Gamma }$ 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.