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We prove the conjecture by Gyenge, N\'emethi and Szendr\H{o}i inarXiv:1512.06844, arXiv:1512.06848 giving a formula of the generating functionof Euler numbers of Hilbert schemes of points $\operatorname{Hilb}^n(\mathbbC^2/\Gamma)$ on a simple singularity $\mathbb C^2/\Gamma$, where $\Gamma$ is afinite subgroup of $\mathrm{SL}(2)$. We deduce it from the claim that quantumdimensions 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 isa special case of a conjecture by Kuniba [Kun93]. Here $h^\vee$ is the dualCoxeter number. We also prove the claim, which was not known for $E_7$, $E_8$before.

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