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

We prove a recent conjecture by Gyenge, Nemethi, and Szendroi giving a formula of the generating function of Euler numbers of Hilbert schemes of points Hilb(N) (C-2/Gamma) on a simple singularity C-2/Gamma, where F is a finite subgroup of 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(2 pi root-1/2(h(boolean OR)+1)) are always 1, which is a special case of an earlier conjecture by Kuniba. Here h(boolean OR) is the dual Coxeter number. We also prove the claim, which was not known for E-7, E-8 before.