We prove a recent conjecture by Gyenge, Némethi, and Szendrői giving a formula of the generating function of Euler numbers of Hilbert schemes of points on a simple singularity , where Γ is a finite subgroup of . We deduce it from the claim that quantum dimensions of standard modules for the quantum affine algebra associated with Γ at are always 1, which is a special case of an earlier conjecture by Kuniba. Here is the dual Coxeter number. We also prove the claim, which was not known for , before.
To Fukaya san on the occasion of his 60th birthday
"Euler numbers of Hilbert schemes of points on simple surface singularities and quantum dimensions of standard modules of quantum affine algebras." Kyoto J. Math. 61 (2) 377 - 397, June 2021. https://doi.org/10.1215/21562261-2021-0006