  This paper is based on a talk at the conference `The McKay correspondence,
mutation and related topics' from July 2020. We provide an introduction to
joint work of the author with S{\o}ren Gammelgaard, \'{A}d\'{a}m Gyenge and
Bal\'{a}zs Szendr\H{o}i that constructs the reduced scheme underlying the
Hilbert scheme of $n$ points on an ADE singularity as a Nakajima quiver variety
for a particular stability parameter. After drawing a parallel with two
well-known constructions of the Hilbert scheme of $n$ points in $\mathbb{A}^2$,
we summarise results of the author and Gwyn Bellamy before describing the main
result by cornering a noncommutative algebra obtained from the preprojective
algebra of the framed McKay graph.
