Sheaves on ALE spaces and quiver varieties

We identify a quiver variety of an affine type with a framed moduli space of torsion free sheaves on an ALE space, a fiber of a simultaneous resolution of the semi-universal deformation of $\mathbb C^2/\Gamma$. This result is an analog of a similar identification for a framed moduli space of anti-self-dual connections on an ALE space, given by Kronheimer and the author. It simultaneously generalizes the description on $\mathbb C^2$ given in Chapter 2 of my “Lectures on Hilbert schemes of points on surfaces” to arbitrary ALE space, and also the description of the Hilbert schemes of points on the ALE space by Kuznetsov to arbitrary torsion free sheaves.