Diagram automorphisms of quiver varieties (Q462287)

Given a finite quiver \(Q\) and two dimension vectors \(\mathbf{v}\) and \(\mathbf{w}\) one can construct a quiver variety \(\mathfrak{M}\left( \mathbf{v,w}\right) .\) To any any admissible automorphism \(a\) of \(Q\) one can also construct a diagram automorphism \(\theta\) of \(\mathfrak{M}\left( \mathbf{v,w}\right) \) and consider the subvariety of fixed points \(\mathfrak{M}\left( \mathbf{v,w}\right) ^{\theta}.\) The main result in this work is a description of this subvariety, showing that, in the case where the quiver is type \(A\), the subvariety \(\mathfrak{M}\left( \mathbf{v,w}\right) ^{\theta}\) is a disjoint union of quiver varieties for quivers of type \(D\). Moreover, if we take a quiver of type \(A_{2n-1}\) and pick \(\mathbf{v}\) and \(\mathbf{w}\) to be symmetric under the Dynkin diagram involution, then \(\mathfrak{M}\left( \mathbf{v,w}\right) ^{\theta}\) is isomorphic to a resolution of a closed subvariety of a certain Slodowy slice. This allows for a description of the Springer resolution of the corresponding Slodowy slices as a quiver variety of type \(D.\)