Diagram automorphisms of quiver varieties

TL;DR

This paper investigates how diagram automorphisms affect Nakajima quiver varieties, revealing that fixed points form unions of varieties related to split-quotient quivers, with connections to classical involutions and Slodowy varieties.

Contribution

It demonstrates that fixed-point subvarieties under diagram automorphisms are unions of quiver varieties for split-quotient quivers, linking type D quiver varieties to Slodowy varieties of classical Lie algebras.

Findings

Fixed points form unions of split-quotient quiver varieties.

Type D quiver varieties relate to Slodowy varieties for classical Lie algebras.

Diagram involutions correspond to classical involutions in specific cases.

Abstract

We show that the fixed-point subvariety of a Nakajima quiver variety under a diagram automorphism is a disconnected union of quiver varieties for the `split-quotient quiver' introduced by Reiten and Riedtmann. As a special case, quiver varieties of type D arise as the connected components of fixed-point subvarieties of diagram involutions of quiver varieties of type A. In the case where the quiver varieties of type A correspond to small self-dual representations, we show that the diagram involutions coincide with classical involutions of two-row Slodowy varieties. It follows that certain quiver varieties of type D are isomorphic to Slodowy varieties for orthogonal or symplectic Lie algebras.