A study of irreducible components of Springer fibers using quiver varieties

TL;DR

This paper provides an explicit method to compute the image of complete flags in Slodowy varieties under the Maffei--Nakajima isomorphism, enabling a detailed description of Springer fiber components via quiver representations.

Contribution

It constructs solutions to the isomorphism system explicitly, facilitating the analysis of Springer fibers through quiver varieties.

Findings

Explicit computation method for the Maffei--Nakajima isomorphism.

Description of Springer fiber components using kernel relations.

Enhanced understanding of the geometry of Slodowy and Springer varieties.

Abstract

It is a remarkable theorem by Maffei--Nakajima that the Slodowy variety, which is a subvariety of the resolution of the nilpotent cone, can be realized as a Nakajima quiver variety of type A. However, the isomorphism is rather implicit as it takes to solve a system of equations in which the variables are linear maps. In this paper, we construct solutions to this system under certain assumptions. This establishes an explicit and efficient way to compute the image of a complete flag contained in the Slodowy variety under the Maffei--Nakajima isomorphism and describe these flags in terms of quiver representations. As Slodowy varieties contain Springer fibers naturally, we can use these results to provide an explicit description of the irreducible components of two-row Springer fibers in terms of a family of kernel relations via quiver representations.