Singular components of the Springer fiber in the two-columns case

TL;DR

This paper characterizes the singularity of components of Springer fibers associated with nilpotent endomorphisms of order 2, providing a criterion based on fixed points under a torus action.

Contribution

It offers a new description of fixed points in Springer fiber components and establishes a complete criterion for their singularity in the two-columns case.

Findings

Describes fixed points of Springer fiber components under torus action.

Provides a necessary and sufficient condition for component singularity.

Advances understanding of Springer fibers in the two-columns case.

Abstract

We consider the Springer fiber ${\mathcal B}_u$ corresponding to a nilpotent endomorphism $u$ of nilpotent order 2. As a first result, we give a description of the elements of a given component of ${\mathcal B}_u$ which are fixed by the action of the standard torus relative to some Jordan basis of $u$. By using this result, we establish a necessary and sufficient condition of singularity for the components of ${\mathcal B}_u$.