On the singularity of some special components of Springer fibers

TL;DR

This paper investigates the singularity properties of special components of Springer fibers associated with nilpotent endomorphisms, characterizing singular Bala-Carter components and introducing two new families of components with distinct geometric structures.

Contribution

It characterizes singular Bala-Carter components via forbidden configurations and introduces two new families of components with duality properties and specific geometric features.

Findings

Singular Bala-Carter components are characterized by minimal forbidden configurations.

Two new families of components are introduced, one with dense orbits and another as iterated fiber bundles.

The new families extend the understanding of the structure of Springer fiber components.

Abstract

Let $u\in\mathrm{End}(\mathbb{C}^n)$ be nilpotent. The variety of $u$-stable complete flags is called the Springer fiber over $u$. Its irreducible components are parameterized by a set of standard Young tableaux. The Richardson (resp. Bala-Carter) components of Springer fibers correspond to the Richardson (resp. Bala-Carter) elements of the symmetric group, through Robinson-Schensted correspondence. Every Richardson component is isomorphic to a product of standard flag varieties. On the contrary, the Bala-Carter components are very susceptible to be singular. First, we characterize the singular Bala-Carter components in terms of two minimal forbidden configurations. Next, we introduce two new families of components, wider than the families of Bala-Carter components and Richardson components, and both in duality via the tableau transposition. The components in the first family are characterized by the fact that they have a dense orbit of special type under the action of the stabilizer of $u$, whereas all components in the second family are iterated fiber bundles over projective spaces.