On the singularity of some special components of Springer fibers

Details On the singularity of some special components of Springer fibers On the singularity of some special components of Springer fibers

2010-04-27

arxiv.org/abs/1004.4839v1

Mathematics

Algebraic Geometry

33 pages

Scientific paper

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.

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