Journal of Lie Theory 21 (2011), No. 1, 205--242
Copyright Heldermann Verlag 2011
On the Singularity of some Special Components of Springer Fibers
Department of Mathematics, Weizmann Institute of Science, 76100 Rehovot, Israel
Let V be an n-dimensional C-vector space and let u from V to V be a nilpotent endomorphism. 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 (respectively, 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. By contrast, 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.
Keywords: Springer fibers, Richardson components, Bala-Carter components, singularity criteria, iterated bundles.
MSC: 14M15; 05E10, 20G05
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