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Singular components of Springer fibers in the two-column case. (English) Zbl 1191.14060
The author studies the singularity of the components of the Springer fiber $$\mathcal{B}_{u}$$ corresponding to a nilpotent endomorphism $$u$$ of order 2. Given a nilpotent endomorphism $$u$$ of a complex vector space, let $$\mathcal{B}_{u}$$ be the set of $$u$$ stable complete flags: it is a closed subvariety of the variety $$GL(V)/B$$ of complete flags, called the Springer fiber. Indeed, it is the fiber over $$u$$ of the Springer resolution of singularities of the cone of nilpotent endomorphisms of $$V$$.
Many problems on the geometry of $$\mathcal{B}_{u}$$ are still unsolved. In particular the problem to determine the singular components has been solved only in two cases. The geometry of $$\mathcal{B}_{u}$$ depends on the Jordan form of $$u$$. It is known that every components of $$\mathcal{B}_{u}$$ is non-singular when: i) at most one Jordan block of $$u$$ has size greater than one or ii) the canonical form of $$u$$ has only two blocks. The author gives necessary and sufficient conditions for the case where all the Jordan blocks have size at most 2.
In general, to verify that a component $$\mathcal{K}^{T}$$ of $$\mathcal{B}_{u}$$ is non-singular it is sufficient to check that $$\mathcal{K}^{T}$$ is non-singular at the points of $$(G/B)^{H}$$ contained in $$\mathcal{T}^{T}$$. Here $$H$$ is the maximal torus of $$GL(V)$$ composed by diagonal matrices in a fixed Jordan basis of $$u$$. Remark that $$\mathcal{B}_{u}$$ is not stabilized by $$H$$. When $$u$$ has order 2, the author proves that all the components of $$\mathcal{B}_{u}$$ contains an explict flag $$F_{\overline{T}}$$ and it sufficients to verify the nonsingularity of $$\mathcal{K}^{T}$$ in such a flag. The author describes the set of points of $$(G/B)^{H}$$ contained in $$\mathcal{K}^{T}$$. All this object have a combinatorial description in terms of tableaux of Yang. The authors defines a subset $$\chi(Y)$$ of $$\mathcal{B}_{u}\cap (G/B)^{H}$$ such that the tableaux associated to the elements of this subset are obtained from the tableau of $$F_{\overline{T}}$$ in a described way. His prove that $$\mathcal{K}^{T}$$ is nonsingular if and only if it contains the least possible number of element of $$\chi(Y)$$.

MSC:
 14M15 Grassmannians, Schubert varieties, flag manifolds 14B05 Singularities in algebraic geometry
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References:
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