Singular components of Springer fibers in the two-column case.

*(English)*Zbl 1191.14060The 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 \(\chi(Y)\).

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 \(\chi(Y)\).

Reviewer: Alessandro Ruzzi (Roma)

##### MSC:

14M15 | Grassmannians, Schubert varieties, flag manifolds |

14B05 | Singularities in algebraic geometry |

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\textit{L. Fresse}, Ann. Inst. Fourier 59, No. 6, 2429--2444 (2009; Zbl 1191.14060)

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