Generic singularities of nilpotent orbit closures

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• 作者：Baohua Fu^a ; ^{bhfu@math.ac.cn" class="auth_mail" title="E-mail the corresponding author} ; Daniel Juteau^b ; ^{daniel.juteau@unicaen.fr" class="auth_mail" title="E-mail the corresponding author} ; Paul Levy^c ; ^{p.d.levy@lancaster.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; Eric Sommers^d ; ^{esommers@math.umass.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Nilpotent orbits ; Symplectic singularities ; Slodowy slice
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：10 January 2017
• 年：2017
• 卷：305
• 期：Complete
• 页码：1-77
• 全文大小：1268 K

文摘

According to a theorem of Brieskorn and Slodowy, the intersection of the nilpotent cone of a simple Lie algebra with a transverse slice to the subregular nilpotent orbit is a simple surface singularity. At the opposite extremity of the poset of nilpotent orbits, the closure of the minimal nilpotent orbit is also an isolated symplectic singularity, called a minimal singularity. For classical Lie algebras, Kraft and Procesi showed that these two types of singularities suffice to describe all generic singularities of nilpotent orbit closures: specifically, any such singularity is either a simple surface singularity, a minimal singularity, or a union of two simple surface singularities of type A_2k−1$A_{2k-1}$. In the present paper, we complete the picture by determining the generic singularities of all nilpotent orbit closures in exceptional Lie algebras (up to normalization in a few cases). We summarize the results in some graphs at the end of the paper.

In most cases, we also obtain simple surface singularities or minimal singularities, though often with more complicated branching than occurs in the classical types. There are, however, six singularities that do not occur in the classical types. Three of these are unibranch non-normal singularities: an SL₂(C)${\mathrm{SL}}_2(\mathbb{C})$-variety whose normalization is 131b1d4952f5e789ccf64" title="Click to view the MathML source">A²${\mathbb{A}}^2$, an Sp₄(C)${\mathrm{Sp}}_4(\mathbb{C})$-variety whose normalization is A⁴${\mathbb{A}}^4$, and a two-dimensional variety whose normalization is the simple surface singularity A₃$A_3$. In addition, there are three 4-dimensional isolated singularities each appearing once. We also study an intrinsic symmetry action on the singularities, extending Slodowy's work for the singularity of the nilpotent cone at a point in the subregular orbit.