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Generic singularities of nilpotent orbit closures
Fu, Baohua1,2; Juteau, Daniel3; Levy, Paul4; Sommers, Eric5
AbstractAccording 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). 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)-variety whose normalization is A(2), an Sp(4)(C)-vari-ety whose normalization is A(4), and a two-dimensional variety whose normalization is the simple surface singularity 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. (C) 2016 The Author(s). Published by Elsevier Inc.
KeywordNilpotent orbits Symplectic singularities Slodowy slice
Funding ProjectNSFC[11225106] ; NSFC[11321101] ; KIAS Scholar Program ; Engineering and Physical Sciences Research Council[EP/K022997/1] ; NSA[H98230-11-1-0173] ; National Science Foundation Independent Research and Development plan ; CNRS ; AMSS of Chinese Academy of Sciences
WOS Research AreaMathematics
WOS SubjectMathematics
WOS IDWOS:000406169200001
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Document Type期刊论文
Corresponding AuthorSommers, Eric
Affiliation1.Chinese Acad Sci, Hua Loo Keng Key Lab Math, 55 ZhongGuanCun East Rd, Beijing 100190, Peoples R China
2.Chinese Acad Sci, AMSS, 55 ZhongGuanCun East Rd, Beijing 100190, Peoples R China
3.Univ Caen Basse Normandie, CNRS, LMNO, BP 5186, F-14032 Caen, France
4.Univ Lancaster, Dept Math & Stat, Fylde Coll, Lancaster LA1 4YF, England
5.Univ Massachusetts, Dept Math & Stat, Amherst, MA 01003 USA
Recommended Citation
GB/T 7714
Fu, Baohua,Juteau, Daniel,Levy, Paul,et al. Generic singularities of nilpotent orbit closures[J]. ADVANCES IN MATHEMATICS,2017,305:1-77.
APA Fu, Baohua,Juteau, Daniel,Levy, Paul,&Sommers, Eric.(2017).Generic singularities of nilpotent orbit closures.ADVANCES IN MATHEMATICS,305,1-77.
MLA Fu, Baohua,et al."Generic singularities of nilpotent orbit closures".ADVANCES IN MATHEMATICS 305(2017):1-77.
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