TY - JOUR

T1 - Generic singularities of nilpotent orbit closures

AU - Fu, Baohua

AU - Juteau, Daniel

AU - Levy, Paul

AU - Sommers, Eric

N1 - This is the author’s version of a work that was accepted for publication in Advances in Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Advances in Mathematics, 305, 2016 DOI: 10.1016/j.aim.2016.09.010

PY - 2017/1/10

Y1 - 2017/1/10

N2 - 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 sufficeto 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 SL 2 (C)-variety whose normalization is A 2 , an Sp 4 (C)-variety 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.

AB - 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 sufficeto 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 SL 2 (C)-variety whose normalization is A 2 , an Sp 4 (C)-variety 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.

KW - Nilpotent orbits

KW - Symplectic singularities

KW - Slodowy slice

U2 - 10.1016/j.aim.2016.09.010

DO - 10.1016/j.aim.2016.09.010

M3 - Journal article

VL - 305

SP - 1

EP - 77

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -