Generic singularities of nilpotent orbit closures

Understand

According to a well-known 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 nilpotent cone, 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).