# The relation of nilpotent orbits and simple singularities, for orbits smaller than subregular ones

Pick a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. There is a partial ordering among nilpotent orbits defined by $O\geq O'$ iff $\bar O\supset O'$.

The unique maximal element under this partial order is the regular nilpotent orbit, and the unique sub-maximal element is the subregular nilpotent orbit. Denote the former by $O$ and the latter by $O'$. Then $\dim O-\dim O'=2$; Brieskorn and Slodowy showed that the transversal slice to $O'$ inside $O$ is the simple singularity of type $\mathfrak{g}$.

Now, there are many pairs of nilpotent orbits $O$ and $O'$ such that $O\geq O'$ and $\dim O-\dim O'=2$. The transversal slice to $O'$ inside $O$ should be a simple simgularity of some type.

How can I determine the type, given $\mathfrak{g}$, $O$ and $O'$?

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In the case of classical groups the answer is known due to the work of Kraft and Procesi (see their papers "Minimal singularities in $GL_n$" and "On the geometry of conjugacy classes in classical groups"). As far as I know for the exceptional groups the answer is not completely known at this time.

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This type of question is intriguing but probably very difficult to analyze, given how little is known geometrically about the situation here. There might not be any helpful literature, though such questions have probably been raised. I'm definitely a non-expert, so my "answer" is mainly cautionary. The development due to Grothendieck, Brieskorn, and Slodowy involves a lot of delicate machinery related to a simple algebraic group and its Lie algebra (over $\mathbb{C}$ or other algebraically closed field of "good" characteristic). In this work one realizes concretely the various simple singularities within a simple Lie algebra of type A, D, or E; for other types some "unfolding" of the Dynkin diagram is then needed.