TITLE: The relation of nilpotent orbits and simple singularities, for orbits smaller than subregular ones
QUESTION [7 upvotes]: 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 $O'$?

REPLY [6 votes]: I just saw this question. See
Baohua Fu, Daniel Juteau, Paul Levy, Eric Sommers.
"Generic singularities of nilpotent orbit closures"
 arXiv:1502.05770 
where we do the same job as Kraft and Procesi, but for exceptional types. We posted the preprint only this year, but we already had most of the results when the question was asked (and gave talks about this).
So most of the time, it is indeed a simple singularity, but sometimes there are several branches, and there is also a nonnormal singularity with smooth normalization which occurs in all exceptional types.
Let me give some information about the methods. Baohua Fu had studied in previous work Q-factorial terminalizations of nilpotent orbit closures. It turns out that they are all "generalized Springer maps", or just the normalization. Restricting to a slice, they provide a resolution of the surface singularity (a Q-factorial terminal singularity is smooth in codimension 2), and actually it is the minimal resolution. If the terminalization corresponds to inducing some nilpotent orbit from a smaller Lie algebra ("generalized Springer map"), then one can understand the exceptional fiber using a formula of Borho and MacPherson. This is sufficient in most cases, but for the singularities between subregular and subsubregular, we also had to use orbital varieties... See our preprint for more details.