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 $O'$?



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


