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

Question

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 $O'$?

Answer 1

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.

Answer 2

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.

Answer 3

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.

The big advantage of working with the subregular orbit and related 2-dimensional affine Slodowy slice is that so much is known concretely from the study of the Springer resolution of the nilpotent variety (in terms of the cotangent bundle of the flag variety). In particular, a fiber of this resolution is a "Dynkin curve": a union of projective lines indexed by vertices of the Coxeter/Dynkin diagram and having an incidence pattern described by that diagram. This picture fits naturally into the bigger picture giving the desired singularity along with a semiuniversal deformation. It's already a fairly long story, told carefully by Slodowy in his thesis and various later lectures.

Beyond the subregular case, there are indeed pairs of nilpotent orbits of dimensions differing again by 2, but almost nothing can be said about the geometry of the associated Springer fibers. There have been a couple of dissertations over the years (by Lorist, Fung) approaching this topic and getting partial results, but so far progress has been very limited. For example, there are 2-dimensional Springer fibers in small rank cases which I strongly suspect are isomorphic as varieties, but this is hard to approach even though the dimensions of their cohomology spaces are known to coincide.

For the question stated, it's necessary in any case to make precise the language about "transversal slice" to one nilpotent orbit inside another, within the algebraic group and Lie algebra framework. Then one probably has to look first at small rank cases where the dimension difference is 2. It's easy to locate such cases but not so easy to take the next step.