Given a simple complex Lie group $G$ (I might say upfront that I am mostly interested with exceptional Lie algebras) and a nilpotent orbit $\mathcal{O}\subset G$ I would like to describe the intersection of $\mathcal{O}$ with Levi subgroups (or probably, their derived groups) and Pseudo-Levi subgroups.
I guess that there is a way to take the weighted Dynkin diagram of $\mathcal{O}$ (or the nilpotent orbit associated to it in the Lie algebra) and construct the weighted Dynkin diagram of the intersection.
As an example to what I'm trying to figure out, I went to Birne Binegar's UMRK database (at http://umrk.dynns.com:800/UMRK/UMRK.html) and looked at the data for Nilpotent orbits of the Lie algebra of type $E_7$.
For the orbit $A_4+A_1$ it says that the producing pseudo-Levi subalgebra is generated by the roots [0, 1, 2, 3, 5, 6, 7] (so the negative highest roots together with roots number 1,2,3,5,6,7 in the Dynkin diagram, enumeration as in Bourbaki I believe). The producing pseudo-Levi subalgebra means here (as far as I understand) a standard pseudo-Levi subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ such that the intersection of $A_4+A_1$ with $\mathfrak{h}$ is distinguished.
My question is, what orbit of $\mathfrak{h}$ is the intesection?
In particular, I wish to know if and how can I find a subalgebra such that the intersection is the principal orbit.
A closely related follow-up question is, can I then go about determining the stabilizer of the orbit from this data (say, the stabilizer of the intersection in $\mathfrak{h}$ and somethineg like $\mathrm{Aut}(\mathrm{Dyn}(\mathfrak{h}))$?
A bonus question, if someone happens to know: The UMRK attaches a further integer to the pseudo-Levi subalgebra on top of the list of roots (it's 4 in the case), what does this integer represents?
Best Answer
You can't always find a pseudo-Levi such that the intersection is a principal = regular orbit, but you can always find a Levi subalgebra such that the intersection is distinguished, i.e. the connected centraliser in the derived subgroup is unipotent. This is the Bala-Carter theorem.
Now, there is an improvement to the Bala-Carter theorem due to Sommers, see "A generalisation of the Bala-Carter theorem for nilpotent orbits" in IMRN in 1998 (can't put my hands on the precise reference at the moment). We consider simple groups of adjoint type. For any nilpotent element $e$ of ${\mathfrak g}$ let $A(e)=Z_G(e) /Z_G(e)^\circ$. Each coset in $A(e)$ has a semisimple representative $s$; $s$ generates $Z(L)/Z(L)^\circ$ where $L$ is the centraliser of $s$ (a pseudo-Levi subgroup); finally, this induces something close to a bijection between conjugacy classes in $A(e)$ and pseudo-Levi subalgebras of ${\mathfrak g}$ (up to conjugacy) which contain $e$ (perhaps modulo some issues with graph automorphisms). In particular, you can improve on "$e$ is distinguished in a Levi subalgebra" by instead saying "$e$ is semiregular in a pseudo-Levi subalgebra". Here semiregular means the centraliser is connected and unipotent (a stronger condition than distinguished).
For almost all cases, semiregular will imply regular, but e.g. I think from memory that there are three semiregular orbits in type $E_6$.