Good even grading and principal Levi type

Question

Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$ and let $e$ be a nilpotent element in it. In the theory of finite W-algebras one often encounters the following two conditions:

1) $e$ is principal in some Levi subalgebra $\mathfrak l$ of $\mathfrak g$.

2) There exists a good even grading on $\mathfrak g$ (recall that a grading on $\mathfrak g$ is called good for e if $e \in \mathfrak g_2$ and the linear map $ad~ e : \mathfrak g_j → \mathfrak g_{j+2}$ is injective for $j \leq −1$ and surjective for $j\geq −1$).

My question is this: is there any relation between these conditions, or are they completely independent?

Answer 1

If $e$ is principal in a proper Levi subalgebra whose Dynkin diagram involves a component of type $A_k$ with $k$ odd, then e is not even. This is very easy to see by writing down an explicit $sl_2$-triple containing $e$. Occasionally, one can find another good grading for such $e$ (if it is Richardson) but this is quite rare outside type $A$. Also, if $e\ne 0$ is rigid in the sense of Lusztig-Spaltenstein then $e$ is never even (nor Richardson) and for $g$ exceptional all rigid nilpotent orbits except two (I think) are principal in Levi subalgebras. From the representation-theoretic viewpoint, rigid nilpotent elements give rise to the most interesting finite $W$-algebras.

Answer 2

Short answer: The two conditions on $e$ are logically unrelated. For example, a long root vector in type $B_2$ (generating the minimal nonzero nilpotent orbit) satisfies 1) but not 2), while an element in the subregular nilpotent orbit in type $G_2$ satisfies 2) but not 1).

Details. A basic source is the paper [EK] by Elashvili and Kac in a 2006 AMS book series, which Ben reminds me was posted on arXiv here. They classify case-by-case the possible good gradings for $e$. One of these is the Dynkin grading, which is even iff the labels on the Dynkin diagram of $e$ (or its orbit) are all even. In some cases but not others there are other good gradings.

A minimal $e$ in type $B_2$ (indeed all four nilpotent orbits here) is regular/standard in a Levi subalgebra of a parabolic. But its Dynkin diagram has an odd label, so the Dynkin grading is not even. In the fourth paragraph of their Section 2, [EK] point out that except for type $A_n$ the only possible good grading for a minimal $e$ is the Dynkin grading.

On the other hand, Theorem 7.1 in [EK] says that in types $G_2$ and $F_4$ only the Dynkin grading is good for each $e$. A subregular nilpotent in type $G_2$ fails to be of standard Levi type: there are only four types of parabolic subalgebras and four corresponding standard Levi type orbits here, the regular, zero, minimal, and "middle" orbits. But a subregular $e$ has even Dynkin labels, so there is an even good grading for $e$.

Discussion. Sorting out nilpotent orbits with various properties gets quite complicated. For instance, all even orbits are Richardson, but some Richardson orbits are odd, such as the orbit labelled $C_3$ in type $F_4$ (if my old computations of Richardson orbits for $F_4$ are correct). Coupled with Theorem 7.1 in [EK] this gives an odd Richardson element having no even good grading. (But in general, when some even grading does exist with $e \in \mathfrak{g}_2$, Proposition 2.1 in [EK] shows that the grading is good for e if and only if $e$ is Richardson.)

Philosophically, the problem here is that standard Levi type and Richardson are in a sense dual ideas (which coincide only in type $A_n$). While "most" nilpotent orbits are of both types, there are exceptions that make the independent technical conditions 1) and 2) in the question both useful to impose for some purposes in the study of finite $W$-algebras.

Answer 3

This was wrong. If you really want to read a wrong answer, look in the edit history.