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?



Good even grading and principal Levi type
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.


