Return to Revisions

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 $e$. Also, if $e$ is rigid in the sense of Lusztig-Spaltenstein then $e$ is never even 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.