Nilpotent orbits and mixed gradings of semisimple Lie algebras

Let σ be an involution of a complex semisimple Lie algebra g and = 0 ⊕ 1 the related Z 2 -grading. We study relations between nilpotent G -orbits in respective . If e ∈ is { , h f } ⊂ sl -triple, then element yields -grading Our main tool combined × which called mixed grading. prove, particular, that if regular weighted Dynkin diagram D ( ) has only isolated zeros. It also shown ⋅ ∩ ≠ ∅ Satake black nodes these occur among zeros Using gradings to we define inner ˇ such commute. Here prove diagrams for both have nodes.