Nilpotent orbits and mixed gradings of semisimple Lie algebras

Abstract

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