Nilpotent orbits and mixed gradings of semisimple Lie algebras

Abstract

Let σ be an involution of a complex semisimple Lie algebra g and g=g0⊕g1 the related Z2-grading. We study relations between nilpotent G0-orbits in g0 and the respective G-orbits in g. If e∈g0 is nilpotent and {e,h,f}⊂g0 is an sl2-triple, then the semisimple element h yields a Z-grading of g. Our main tool is the combined Z×Z2-grading of g, which is called a mixed grading. We prove, in particular, that if eσ is a regular nilpotent element of g0, then the weighted Dynkin diagram of eσ, D(eσ), has only isolated zeros. It is also shown that if G⋅eσ∩g1≠∅, then the Satake diagram of σ has only isolated black nodes and these black nodes occur among the zeros of D(eσ). Using mixed gradings related to eσ, we define an inner involution σˇ such that σ and σˇ commute. Here we prove that the Satake diagrams for both σˇ and σσˇ have isolated black nodes.