Distinguished nilpotent orbits, Kostant pairs and normalizers of Lie algebras

A pair of Lie algebras (g, g(1)) will be called a Kostant pair if g is semisimple, g is reductive in g and the restriction of the Killing form B-g to g(1) is nondegenerate. We study the class of such (nonsymmetric) pairs and obtain some useful and new structural results. We study the structure of the normalizers N-g(g(1)), and as a consequence we obtain some corresponding worthy results about algebraic groups. In particular we consider an interesting case when g(1) is a distinguished sl(2)-subalgebra of g. Combined with the research due to V.L. Popov we observe that the notions of self-normalizing (reductive) subalgebras of a semisimple Lie algebra and projective self-dual algebraic subvarieties of the usual nilpotent cones are closely related. (C) 2014 Elsevier Inc. All rights reserved.