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 1 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. [ABSTRACT FROM AUTHOR]