Pairs of Lie Algebras and their Self-Normalizing Reductive Subalgebras

We consider a class P of pairs (g, g(1)) of K-Lie algebras g(1) subset of g satisfying certain "rigidity conditions"; here K is a field of characteristic 0, g is semisimple, and 91 is reductive. We provide some further evidence that P contains a number of nonsymmetric pairs that are worth studying; e.g., in some branching problems, and for the purposes of the geometry of orbits. In particular, for an infinite series (g, g(1)) = (sI(n + 1), sI(2)) we show that it is in P, and precisely describe a g(1)-module structure of the Killing-orthogonal p(n) of g(1) in g. Using this and the Kostant's philosophy concerning the exponents for (complex) Lie algebras, we obtain two more results. First; suppose K is algebraically closed, g is semisimple all of whose factors are classical, and s is a principal TDS. Then (g,s) belongs to P. Second; suppose (g, g(1)) is a pair satisfying certain technical condition (C), and there exists a semisimple s subset of g(1) such that (g, s) is from P (e.g., s is a principal TDS). Then (g, g(1)) is from P as well. Finally, given a pair (g, g(1)), we have some useful observations concerning the relationship between the coadjoint orbits corresponding to g and g(1), respectively.