Z-Gradiations of Lie Algebras and Infinitesimal Generators

In this paper we arrive at explicit formulae for the infinitesimal generators of the action of a complex simple Lie group G on the manifold M = G/P where P is a maximal parabolic subgroup.These formulae are obtained by assuming that local coordinates on M are furnished by the nilpotent subalgebra n complementary to the maximal parabolic subalgebra p corresponding to P .For the classical isogeny classes A r , B r , C r , and D r , the components of the infinitesimal generators are never worse than quartic polynomials in the coordinate functions, but for the exceptional cases, G 2 , F 4 , and E r , higher-degree polynomials frequently occur.