A note on the tangent bundle of G/P

Let P be a parabolic subgroup of a complex simple linear algebraic group G. We prove that the tangent bundle T (G/P) is stable

A constructive method for decomposing real representations

A constructive method for decomposing finite dimensional representations of semisimple real Lie algebras is developed. The method is illustrated by an example. We also discuss an implementation of the algorithm in the language of the computer algebra system {\sf GAP}4.Comment: Final version; to appear in "Journal of Symbolic Computation

Semisimple algebras of vector fields on C^N of maximal rank

A classification of semisimple algebras of vector fields on C^N that have a Cartan subalgebra of dimension N is given. The proof uses basic representation theory and the local canonical form of semisimple Lie algebras of vector fields