Leading weight vectors and homomorphisms between generalized Verma modules

Kostant introduced leading weight vectors in his paper on Verma modules and quasi-invariant differential operators. We extend his idea to study homomorphisms between generalized Verma modules. We classify all first order leading weight vectors and thus determine corresponding Hom spaces between generalized Verma modules. Moreover, in light of the correspondence between leading weight vectors and invariant differential operators, we obtain a new and practical approach to the existence of first order invariant differential operators.