Leading Weight Vectors in Generalized Verma Modules

In this thesis, we systematically study the leading weight vectors in generalized Verma modules. As an application, we classify all the first order leading weight vectors and determine the corresponding Hom spaces between generalized Verma modules. Furthermore, in light of the correspondence between leading weight vectors and invariant differential operators, we obtain a practical criterion for the existence of first order invariant differential operators. This is an improvement of the result in [SS].