BGG引入的范畴O可以用来研究很多重要的表示问题。Verma模和广义Verma模是范畴O中的基本研究对象。本项目旨在研究广义Verma模中的首权向量。它为广义Verma模同态问题及齐次向量丛上不变微分算子的研究提供了新思路。. 本项目主要围绕两方面展开。首先是把高阶的首权向量的问题转化为相对低阶的情形，这需要利用范畴O中的子范畴等价以及权空间的组合性质。另一方面，我们也希望为一些低阶的首权向量和微分算子做出简明的刻画，此时需要重点考察首权向量中的支配项。

Many interesting representations can be studied effectively in terms of the BGG category O introduced by Bernstein, Gelfand and Gelfand. Verma modules and generalized Verma modules are the elementary objects in the catergory O. The aim of this proposed research project is to study the leading weight vectors in generalized Verma modules. It provides a new approach to the Hom spaces between generalized Verma modules and invariant differential operators on homogeneous vector bundles. .First, we aim to simplify the question of leading weight vectors to a lower order. We need to apply the category equivalences in O and the combinatorial properties of the weight space. Second, we plan to present an explicit description of the lower order leading weight vectors. The dominant terms of vectors play central roles here.