我们拟研究一般线性群多项式表示理论中如下基本问题：有限维投射内射模的刻画与分类；Weyl模的Jantzen滤过；单模不同实现方式的比较与统一；与一般线性李代数相关的正特征域上的范畴O理论。相关的问题也会一并研究。 有限维投射内射模是一类结构丰富且应用广泛的研究对象。De Visscher-Donkin猜测给出了这类模的完全分类，且仅在低秩情形完全证明。考察(无穷小)Schur代数的控制维数这一衡量投射内射模多少的同调不变量为研究该猜想及相关问题提供了一条可操作途径。 考察正特征域上的范畴O理论和Verma模的Jantzen滤过等不仅有十分明确的重要理论价值，而且对研究席南华的Weyl模Jantzen滤过猜想、推广Brundan-Kleshchev的零特征域上高水平Schur-Weyl对偶都非常有帮助。 比较几种不同的单模实现方式除基本理论意义外，在解决具体问题方面亦有广阔的前景。

We plan to consider the following fundamental problems in the theory of polynomial representations of general linear groups: characterization and classification of finite dimensional projective and injective modules;Jantzen filtration of Weyl modules;comparison and unification of different realizations of simple modules；generalization to positive characteristic field of the theory of category O for the general linear Lie algebras. Finite dimensional projective and injective modules constitute an important class of objects with rich structures and wide applications. De Visscher-Donkin conjecture gives a complete classification of such modules and is only proved in the lower rank cases. Study on the dominant dimension, a homological invariant which detects the sufficency of projective injective modules, of (infinitesimal) Schur algebras will provides an applicable approach to this conjecture and some related problems. Investigating the category O theory and Jantzen filtrations of Verma modules over a positive characterisitic field for the general linear Lie algebra is not only of significant theoretical meaning, but also tremendously helps both to the study of Nanhua Xi's conjecture on a realization of Jantzen filtration of Weyl modules and generalizing Brundan-Kleshchev's higher lever Schur-Weyl duality in characteristic zero. Besides the basic theoretical meaning, comparison of different realization of simple modules has also a wide application prospect in solving some concrete problems.