Let G be a connected and simply connected semisimple algebraic group over an algebraically closed field of positive prime characteristic. Let L([lambda]) and [upside-down triangle]([lambda]) be the simple and induced finite dimensional rational G-modules with p-singular dominant highest weight [lambda].

In this thesis, the concept of Weyl filtration dimension of a finite dimensional rational G-module is studied for some highest weight modules with p-singular highest weights inside

the p2-alcove when G is of type B[subscript]2. In chapter 4, intertwining morphisms, a diagonal

G-module morphism and tilting modules are used to compute the Weyl filtration dimension

of L([lambda]) with [lambda] p-singular and inside the p[superscript]2-alcove. It is shown that the Weyl filtration

dimension of L([lambda]) coincides with the Weyl filtration dimension of [upside-down triangle]([lambda]) for almost all (all but one of the 6 facet types) p-singular weights inside the p[superscript]2-alcove. In chapter 5 we study

some submodule structures of Weyl (and there translations), Vogan, and tilting modules

with both p-regular and p-singular highest weights. Most results are for the p[superscript]2 -alcove only

although some concepts used are alcove independent.