WEYL filtration dimension and submodule structures for B2

Abstract

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 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 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.