  Let $G$ be a simple, simply connected algebraic group over an algebraically
closed field of positive characteristic $p$. In recent work, the authors have
studied a graded analogue of the category of rational $G$-modules. These
gradings are not natural but are "forced" on related algebras though
filtrations, often obtained from appropriate quantum structures. This paper
presents new results on Koszul modules for the graded algebras obtained through
this forced grading process. Most of these results require that the Lusztig
character formula holds for all restricted $p$-regular weights, but the paper
begins to investigate how these and previous results might be established when
the Lusztig character formula is only assumed to hold on a proper poset ideal
in the Jantzen region. This opens up the possibility of inductive arguments.
