Some Koszul properties of standard and irreducible modules

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.