Some Koszul properties of standard and irreducible modules

TL;DR

This paper investigates Koszul properties of modules in a graded setting for algebraic groups over fields of positive characteristic, exploring conditions under which these properties hold and potential inductive approaches.

Contribution

It introduces new results on Koszul modules for graded algebras derived from forced gradings, and examines cases where Lusztig's character formula holds on restricted regions.

Findings

Results depend on Lusztig's character formula for all restricted p-regular weights.

Explores partial validity of Lusztig's formula for inductive proofs.

Provides groundwork for future inductive methods in modular representation theory.

Abstract

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.