Tilting modules and highest weight theory for reduced enveloping algebras

TL;DR

This paper investigates the structure of graded representations of reduced enveloping algebras for reductive algebraic groups in positive characteristic, focusing on tilting modules and highest weight theory, with new developments in canonical flags and wall-crossing functors.

Contribution

It introduces new highest weight theoretic tools for reduced enveloping algebras, including the study of tilting modules, translation functors, and canonical flag structures.

Findings

Analysis of tilting modules in category ${\ m extbf{C}}_\chi$

Development of canonical $\\Delta$-flags and $\overline{\nabla}$-sections

Insights into wall-crossing functors and their effects

Abstract

Let $G$ be a reductive algebraic group over an algebraically closed field of characteristic $p>0$, and let ${\mathfrak g}$ be its Lie algebra. Given $\chi\in{\mathfrak g}^{*}$ in standard Levi form, we study a category ${\mathscr C}_\chi$ of graded representations of the reduced enveloping algebra $U_\chi({\mathfrak g})$. Specifically, we study the effect of translation functors and wall-crossing functors on various highest-weight-theoretic objects in ${\mathscr C}_\chi$, including tilting modules. We also develop the theory of canonical $\Delta$-flags and $\overline{\nabla}$-sections of $\Delta$-flags, in analogy with similar concepts for algebraic groups studied by Riche and Williamson.