Tilting modules and highest weight theory for reduced enveloping algebras

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 $\chi \in {\mathfrak{g}}^{\ast }$ in standard Levi form, we study a category ${\mathcal{C}}_{\chi }$ 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 ${\mathcal{C}}_{\chi }$, including tilting modules. We also develop the theory of canonical $\mathrm{\Delta }$-flags and $\bar{\mathrm{\nabla }}$-sections of $\mathrm{\Delta }$-flags, in analogy with similar concepts for algebraic groups studied by Riche and Williamson.