Quasi-simple modules and Loewy lengths in modular representations of reductive Lie algebras

Let $\mathfrak{g}$ be a reductive Lie algebra over an algebraically closed field of characteristic $p>0$. In this paper, we study the representations of $\frak g$witha$p$-character$\chi$ of standard Levi form associated with a given subset $I$ of the simple root system $\mathrm{\Pi }$ of $\mathfrak{g}$. Let $U_\chi({\frak g})$bethereducedenvelopingalgebraof$\frak g$. A notion “quasi-simple module” (denoted by ${\mathcal{L}}_{\chi }(\lambda )$) is introduced. The properties of such a module turn out to be better than those of the corresponding simple module ${\widehat{L}}_{\chi }(\lambda )$. It enables us to investigate the $U_{\chi }(\mathfrak{g})$-modules from a new point of view, and correspondingly gives rise new consequences. First, we show that the first self extension of ${\mathcal{L}}_{\chi }(\lambda )$ is zero, and the projective dimension of $\mathcal L_\chi(\lambda)$isfinitewhen$\lambda$is$p$-regular. These properties make it significant to rewrite the formula of Lusztig’s Hope (Lusztig’s conjecture on the irreducible characters in the category of $U_{\chi }(\mathfrak{g})$-modules) by replacing ${\widehat{L}}_{\chi }(\lambda )$ by ${\mathcal{L}}_{\chi }(\lambda )$. Second, with the aid of quasi-simple modules, we get a formula on the Loewy lengths of standard modules and proper standard modules over $U_{\chi }(\mathfrak{g})$. And by studying some examples, we formulate some conjectures on the Loewy lengths of indecomposable projective $\mathfrak{g}$-modules, standard modules and proper standard modules.