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

Let $\frak g$ be a reductive Lie algebra over an algebraically closed field of characteristic $p>0$. In this paper, we study the representations of $\frak g$ with a $p$-character $\chi$ of standard Levi form associated with a given subset $I$ of the simple root system $\Pi$ of $\frak g$. Let $U_\chi({\frak g})$ be the reduced enveloping algebra of $\frak g$. 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 $U_\chi({\frak g})$-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 $\mathcal L_\chi(\lambda)$ is finite when $\lambda$ 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 $U_\chi({\frak g})$-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 $U_\chi({\frak g})$. And by studying some examples, we formulate some conjectures on the Loewy lengths of indecomposable projective $\frak g$-modules, standard modules and proper standard modules.