  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.
