  Let $G$ be a semisimple, simply connected algebraic group defined and split
over a prime field ${\mathbb F}_p$ of positive characteristic. For a positive
integer $r$, let $G_r$ be the $r$th Frobenius kernel of $G$. Let $Q$ be a
projective indecomposable (rational) $G_r$-module. The well-known
Humprheys-Verma conjecture (cf. \cite{Ballard}) asserts that the $G_r$-action
on $Q$ lifts to an rational action of $G$ on $Q$. For $p\geq 2h-2$ (where $h$
is the Coxeter number of $G$), this conjecture was proved by Jantzen in 1980,
improving on early work of Ballard. However, it remains open for general
characteristics. In this paper, the authors establish several graded analogues
of the Humphreys-Verma conjecture, valid for all $p$. The most general of our
results, proved in full here, was announced (without proof) in an earlier
paper. Another result relates the Humphreys-Verma conjecture to earlier work of
Alperin, Collins, and Sibley on finite group representation theory. A key idea
in all formulations involves the notion of a forced grading. The latter goes
back, in particular, to the recent work of the authors, relating graded
structures and $p$-filtrations. The authors anticipate that the Humphreys-Verma
conjecture results here will lead to extensions to smaller characteristics of
these earlier papers.
