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    Forced gradings and the Humphreys-Verma conjecture

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    Let GG be a semisimple, simply connected algebraic group defined and split over a prime field Fp{\mathbb F}_p of positive characteristic. For a positive integer rr, let GrG_r be the rrth Frobenius kernel of GG. Let QQ be a projective indecomposable (rational) GrG_r-module. The well-known Humprheys-Verma conjecture (cf. \cite{Ballard}) asserts that the GrG_r-action on QQ lifts to an rational action of GG on QQ. For pβ‰₯2hβˆ’2p\geq 2h-2 (where hh is the Coxeter number of GG), 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 pp. 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 pp-filtrations. The authors anticipate that the Humphreys-Verma conjecture results here will lead to extensions to smaller characteristics of these earlier papers
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