3,321 research outputs found
Forced gradings and the Humphreys-Verma conjecture
Let be a semisimple, simply connected algebraic group defined and split
over a prime field of positive characteristic. For a positive
integer , let be the th Frobenius kernel of . Let be a
projective indecomposable (rational) -module. The well-known
Humprheys-Verma conjecture (cf. \cite{Ballard}) asserts that the -action
on lifts to an rational action of on . For (where
is the Coxeter number of ), 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 . 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 -filtrations. The authors anticipate that the Humphreys-Verma
conjecture results here will lead to extensions to smaller characteristics of
these earlier papers
Extensions, Levi subgroups and character formulas
This paper consists of three interconnected parts. Parts I,III study the
relationship between the cohomology of a reductive group and that of a Levi
subgroup. For example, we provide a necessary condition, arising from
Kazhdan-Lusztig theory, for the natural map on Ext-groups of irreducible
modules to be surjective. In cohomological degree 1, the map is always an
isomorphism, under our hypothesis. These results were inspired by recent work
of Hemmer obtained for general linear groups, and they both extend and improve
upon his work when our condition is met. Part II obtains results on Lusztig
character formulas for reductive groups, obtaining new necessary and sufficient
conditions for such formulas to hold. In the special case of general linear
groups, these conditions can be recast in a striking way completely in terms of
explicit representation theoretic properties of the symmetric group (and the
results improve upon the sufficient cohomological conditions established
recently in the authors)
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