Filtrations in Modular Representations of Reductive Lie Algebras

2007/03/18 by Yiyang Li, Li, Yiyang, Bin Shu +1
Mathematics · #17B10 #17B20 #17B35 #17B50 #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:17B10 #msc:17B20 #msc:17B35 #msc:17B50

paper · pdf · doi:10.48550/arxiv.math/0703528

The current version of this paper will appear in Algebra Colloquium

arxiv created 2007/11/17 · arxiv updated 2011/11/09

Abstract

Let G be a connected reductive algebraic group G over an algebraically closed field k of prime characteristic p, and \ggg=\Lie(G). In this paper, we study modular representations of the reductive Lie algebra \ggg with p-character χ of standard Levi-form associated with an index subset I of simple roots. With aid of support variety theory we prove a theorem that a U_χ(\ggg)-module is projective if and only if it is a strong "tilting" module, i.e. admitting both \cz_Q- and \cz^wI_Q-filtrations (to see Theorem \refTHMFORINV). Then by analogy of the arguments in \citeAK for G₁T-modules, we construct so-called Andersen-Kaneda filtrations associated with each projective \ggg-module of p-character χ, and finally obtain sum formulas from those filtrations.

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