Filtrations on G1T-Modules |
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Authors: | Andersen Henning Haahr; Masaharu Kaneda |
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Institution: | Department of Mathematics, University of Aarhus Building 530, Ny Munkegade, 8000 Aarhus C, Denmark; e-mail: mathha{at}imf.au.dk
Department of Mathematics, Graduate School of Science, Osaka City University 558-8585 Osaka Sumiyoshi-ku, Japan; e-mail: kaneda{at}sci.osaka-cu.ac.jp |
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Abstract: | Let G be an almost simple algebraic group defined over Fp forsome prime p. Denote by G1 the first Frobenius kernel in G andlet T be a maximal torus. In this paper we study certain Jantzentype filtrations on various modules in the representation theoryof G1T. We have such filtrations on the baby Verma modules Z,where is a character of T. They are obtained via a certaindeformation of the natural homomorphism from Z into its contravariantdual Z. Using the same deformation we construct for each projectiveG1T-module Q a filtration of the vector space . We then prove that this filtration may also bedescribed in terms of the above-mentioned homomorphism Z() Z() and this leads us to a sum formula for our filtrations.When Q is indecomposable with highest weight in the bottom alcove(with respect to some special point) we are able to computethe filtrations on F(Q) explicitly for all . This is then thestarting point of an induction which proceeds via wall crossingsto higher alcoves. If our filtrations behave as expected undersuch wall crossings then we obtain a precise relation betweenthedimensions of the layers in the filtrations of F(Q) for an arbitraryindecomposable projective Q and the coefficients in the correspondingKazhdanLusztig polynomials. We conclude the paper byproving that the above results in the G1T theory have some analoguesin the representation theory of G (where, however, we have towork with representations of bounded highest weights) and thecorresponding theory for quantum groups at roots of unity. Theseresults extend previous work by the first author. 2000 MathematicsSubject Classification: 20G05, 20G10, 17B37. |
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Keywords: | baby Verma modules Jantzen filtrations tilting modules |
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