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Loewy series of parabolically induced $G_1T$-Verma modules

Part of: Linear algebraic groups and related topics

Published online by Cambridge University Press:  28 March 2014

Abe Noriyuki  and
Kaneda Masaharu
Abe Noriyuki
Affiliation:
Hokkaido University, Creative Research Institution (CRIS), Japan(abenori@math.sci.hokudai.ac.jp)
Kaneda Masaharu
Affiliation:
Osaka City University, Department of Mathematics, Japan(kaneda@sci.osaka-cu.ac.jp)
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Abstract

We show that the modules for the Frobenius kernel of a reductive algebraic group over an algebraically closed field of positive characteristic $p$ induced from the $p$-regular blocks of its parabolic subgroups can be $\mathbb{Z}$-graded. In particular, we obtain that the modules induced from the simple modules of $p$-regular highest weights are rigid and determine their Loewy series, assuming the Lusztig conjecture on the irreducible characters for the reductive algebraic groups, which is now a theorem for large $p$. We say that a module is rigid if and only if it admits a unique filtration of minimal length with each subquotient semisimple, in which case the filtration is called the Loewy series.

Keywords

Loewy seriesparabolic inductiongraded inductionrigidityFrobenius kernel

MSC classification

Secondary: 20G05: Representation theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 14 , Issue 1 , January 2015 , pp. 185 - 220
Copyright
© Cambridge University Press 2014 

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