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THE LOEWY STRUCTURE OF $G_{1}T$-VERMA MODULES OF SINGULAR HIGHEST WEIGHTS

Part of: Linear algebraic groups and related topics

Published online by Cambridge University Press:  02 October 2015

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

Let $G$ be a reductive algebraic group over an algebraically closed field of positive characteristic, $G_{1}$ the Frobenius kernel of $G$, and $T$ a maximal torus of $G$. We show that the parabolically induced $G_{1}T$-Verma modules of singular highest weights are all rigid, determine their Loewy length, and describe their Loewy structure using the periodic Kazhdan–Lusztig $P$- and $Q$-polynomials. We assume that the characteristic of the field is sufficiently large that, in particular, Lusztig’s conjecture for the irreducible $G_{1}T$-characters holds.

Keywords

Frobenius kernelsingular blockLoewy series

MSC classification

Primary: 20G05: Representation theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 16 , Issue 4 , September 2017 , pp. 887 - 898
Copyright
© Cambridge University Press 2015 

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