Filtrations of the modules for Chevalley groups arising from admissible lattices
type: Item
instance of: scholarly article
title: Filtrations of the modules for Chevalley groups arising from admissible lattices
zbMATH Open document ID: 0734.20020
DOI: 10.1007/BF02571789
published in: Mathematische Zeitschrift
publication date: 1992-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174394
review text: Let \(G\) be the simply connected Chevalley group over a field \(k\) of characteristic \(p > 0\) associated with a semisimple complex Lie algebra \(\mathfrak g\). Let \(M\) be an admissible lattice in a finite-dimensional irreducible \(\mathfrak g\)-module \(V(\lambda)\) and \(\overline M\) a rational \(G\)-module obtained from \(M\) through reduction modulo \(p\). The author proves that \(\overline M\) is indecomposable if all weight spaces of \(V(\lambda)\) are 1-dimensional. He constructs a filtration of \(\overline M\) generalizing the Jantzen filtration for a Weyl module and the Andersen filtration for a cohomology module in the generic situation. It turns out that the filtration layers are self-dual under the transpose dual. In particular they are semisimple if the corresponding Weyl module is multiplicity free. Then the groups of type \(A_ 1\) are considered, in which case a description of the module graph of \(\overline M\) is given. It is shown that the nondirected graph associated to the module graph of \(\overline M\) is independent of \(M\). All the possible graphs for \(\overline M\) can be characterized from the module graph \(H^ 0_ k(\lambda)\) in a purely combinatorial manner. All \(\overline M\) with maximal Loewy length, which is the Loewy length of \(H^ 0_ k(\lambda)\) are classified.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 20G15
zbMATH DE Number: 4214255
zbMATH Keywords: finite group
zbMATH Keywords: modular representation
zbMATH Keywords: invariant lattice
zbMATH Keywords: rational modules
zbMATH Keywords: simply connected Chevalley group
zbMATH Keywords: weight spaces
zbMATH Keywords: natural filtration
zbMATH Keywords: Jantzen filtration
zbMATH Keywords: Weyl module
zbMATH Keywords: semisimple complex Lie algebra
zbMATH Keywords: admissible lattice
zbMATH Keywords: finite-dimensional irreducible \(\mathfrak g\)-module
zbMATH Keywords: Andersen filtration
zbMATH Keywords: cohomology module
zbMATH Keywords: Loewy length
author: Zongzhu Lin
reviewed by: Vladimir L. Popov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4214255