On injective modules for infinitesimal algebraic groups. II
type: Item
instance of: scholarly article
title: On injective modules for infinitesimal algebraic groups. II
zbMATH Open document ID: 0769.20017
DOI: 10.1016/0021-8693(90)90054-R
published in: Journal of Algebra
publication date: 1992-06-25T00:00:00Z
review text: Part I is a joint paper by the author, \textit{B. J. Parshall} and \textit{L. L. Scott} [J. Lond. Math. Soc., II. Ser. 31, 277-291 (1985; Zbl 0567.20024)]. Part II is not a direct continuation, though it deals with some of the same themes. When \(G\) is a semisimple algebraic group, defined and split over the prime field of an algebraically closed field of prime characteristic, the representation theory of \(G\) is intimately related to that of its (nonreduced) subgroup schemes \(TG_ r\), where \(T\) is a maximal (split) torus and \(G_ r\) is the kernel of the \(r\)th power of the Frobenius endomorphism. The author's aim is to study the structure of injective \(TG_ r\)-modules using parabolic subgroup schemes \(TP_ r\) and their Levi factors \(TL_ r\). Besides the usual restriction functors, one has ``lifting'' functors from the category of rational \(TL_ r\)- modules to the category of rational \(TG_ r\)-modules, the composite of inflation to \(TP_ r\) (or its opposite) and induction from \(TP_ r\) to \(TG_ r\). In terms of these functors, the author establishes isomorphisms between certain \(\text{Ext}^ n\) groups for \(TL_ r\) and \(TG_ r\). This applies especially to induced modules. He also studies the relationship between Jantzen's translation functors (for Levi factors) and the lifting functors, showing that lifting often commutes with translation.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 14L35
Mathematics Subject Classification ID: 14L17
zbMATH DE Number: 5934
zbMATH Keywords: semisimple algebraic group
zbMATH Keywords: representation theory
zbMATH Keywords: subgroup schemes
zbMATH Keywords: injective \(TG_ r\)-modules
zbMATH Keywords: Levi factors
zbMATH Keywords: restriction functors
zbMATH Keywords: induced modules
zbMATH Keywords: translation functors
zbMATH Keywords: lifting functors
author: Edward Cline
MaRDI profile type: Publication
Wikidata QID: Q122980118
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full work available at URL: 0021-8693(90)90054-r
OpenAlex ID: W2071654192
description: scientific article; zbMATH DE number 5934