Sheaf homology and complete reducibility
type: Item
instance of: scholarly article
title: Sheaf homology and complete reducibility
zbMATH Open document ID: 0609.20026
DOI: 10.1016/0021-8693(85)90093-6
author: Stephen D. Smith
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: Let G denote a Chevalley group over finite field \(F_ q\) of characteristic p with a fixed Borel subgroup B. Consider an \(F_ qG\)- module V satiyfing \(V=<C_ V(O_ p(B))^ g:\) \(g\in G>\). Following the 0-th homology construction in \textit{M. A. Ronan} and \textit{S. D. Smith} [ibid. 96, 319-346 (1985; Zbl 0604.20043)] one can define a sheaf \({\mathcal F}_ V\) on V in terms of subspaces \(V_ 0\) of \(C_ V(O_ p(P_ 0))\) when \(P_ 0\) runs over the parabolics of G containing B. The problem of splitting of V with only two composition factors S and \(T\cong V/S\) reduces to the splitting of \({\mathcal F}_ V\) in the direct sum of \({\mathcal F}_ S\) and \({\mathcal F}_ T\) and some homological questions for \(H_ 0({\mathcal F}_ T)\). Then the author gives an elegant sufficient condition for complete reducibility of V: all composition factors of V are modules \(T_ i\) with minimal highest weights \(\lambda_ i\) not pairwise adjacent in the Dynkin diagram (if \(p=2\), G cannot be of type \(BC_ 3)\), or all \(T_ i\) are pairwise isomorphic.
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 3985506
zbMATH Keywords: Chevalley group
zbMATH Keywords: Borel subgroup
zbMATH Keywords: composition factors
zbMATH Keywords: splitting
zbMATH Keywords: complete reducibility
zbMATH Keywords: weights
zbMATH Keywords: Dynkin diagram
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90093-6
OpenAlex ID: W2092471744
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description: scientific article; zbMATH DE number 3985506