Extensions for Frobenius kernels.
type: Item
instance of: scholarly article
title: Extensions for Frobenius kernels.
zbMATH Open document ID: 1044.20031
author: Christopher P. Bendel
author: Daniel K. Nakano
author: Cornelius Pillen
published in: Journal of Algebra
publication date: 2004-03-14T00:00:00Z
review text: Let \(G\) be a simply connected simple algebraic group defined and split over a prime field \(\mathbb{F}_p\). We follow the notations of \textit{J. C. Jantzen} [Representations of algebraic groups. (2nd ed. Mathematical Surveys and Monographs 107. Providence, RI: American Mathematical Society) (2003; Zbl 1034.20041)]. The representation theory of the Frobenius kernels \(G_r\) is important for the representation theory of both \(G\) and its finite subgroups \(G(\mathbb{F}_q)\). This paper gives precise information on several \(\text{Ext}\) groups of \(G_r\)-modules, including cases of small characteristic. This inclusiveness necessitates many case distinctions. The authors compute the first \(G_r\)-cohomology of each induced representation \(H^0(\lambda)\) by way of its first \(B_r\)-cohomology. Thanks to this effort we now know all \(H^1(G_r,H^0(\lambda))^{(-r)}\). Each of them has a good filtration. One may hope that the good filtration property holds for higher \(H^i\) too, as predicted by Donkin. Next they study the \(G_r\)-extensions between simple modules. Here the results are less definite. They express the \(G\)-module \(E=\text{Ext}^1_{G_r}(L(\lambda),L(\mu))\) in terms of \(\text{Ext}^1_G\) of suitable \(G\)-modules, but sometimes only as a vector space. Then new bounds are given for the weights that occur in \(E\) or in the auxiliary \(G\)-modules.
reviewed by: Wilberd van der Kallen
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 2055641
zbMATH Keywords: Frobenius kernels
zbMATH Keywords: restricted nullcones
zbMATH Keywords: induced representations
zbMATH Keywords: cohomology
zbMATH Keywords: good filtrations
zbMATH Keywords: weights
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.04.003
OpenAlex ID: W2020680256
cites work: Extensions of Modules for Algebraic Groups
cites work: Extensions of simple modules for finite Chevalley groups
cites work: Cohomology of induced representations for algebraic groups
cites work: On comparing the cohomology of algebraic groups, finite Chevalley groups and Frobenius kernels
cites work: Extensions for finite Chevalley groups II
cites work: Q4422481
cites work: Q3804882
cites work: Q3792890
cites work: Cohomology of Lie Algebras and Algebraic Groups
cites work: Q3797389
cites work: Q4012103
cites work: Frobenius splitting of cotangent bundles of flag varieties
cites work: Support varieties for algebraic groups
cites work: Infinitesimal fixed points in modules with good filtration
DOI: 10.1016/J.JALGEBRA.2003.04.003
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description: scientific article; zbMATH DE number 2055641