A note on the adjoint action of a semisimple group on the coordinate algebra of an infinitesimal subgroup
type: Item
instance of: scholarly article
title: A note on the adjoint action of a semisimple group on the coordinate algebra of an infinitesimal subgroup
zbMATH Open document ID: 1405.20034
author: Stephen Donkin
published in: Journal of Algebra
publication date: 2018-12-04T00:00:00Z
review text: Let \(G\) be a connected, semisimple, simply connected affine algebraic group over an algebraically closed field \(K\) of prime characteristic \(p\). Let \(T\) be a maximal torus, \(B \supset T\) be a Borel subgroup corresponding to the negative roots, and \({\mathfrak g}\) denote the Lie algebra of \(G\). Let \(X^{+}(T)\) denote the dominant weights. Each \(\lambda \in X^{+}(T)\) determines a one-dimensional \(B\)-module \(K_{\lambda}\). Inducing to \(G\), one obtains a costandard module \(\nabla(\lambda) := \text{ind}_{B}^{G}(K_{\lambda})\), which is dual to a Weyl module. A finite-dimensional rational \(G\)-module \(M\) is said to have a good filtration if there exists a filtration \(0 = M_0 \leq M_1 \leq M_2 \leq \cdots \leq M_n = M\), such that each (non-zero) quotient \(M_{i}/M_{i-1} \cong \nabla (\lambda_i)\) for some \(\lambda_i \in X^{+}(T)\). If the dual of the module \(M\) also admits a good filtration (or, equivalently, if \(M\) admits a Weyl filtration), then \(M\) is said to be a tilting module. Let \(K[G]\) denote the coordinate algebra of \(G\). Given a positive integer \(r\), let \(G_r\) denote the \(r\)th Frobenius kernel and \(K[G_r]\) denote its coordinate algebra, which may be identified with a quotient of \(K[G]\). The conjugation action of \(G\) on itself, gives \(K[G]\) and, hence, \(k[G_r]\) the structure of \(G\)-modules. Let \(\rho\) denote the half-sum of the positive roots and set \(\text{St}_r := \nabla((p^r - 1)\rho)\), the \(r\)th Steinberg module. The module \(\text{St}_r\) is simple as a \(G\)-module and projective (and injective) over \(G_r\). Given a finite-dimensional rational \(G\)-module \(M\), the author has a long-standing conjecture on a condition for the tensor product \(\text{St}_r\otimes M\) to admit a good filtration. While investigating that conjecture, \textit{D. Nakano} et al. [``On tensoring with the Steinberg module'', Transform. Groups, to appear, \url{arXiv:1804.00613}] raised the question of whether the module \(…
reviewed by: Christopher P. Bendel
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 16W60
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 16G30
Mathematics Subject Classification ID: 17B01
zbMATH DE Number: 6988690
zbMATH Keywords: algebraic group
zbMATH Keywords: infinitesimal subgroup
zbMATH Keywords: good filtration
zbMATH Keywords: tilting module
zbMATH Keywords: coordinate algebra
zbMATH Keywords: Steinberg module
zbMATH Keywords: general linear group
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2018.11.001
OpenAlex ID: W2899260658
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Wikidata QID: Q128969407
DOI: 10.1016/J.JALGEBRA.2018.11.001
description: scientific article; zbMATH DE number 6988690