{"title":"Geometric rigidity of simple modules for algebraic groups","authors":"Michael Bate, David I. Stewart","doi":"arxiv-2409.05221","DOIUrl":null,"url":null,"abstract":"Let k be a field, let G be a smooth affine k-group and V a finite-dimensional\nG-module. We say V is \\emph{rigid} if the socle series and radical series\ncoincide for the action of G on each indecomposable summand of V; say V is\n\\emph{geometrically rigid} (resp.~\\emph{absolutely rigid}) if V is rigid after\nbase change of G and V to \\bar k (resp.~any field extension of k). We show that\nall simple G-modules are geometrically rigid, though not in general absolutely\nrigid. More precisley, we show that if V is a simple G-module, then there is a\nfinite purely inseparable extension k_V/k naturally attached to V such that\nV_{k_V} is absolutely rigid as a G_{k_V}-module. The proof for connected G\nturns on an investigation of algebras of the form K\\otimes_k E where K and E\nare field extensions of k; we give an example of such an algebra which is not\nrigid as a module over itself. We establish the existence of the purely\ninseparable field extension k_V/k through an analogous version for artinian\nalgebras. In the second half of the paper we apply recent results on the structure and\nrepresentation theory of pseudo-reductive groups to gives a concrete\ndescription of k_V. Namely, we combine the main structure theorem of the\nConrad--Prasad classification of pseudo-reductive G together with our previous\nhigh weight theory. For V a simple G-module, we calculate the minimal field of\ndefinition of the geometric Jacobson radical of \\End_G(V) in terms of the high\nweight of G and the Conrad--Prasad classification data; this gives a concrete\nconstruction of the field k_V as a subextension of the minimal field of\ndefinition of the geometric unipotent radical of G. We also observe that the Conrad--Prasad classification can be used to hone\nthe dimension formula for G we had previously established; we also use it to\ngive a description of \\End_G(V) which includes a dimension formula.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"57 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.05221","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Let k be a field, let G be a smooth affine k-group and V a finite-dimensional
G-module. We say V is \emph{rigid} if the socle series and radical series
coincide for the action of G on each indecomposable summand of V; say V is
\emph{geometrically rigid} (resp.~\emph{absolutely rigid}) if V is rigid after
base change of G and V to \bar k (resp.~any field extension of k). We show that
all simple G-modules are geometrically rigid, though not in general absolutely
rigid. More precisley, we show that if V is a simple G-module, then there is a
finite purely inseparable extension k_V/k naturally attached to V such that
V_{k_V} is absolutely rigid as a G_{k_V}-module. The proof for connected G
turns on an investigation of algebras of the form K\otimes_k E where K and E
are field extensions of k; we give an example of such an algebra which is not
rigid as a module over itself. We establish the existence of the purely
inseparable field extension k_V/k through an analogous version for artinian
algebras. In the second half of the paper we apply recent results on the structure and
representation theory of pseudo-reductive groups to gives a concrete
description of k_V. Namely, we combine the main structure theorem of the
Conrad--Prasad classification of pseudo-reductive G together with our previous
high weight theory. For V a simple G-module, we calculate the minimal field of
definition of the geometric Jacobson radical of \End_G(V) in terms of the high
weight of G and the Conrad--Prasad classification data; this gives a concrete
construction of the field k_V as a subextension of the minimal field of
definition of the geometric unipotent radical of G. We also observe that the Conrad--Prasad classification can be used to hone
the dimension formula for G we had previously established; we also use it to
give a description of \End_G(V) which includes a dimension formula.
设 k 是一个域,G 是一个光滑仿射 k 群,V 是一个有限维 G 模块。如果 G 对 V 的每个不可分解和子的作用的索序列和根序列一致,我们就说 V 是 \emph{刚性的(rigid);如果在把 G 和 V 改为 \bar k(respect.~任何 k 的域扩展)之后,V 仍然是刚性的,我们就说 V 是 \emph{几何刚性的(geometrically rigid)(respect.~\emph{绝对刚性的(absolutely rigid))。我们证明了所有简单 G 模块都是几何刚性的,尽管一般来说不是绝对刚性的。更确切地说,我们证明了如果 V 是一个简单 G 模块,那么有一个无限的纯不可分的扩展 k_V/k 自然地连接到 V,使得 V_{k_V} 作为 G_{k_V} 模块是绝对刚性的。对连通 G 的证明依赖于对 K/otimes_k E 形式的代数的研究,其中 K 和 E 都是 k 的域扩展;我们给出了这样一个代数的例子,它作为自身的模块是不刚性的。我们通过对artinian代数的类似版本,建立了纯不可分场扩展k_V/k的存在性。在论文的后半部分,我们应用伪还原群的结构和表示理论的最新成果,给出了 k_V 的具体描述。也就是说,我们将伪还原 G 的康拉德--普拉萨德分类的主要结构定理与之前的高权重理论结合起来。对于简单的 G 模块 V,我们根据 G 的高权重和康拉德--普拉萨德分类数据计算了 \End_G(V) 的几何雅各布森根的最小定义域;这给出了 k_V 作为 G 的几何单能根的最小定义域的子扩展的具体构造。我们还观察到康拉德--普拉萨德分类法可以用来兑现我们之前建立的 G 的维度公式;我们还用它给出了包含维度公式的 \End_G(V) 的描述。