A Classification of the Representations of Reductive Algebraic Groups Which Admit Only a Finite Number of Orbits
A Classification of the Representations of Reductive Algebraic Groups Which Admit Only a Finite Number of Orbits
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只承认有限轨道数的还原代数群表示的分类
DOI：
10.2307/2374658
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发表时间：
1986
影响因子：
1.7
通讯作者：
Osami Yasukura
中科院分区：
文献类型：
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作者：
Tatsuo Kimura;S. Kasai;Osami Yasukura
导论.设p '：G' -* GL（V）是定义在C上的半单代数群G '的有限维有理表示.那么，我们有p' = Pi？）... (Epe V= V1I？其中pi是G'在Vi上的不可约表示，i = 1，. . ., F .设p是G = GL（1）IXG '对V的作用，V是p'与标量乘GL（1）Ion的合成，每个不可约分支Vi（i = 1，. . .,（f）。本文将对所有的三元组（G，p，V）进行分类，使得V分解为G-轨道的有限并。V.G.教授Kac（[3]，[4]）和第一作者（[7]，[14]，但[14]中只写了必要条件）已经独立地完成了p不可约的情况，即，f = 1. V.G.教授Kac还完成了每个不可约分支为（GL（n）× GL（m），Al？Al，V（n）？V（m））（见[5]），它是P. Gabriel教授（[1]）结果的发展。注意，如果V分解为G-轨道的有限并，则其中一个轨道必须是Zerkiki稠密的，因此（G，p，V）必须是预齐次向量空间（焦v. P. V.）。我们称这样的向量空间为有限预齐次向量空间（fixvFP）。本文件由以下几个部分组成。
Introduction. Let p': G' -* GL(V) be a finite-dimensional rational representation of a semi-simple algebraic group G', all defined over C. Then, we have p' = Pi ?) ... (Epe and V= V1I ?D * ** Ve where pi is an irreducible representation of G' on Vi for i = 1, . . ., f . Let p be the action of G = GL(1) I X G' on V which is the composition of p' and scalar multiplications GL(1) Ion each irreducible component Vi (i = 1, . . ., f). In this paper, we shall classify all the triplet (G, p, V) such that V decomposes into a finite union of G-orbits. Prof. V. G. Kac ([3], [4]) and the first author ([7], [14], but only the necessary conditions are written in [14]) have already completed independently the case when p is irreducible, i.e., f = 1. Prof. V. G. Kac has also completed the case when each irreducible components are (GL(n) X GL(m), Al ? Al, V(n) ? V(m)) (see [5]) which is the development of the results of Prof. P. Gabriel ([1]). Note that if V decomposes into the finite union of G-orbits, then one of the orbits must be Zariski-dense, and hence (G, p, V) must be a prehomogeneous vector space (abbrev. P. V.). We call such a P. V. a finite prehomogeneous vector space (abbrev. F.P.). This paper consists of the following sections.
预齐次向量空间的有理轨道
DOI：
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发表时间：
2020
期刊：
Algebraic number theory and related topics 2016, RIMS K\^{o}ky\^{u}roku Bessatsu
影响因子：
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作者：
駒場敦;城野悠志;中本和典;山崎愛一;A. Yukie
通讯作者：
A. Yukie