On geometric invariant theory for infinite-dimensional groups
On geometric invariant theory for infinite-dimensional groups
复制标题
论无限维群的几何不变量理论
DOI：
10.1007/bfb0079235
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发表时间：
1987
期刊：
影响因子：
--
通讯作者：
D. Peterson
中科院分区：
文献类型：
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作者：
V. Kac;D. Peterson
导论.设G是有限维向量空间V上的复约化代数群，给定一个I-参数子群a：eX G，若lima（t）· v= O，则称点veV为a-不稳定的.对于某个α不稳定的所有点的集合NO包含在零锥N={ve VIO E IT 7V}中。希尔伯特-芒福德定理说，实际上，N= N 0，肯普夫[10]对此进行了详细阐述。他构造了一个G-等变映射：其中PN表示N的射影化，表示G的所有真抛物子群的集合，G在这些真抛物子群上共轭作用。给定ev e PN，其相应的抛物子群的极大约化子群的集合？(ev)由所有的”最佳”的I-参数子群的中心化，其中v是a-不稳定的;？(ev)是由这个唯一决定的。他的工作的主要目的是显示存在的一个领域的定义。然而，人们直接从映射的存在性得到另一个推论：任何ev e PN的稳定子包含在G的真抛物子群中（即，在？(ev».设G（A）是与A相关联的复Kac-Moody群，
Introduction. Let G be a complex reductive algebraic group operating on a finite-dimensional vector space V. Given a I-parameter subgroup a: eX G, a point v e V is called a-unstable if lim a (t)· v= O. The set NO of all points which are a-unstable for some a is contained in the null-cone N={ve VIO E IT7V}. The Hilbert-Mumford theorem says that, in fact, N= NO'Kempf [lOJ elaborated on this. He constructed a G-equivariant map?: PN where PN denotes the projectivization of N and denotes the set of all proper parabolic subgroups of G, on which G acts by conjugation. Given ev e PN, the set of maximal reductive subgroups of the associated parabolic subgroup?(ev) consists of the centralizers of all" optimal" I-parameter subgroups a for which v is a-unstable;?(ev) is uniquely determined by this. The main purpose of his work was to show the existence of a over the field of definition. However, one obtains immediately another corollary, just from the existence of the map: the stabilizer of any ev e PN is contained in a proper parabolic subgroup of G (namely, in?(ev». Now let G (A) be the complex Kac-Moody group associated to a