Existence of irreducible R-regular elements in Zariski-dense subgroups
Existence of irreducible R-regular elements in Zariski-dense subgroups
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Zariski 密集子群中不可约 R 正则元素的存在
DOI：
10.4310/mrl.2003.v10.n1.a3
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发表时间：
2003
影响因子：
1
通讯作者：
A. Rapinchuk
中科院分区：
文献类型：
--
作者：
Gopal Prasad;A. Rapinchuk
设G是定义在真实的数域R上的连通半单代数群. G（R）的一个元素x称为R-正则的，如果Adx的模1的特征值的重数是最小可能的。(If G是R-各向异性的，即，群G（R）是紧的，G（R）的每个元素都是sR正则的。证明了G（R）的任意子半群Γ中R-正则元的存在性，且该子半群是G中的Zagliki-稠密的. Benoist和F. Labourie [3]利用Oseledet的乘法遍历定理，然后由第一作者[15]用直接论证加以反驳。最近，G。马古利斯和GA Soifer问了我们一个问题，这个问题是在他们与H的联合工作中提出的。Abels关于Auslander问题，关于具有某些特殊性质的R-正则元的存在性。本说明的目的是对他们的问题作出肯定的答复。在形成结果之前，我们回顾（cf. [16]，注1.6（1）），R-正则元x必然是半单的，所以如果它是正则的，则T：= ZG（x）≠是极大环面;而且，x属于T（见[4]，推论11.12）。
Let G be a connected semisimple algebraic group defined over the field R of real numbers. An element x of G(R) is called R-regular if the number of eigenvalues, counted with multiplicity, of modulus 1 of Ad x is minimum possible. (If G is R-anisotropic, i.e., the group G(R) is compact, every element of G(R )i sRregular.) The existence of R-regular elements in an arbitrary subsemigroup Γ of G(R) which is Zariski-dense in G was proved by Y. Benoist and F. Labourie [3] using Oseledet’s multiplicative ergodic theorem, and then reproved by the firstnamed author [15] by a direct argument. Recently G.A. Margulis and G.A. Soifer asked us a question, which arose in their joint work with H. Abels on the Auslander problem, about the existence of R-regular elements with some special properties. The purpose of this note is to answer their question in the affirmative. Before formulating the result, we recall (cf. [16], Remark 1.6(1)) that an R-regular element x is necessarily semisimple, so if in addition it is regular, then T := ZG(x) ◦ is a maximal torus; moreover, x belongs to T (see [4], Corollary 11.12).