摘要翻译：
设G是特征p>0的代数闭域上的一个约化线性代数群。如果G的一个子群的全局和无穷小中心化子维数相同，则称它在G中是可分的。研究了可分性概念与Serre关于G的子群的G-完全可约性概念之间的相互作用。可分性假设出现在许多关于G-完全可约性的一般定理中。我们证明，如果没有这个假设，这些结果中的许多都是失败的。另一方面，我们证明了如果G是连通还原群，p对G是很好的，那么G的任何子群都是可分的；在这些假设下，如果G的李代数是半单的H-模，则G的子群H是G-完全可约的。最近，Guralnick证明了如果H是G的一个还原子群，C是G的一个共轭类，则C与H的交是H-共轭类的有限并。对于泛型p--当某些额外的假设成立时，包括可分性--这是从Richardson的一个众所周知的切空间论证中得出的，但一般而言，它基于Lusztig的深刻结果，即连通还原群只有有限多个单幂共轭类。我们证明了当考虑n元组的共轭类时，Guralnick的类似结果是假的。
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英文标题：
《Complete Reducibility and Separability》
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作者：
Michael Bate, Benjamin Martin, Gerhard Roehrle, and Rudolf Tange
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最新提交年份：
2008
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分类信息：

一级分类：Mathematics        数学
二级分类：Group Theory        群论
分类描述：Finite groups, topological groups, representation theory, cohomology, classification and structure
有限群、拓扑群、表示论、上同调、分类与结构
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一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p > 0. A subgroup of G is said to be separable in G if its global and infinitesimal centralizers have the same dimension. We study the interaction between the notion of separability and Serre's concept of G-complete reducibility for subgroups of G. The separability hypothesis appears in many general theorems concerning G-complete reducibility. We demonstrate that many of these results fail without this hypothesis. On the other hand, we prove that if G is a connected reductive group and p is very good for G, then any subgroup of G is separable; we deduce that under these hypotheses on G, a subgroup H of G is G-completely reducible provided the Lie algebra of G is semisimple as an H-module.   Recently, Guralnick has proved that if H is a reductive subgroup of G and C is a conjugacy class of G, then the intersection of C and H is a finite union of H-conjugacy classes. For generic p -- when certain extra hypotheses hold, including separability -- this follows from a well-known tangent space argument due to Richardson, but in general, it rests on Lusztig's deep result that a connected reductive group has only finitely many unipotent conjugacy classes. We show that the analogue of Guralnick's result is false if one considers conjugacy classes of n-tuples of elements from H for n > 1.
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PDF链接：
https://arxiv.org/pdf/0709.3803