Reflection Groups, Braid Groups, Hecke Algebras, Finite Reduction Groups
Reflection Groups, Braid Groups, Hecke Algebras, Finite Reduction Groups
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反射群、辫群、赫克代数、有限约简群
DOI：
10.4310/cdm.2000.v2000.n1.a1
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发表时间：
2000
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通讯作者：
M. Broué
中科院分区：
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作者：
M. Broué
由反射生成的GLn（Q）的有限子群，称为Weyl群，将简单复李群和简单代数群分类。他们还为许多其他重要的数学对象，如辫子群和Hecke代数奠定了基础。通过最近的工作表示约化群在有限领域的基础上乔治Lusztig的基本工作，并激发了有关模块表示一般有限群，它已变得越来越清楚，有限子群GLn（C）所产生的伪反射（“复反射群”）的行为非常像外尔群，甚至可能是重要的。我们在这里提出了一个串联的一些最近的工作（主要是由D。Bessis，G. Malle，J. Michel，R. Rouquier和作者）对复反射群、辫子群和Hecke代数的研究，强调了Weyl群的一般性质，推广了Weyl群的基本性质。在许多方面，有限域上具有q个元素的有限群G（q）族表现得好像它们是依赖于不确定x的对象在x = q处的特化。令人信服的指标往往表明，虽然复反射群不是外尔群，但它们不能定义有限域上的有限群，它们可能与类似的神秘对象有关。我们在此介绍了该机制的一些方面，以强调这一观点。我们用这个机器的状态一般的陈述有关的陈述有限还原群的进环，十年前，起源这项工作。1991年数学学科分类。小学16 G99、20 C20、20 C30、20 G 05;中学17 B67、05 E10。我感谢Gunter Malle、Jean Michel和Raphaël Rouquier多年来的长期合作，并允许我自由使用我们的共同作品[BrMa 1]、[BrMa 2]、[BMM 2]和[BMR]。我感谢大卫贝西斯、迈诺夫·凯克和乔治·卢斯蒂格富有成效的对话。c ©International Press，2001 1
Finite subgroups of GLn(Q) generated by reflections, known as Weyl groups, classify simple complex Lie Groups as well as simple algebraic groups. They are also building stones for many other significant mathematical objects like braid groups and Hecke algebras. Through recent work on representations of reductive groups over finite fields based upon George Lusztig’s fundamental work, and motivated by conjectures about modular representations of general finite groups, it has become clearer and clearer that finite subgroups of GLn(C) generated by pseudo–reflections (“complex reflection groups”) behave very much like Weyl groups, and might even be as important. We present here a concatenation of some recent work (mainly by D. Bessis, G. Malle, J. Michel, R. Rouquier and the author) on complex reflection groups, their braid groups and Hecke algebras, emphasizing the general properties which generalize basic properties of Weyl groups. By many aspects, the family of finite groups G(q) over finite fields with q elements behave as if they were the specialisations at x = q of an object depending on an indeterminate x. Convincing indices tend to show that, although complex reflection groups which are not Weyl groups do not define finite groups over finite fields , they might be associated to similar mysterious objects. We present here some aspects of the machinery allowing to emphasize this point of view. We use this machinery to state the general conjectures about representations of finite reductive groups over –adic rings which, ten years ago, originated this work . 1991 Mathematics Subject Classification. Primary 16G99, 20C20, 20C30, 20G05 ; Secondary 17B67, 05E10. I thank Gunter Malle, Jean Michel and Raphaël Rouquier for years of permanent cooperation, and for having allowed me to use freely our common works [BrMa1], [BrMa2], [BMM2] and [BMR]. I thank David Bessis, Meinolf Geck and George Lusztig for fruitful conversations. c ©International Press, 2001 1
DOI：
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发表时间：
2007
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作者：
F.R.Cohen;T.Kohno;M.A.Xicont'encatl;森田茂之;Toshitake Kohno
通讯作者：
Toshitake Kohno
白井三平：星号。
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