Hecke Algebras and Complex Reflection Groups

复反射群是由反射生成的复向量空间的有限变换群（即，定定点超平面的有限阶变换）。最近一些人的工作揭示了这些群与真实的反射群，甚至Weyl群之间惊人的相似之处：具体来说，它们产生了许多表征理论对象，如Hecke代数和一般度，尽管背景中没有描述表征理论的代数群。布鲁和其他一些人推测，这些现象的来源是被称为“物种”的神秘未知物体的存在。这个项目的目的是扩展这个观点：一个特定的目标是为复杂的反射群构建一个类似的Springercorrespondence。我们的技术应该能够对来自复杂反射群的hecke代数的某些结构给出统一的解释，或许还能根据科克斯特理论的精神，为研究这些群指明一条发展一般框架的道路。如果(A)一个方阵的某个幂是单位矩阵，并且(b)它的特征值除了一个以外都是1，那么这个方阵就叫做反射矩阵。反射群是矩阵的有限群，其中的每个元素都可以写成群中反射矩阵的乘积。实值矩阵的反射群，最早由Coxeter在20世纪30年代进行了深入研究，长期以来在数学的许多领域发挥着至关重要的作用：也许最重要的是，一大批李群的结构和表示与某些实反射群（所谓的Weylgroups）密切相关。相比之下，复值矩阵的反射群的研究要少得多，但最近一些人的工作表明，它们与真实的反射群，甚至与Weyl群，表现出许多不同的、令人惊讶的相似之处。在这个项目中，我们建议进一步发展与真实反射群的类比，特别是通过研究某些称为Hecke代数的相关对象的结构。在这个过程中，我们希望能找到对这些现象的统一解释，也许能找到复杂反射群本身的结构理论。

A complex reflection group is a finite group of transformations of acomplex vector space that is generated by reflections (i.e.,finite-order transformations that fix some hyperplane pointwise).Recent work by a number of people has revealed surprising parallelsbetween these groups and real reflection groups, or even Weyl groups:specifically, they give rise to a number of representation theoreticobjects, such as Hecke algebras and generic degrees, even though thereis no algebraic group in the background whose representation theory isbeing described. Broue and others have conjectured the existence ofmysterious, unknown objects called "spetses" as the source of thesephenomena. The aim of this project is to extend this view: onespecific goal is to construct an analogue of the Springercorrespondence for complex reflection groups. Our techniques ought toyield uniform accounts of certain structures coming from Heckealgebras of complex reflection groups, and perhaps point the way todeveloping a general framework in which to study these groups, in thespirit of Coxeter theory.A square matrix is called a reflection matrix if (a) some power of itis the identity matrix, and (b) all but one of its eigenvalues is 1.A reflection group is a finite group of matrices in which everyelement can be written as a product of reflection matrices that arealso in the group. Reflection groups of real-valued matrices, firststudied in depth by Coxeter in the 1930's, have long played a vitalrole in many areas of mathematics: perhaps most importantly, thestructure and representations of a large class of Lie groups areclosely governed by certain real reflection groups (the so-called Weylgroups). Reflection groups of complex-valued matrices, in contrast,are much less well-studied, but recent work by a number of people hasshown that they exhibit a number of disparate and surprising parallelswith real reflection groups, or even with Weyl groups. In thisproject, we propose to further develop the analogy with realreflection groups, specifically by studying the structure of certainrelated objects called Hecke algebras. In the process, we hope thepoint the way to a uniform explanation for these phenomena, andperhaps to a structure theory for the complex reflection groupsthemselves.