Hecke Algebras and Complex Reflection Groups

复反射群是由反射生成的复向量空间的变换的有限群（即，一些人最近的工作揭示了这些群与真实的反射群，甚至Weyl群之间令人惊讶的相似之处：具体地说，它们产生了许多表示理论，如Hecke代数和一般度，即使在背景中没有代数群的表示理论被描述。 布鲁和其他人已经证实了神秘的，未知的物体的存在，称为“斯佩策”作为这些现象的来源。 这个项目的目的是扩展这个观点：一个具体的目标是为复杂的反射群构建一个类似的Springer对应。 我们的方法应该能对复反射群的Heck代数的某些结构给出统一的解释，也许还能指出一条发展一个研究这些群的一般框架的道路，以Coxeter理论的精神。一个方阵称为反射矩阵，如果（a）它的某个幂是单位矩阵，（B）它的特征值除一个外都是1.反射群是一个有限的矩阵群，其中每个元素都可以写成也在群中的反射矩阵的乘积. 实值矩阵的反射群在20世纪30年代首先由Coxeter深入研究，长期以来在数学的许多领域中发挥着重要作用：也许最重要的是，一大类李群的结构和表示密切地受某些真实的反射群（所谓的Weyl群）的支配。 与此相反，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和令人惊讶的平行性。 在这个项目中，我们建议进一步发展与实反射群的类比，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，我们希望能为这些现象的统一解释指明道路，也许还能为复杂反射群本身的结构理论指明道路。

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.