Hecke Algebras and Complex Reflection Groups

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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.