On the Cohomology of complements of complex reflection arrangements

点击翻译按钮获取中文摘要

The theory of hyperplane arrangements has been a driving force in mathematics over many decades. It naturally lies at the crossroads of algebra, combinatorics, algebraic geometry, representation theory and topology. This proposal in turn lies at the very heart of these subject matters. Deep and remarkable connections have been discovered over the years between the topology of the complement M(A) of an arrangement A, the freeness of the module of derivations D(A) of A and the combinatorics of the intersection lattice L(A) of A consisting of the subspaces arising as intersections of hyperplanes from A.The study of the complement of complex hyperplane arrangements has a long and remarkably rich history. To this day it is a very active field of research. Frequently, questions in hyperplane arrangements relating to reflection arrangements A, where A consists of the reflecting hyperplanes of an underlying reflection group, arose first for symmetric groups, then were extended to the remaining finite Coxeter groups and finally embraced the entire class of complex reflection groups. A prime example of this phenomenon is the question about the topological nature of the complement M(A) of the union of the hyperplanes in the reflection arrangement A which was settled after a development streching more than 50 years by Bessis in 2015.In this research proposal we traverse a similar route concerning questions on the cohomology of the complement of a complex reflection arrangement. In recent joint work with Douglass and Pfeiffer we refined Brieskorn's study of the cohomology of the complement of a Coxeter arrangement A(W). As a result of our study we derive a conjecture due to Felder and Veselov from 2005 on the structure of the W-invariants of the Orlik-Solomon algebra of a finite Coxeter group W.One of the aims of this proposal is to investigate an analogue of the conjecture of Felder and Veselov for the more general case of complex reflection groups.In 1986 Lehrer and Solomon have described the representation of W on the Orlik-Solomon algebra of W as a sum of representations induced from linear characters of centralizers of elements in W when W is a symmetric group. They have conjectured that there is such a decomposition for general Coxeter groups. Indeed a refined version of this conjecture was established in a series of joint papers with Douglass and Pfeiffer for symmetric groups and all irreducible Coxeter groups W up to rank 8. In our second research strand we aim to investigate an analogue of the Lehrer-Solomon Conjecture for the more general class of complex reflection groups.