On the Cohomology of complements of complex reflection arrangements

几十年来，超平面排列理论一直是数学的推动力。它自然地处于代数、组合学、代数几何、表示理论和拓扑学的十字路口。这一建议反过来又处于这些问题的核心。近年来，在排列A的补M(A)的拓扑结构、A的导数模D(A)的自由度以及由A的超平面的相交产生的子空间组成的A的交格L(A)的组合学之间，已经发现了深刻而显著的联系。直到今天，它还是一个非常活跃的研究领域。通常，与反射排列A相关的超平面排列中的问题首先出现在对称群中，然后扩展到剩余的有限Coxeter群，最后包括整个复反射群类。这种现象的一个主要例子是关于反射排列A中超平面并集的补M(A)的拓扑性质的问题，该问题在贝西斯（Bessis）经过50多年的发展后于2015年得到解决。在这个研究计划中，我们穿越一个类似的路线，关于一个复杂反射排列的补上同调的问题。在最近与Douglass和Pfeiffer的合作中，我们改进了Brieskorn对Coxeter排列a (W)的补上同调的研究。作为我们研究的结果，我们从2005年推导出一个关于有限Coxeter群w的ork - solomon代数的w不变量结构的由于Felder和Veselov的猜想。本建议的目的之一是研究Felder和Veselov猜想在更一般的复反射群情况下的类比。1986年，Lehrer和Solomon将W在orliko -Solomon代数上的表示描述为W是对称群时W中元素中心点的线性特征所导出的表示的和。他们推测在一般的考克斯特群体中存在这样的分解。事实上，在与Douglass和Pfeiffer的一系列联合论文中，这个猜想的一个改进版本被建立在对称群和所有不可约的Coxeter群W上，直到秩8。在我们的第二个研究链中，我们的目标是研究更一般类的复杂反射群的Lehrer-Solomon猜想的模拟。

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.