论文标题
$ \ ell $ -compact群和Spets的重量猜想
Weight conjectures for $\ell$-compact groups and spetses
论文作者
论文摘要
更确切地说，有限群体的模块化表示理论的基本猜想，阿尔珀林的体重猜想和鲁滨逊的普通重量猜想可以用融合系统来表达。我们使用融合系统将谎言类型有限群体的模块化表示理论与$ \ ell $ -compact组的理论联系起来。在某些温和的条件下，我们证明与$ \ ell $ -compact组相关的融合系统满足了一个方程，该方程式对于有限的谎言类型组等于Alperin的重量猜想。对于有限的还原组，鲁滨逊的普通重量猜想与卢斯蒂格（Lusztig）的约旦（Jordan）角色分解以及Brauer $ \ ell $ -Blocks的相应结果密切相关。在此激励的情况下，我们使用与$ \ ell $ -compact组相关的融合系统定义了连接到Spsitial $ {\ Mathbb Z} _ \ Ell $反射组的SPES的主要块，并制定了Robinson对此块的构想的类似物。我们证明了无限案件家族以及某些特殊类型的组的表述。我们的结果不仅为重量猜想的有效性提供了进一步的有力证据，而且还指出了纯粹在融合系统框架中的一些但未知的结构解释。
Fundamental conjectures in modular representation theory of finite groups, more precisely, Alperin's Weight Conjecture and Robinson's Ordinary Weight Conjecture, can be expressed in terms of fusion systems. We use fusion systems to connect the modular representation theory of finite groups of Lie type to the theory of $\ell$-compact groups. Under some mild conditions we prove that the fusion systems associated to homotopy fixed points of $\ell$-compact groups satisfy an equation which for finite groups of Lie type is equivalent to Alperin's Weight Conjecture. For finite reductive groups, Robinson's Ordinary Weight Conjecture is closely related to Lusztig's Jordan decomposition of characters and the corresponding results for Brauer $\ell$-blocks. Motivated by this, we define the principal block of a spets attached to a spetsial ${\mathbb Z}_\ell$-reflection group, using the fusion system related to it via $\ell$-compact groups, and formulate an analogue of Robinson's conjecture for this block. We prove this formulation for an infinite family of cases as well as for some groups of exceptional type. Our results not only provide further strong evidence for the validity of the weight conjectures, but also point toward some yet unknown structural explanation for them purely in the framework of fusion systems.
