Unipotent and nilpotent classes in characteristic two

我们的建议涉及到单代数群和李代数中的幂零类和幂零类。幂等元是李型代数群和有限群理论的基础，在结构理论和表示理论中都起着重要的作用。一般的幂等元理论的某些部分(如Steinberg、Springer、Richardson、Carter和Spalstein的结果)相当漂亮，而另一些部分则处于不令人满意的状态。例如，文献中确实存在例外Lie类型群的共轭类的基本列表和幂等元的中心化子阶，但结果散布在许多论文中，使用各种技巧和符号，大多基于大量的逐个案例分析，并且几乎没有提供整体的概念理解。此外，Holt和Spalstein使用重型机器计算对具有不良特征的幂零类进行的分类，留下了一些基本问题没有得到回答，例如中心化的完整结构。这是一个需要重大修改和发展的领域，这是我们的目标。在2004-2006年的一系列访问中，在我们之前的拨款的资助下，Seitz和我开发了一个新的统一方法来研究简单代数群和李代数中的幂零类，假设基本特征不是2。我们在这个项目中的主要目标是当基础特征为2时执行我们的新的统一方法。这将需要在我们以前关于奇数特征的工作的基础上有实质性的新想法。例如，即使对于经典李代数，特征2中仍然有一些基本的未回答的问题，例如幂零元素的中心化的精确结构。

Our proposal concerns the unipotent and nilpotent classes in simple algebraic groups and Lie algebras. Unipotent elements are fundamental to the theory ofalgebraic groups and finite groups of Lie type, and play an important role inboth the structure and representation theory. Some parts of the generaltheory of unipotent elements (e.g. results of Steinberg, Springer, Richardson, Carter, and Spaltenstein) are quite beautiful, while other parts are in anunsatisfactory state. For example, basic lists of conjugacy classesand centralizer orders of unipotent elements of groups of exceptional Lie type do exist in the literature, but the results are spread overmany papers using a variety of techniques and notations, mostly based on a massive case-by-case analysis and offering little overall conceptual understanding. Moreover, the classification of nilpotent classes in bad characteristics carried out by Holt and Spaltenstein using heavy machine computation, leaves some basic questions unanswered, such as the full structure of the centralizers. This is an area that is in need of major revision and development, and it is our goal to carrythis out. In a series of visits in 2004-6, funded by our previous grant, Seitz and I developed a new and unified approach to the unipotent and nilpotent classes in simple algebraic groups and Lie algebras, assuming that the underlying characteristic is not 2. Our main goal in this project is to carry out our new unified approach whenthe underlying characteristic is 2. This will require substantial new ideas on top of our previous work for odd characteristics. For example, even for classical Lie algebras there are still basic unanswered questions in characteristic 2, such as the precise structure of the centralizers of nilpotent elements.