Unipotent and nilpotent classes in characteristic two

我们的建议涉及单代数群和李代数中的幂单类和幂零类。幂幺元是代数群和李型有限群理论的基础，在结构论和表示论中都起着重要的作用。幂幺元的一般理论的某些部分（如Steinberg，Springer，Richardson，Carter和Spaleen的结果）是相当漂亮的，而另一些部分则处于不令人满意的状态。例如，例外李型群的幂幺元的共轭类和中心化子阶的基本列表确实存在于文献中，但结果散布在许多论文中，使用各种技术和符号，大多基于大量的逐案分析，并提供很少的整体概念理解。此外，Holt和Spaleans使用重型机器计算对具有不良特征的幂零类进行了分类，留下了一些基本问题没有回答，例如中心化器的完整结构。这是一个需要进行重大修改和发展的领域，我们的目标是实现这一目标。在2004年至2006年的一系列访问中，由我们以前的资助，塞茨和我开发了一种新的统一方法来研究简单代数群和李代数中的幂单和幂零类，假设基本特征不是2。我们在这个项目中的主要目标是实现我们新的统一方法，当基本特征是2。这就需要在我们以前对奇异特性的研究基础上，有大量的新想法。例如，即使对于经典李代数，在特征2中仍然有一些基本的未解问题，例如幂零元的中心化子的精确结构。

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.