Overgroups of distinguished unipotent elements in reductive groups

还原基团中杰出单能元素的超群

• 批准号: 498503969
• 负责人: Professor Dr. Gerhard Röhrle
• 金额: --
• 依托单位: Lehrstuhl Algebra/Zahlentheorie
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/498503969?language=en
• 关键词: Overgroups; distinguished; unipotent; elements; reductive

The proposal is in a core area of algebraic group theory and at the interdisciplinary cross roads of algebra, representation theory, and geometric invariant theory. It is a contribution to the study of the subgroup structure of reductive algebraic groups. Specifically, the aim of this proposal is to generalize the main results from recent joint work with Bate and Martin on overgroups of regular unipotent elements and work of Korhonen to reductive overgroups of distinguished unipotent elements of reductive groups (both possibly non-connected).By virtue of the central importance of Bala-Carter Theory, distinguished unipotent elements of reductive groups G and distinguished nilpotent elements of their Lie algebras Lie(G) play a pivotal role in the description and study of unipotent classes in G and nilpotent G-orbits in Lie(G), respectively. In turn, nilpotent orbits in positive characteristic are important in representation theory of finite groups of Lie type and reduced enveloping algebras. Therefore, an understanding of the properties of reductive overgroups H of such elements in G is of particular interest. We aim to show that under suitable natural circumstances in this setting such subgroups H are very special in that they are G-irreducible in the sense defined by J-P. Serre, that is, they are not contained in any proper parabolic subgroup of G. We intend to demonstrate that much of the machinery used to derive the principal results in our earlier paper, where the special case of overgroups H of regular unipotent elements is handled, can be employed in this more general setting to address the same question about G-irreducibility. Indeed, many of the key properties of reductive overgroups of regular unipotent elements of G are paralleled for their counterpart overgroups of distinguished unipotent elements. Central are the concepts of G-complete reducibility and optimal parabolic subgroups.However, we need to stress that this extension is far from mere routine, as the powerful pivotal theorems for regular unipotent elements from work of Steinberg and Spaltenstein are not available in this setting.

该建议是在一个核心领域的代数群理论和交叉学科的道路代数，表示理论和几何不变理论。这是对约化代数群的子群结构研究的一个贡献。具体地说，这个建议的目的是将最近与Bate和Martin关于正则幂幺元的扩群以及Korhonen的工作的主要结果推广到约化群的特殊幂幺元的约化扩群（两者都可能是非连接的）。凭借巴拉-卡特理论的核心重要性，约化群G的特殊幂幺元及其李代数的特殊幂零元Lie（G）在描述和研究G中的幂幺类和幂零类G中起着关键的作用。Lie（G）中的轨道。反过来，正特征的幂零轨道在李型有限群和约化包络代数的表示论中也很重要。因此，了解G中这些元素的还原扩群H的性质是特别有意义的。我们的目的是证明，在适当的自然条件下，这类子群H是非常特殊的，因为它们在J-P. Serre定义的意义下是G-不可约的，也就是说，它们不包含在G的任何真抛物子群中. 我们打算证明，在我们早先的论文中，处理了正则幂幺元的扩群H的特殊情况，用于导出主要结果的许多机制，可以在这个更一般的设置中使用，以解决关于G-不可约性的相同问题。事实上，G的正则幂幺元的约化扩群的许多关键性质对于它们的对应的杰出幂幺元的扩群来说是不确定的。中心是G-完全约化和最优抛物子群的概念。然而，我们需要强调的是，这种扩展远不是简单的常规，因为Steinberg和Spalbergy工作中关于正则幂幺元的强大的关键定理在这种情况下是不可用的。