Overgroups of distinguished unipotent elements in reductive groups

这项提议是在代数群论的核心领域，在代数、表示理论和几何不变量理论的交叉学科交叉路口上提出的。这是对约化代数群的子群结构研究的一个贡献。具体地说，这一建议的目的是将最近与Bate和Martin关于正则幂等元的上群的主要结果和Korhonen的工作推广到还原群的可区别的酉元的约化上群(两者都可能是非连通的).借助于Bala-Carter理论的中心重要性，还原群G的可区别的酉元及其李代数Lie(G)的可区别的幂零元分别在描述和研究G中的酉类和Lie(G)中的幂零G-轨道中起着关键作用.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理论中也是重要的。因此，对G中这类元素的约化超群H的性质的理解是特别有趣的。我们的目的是证明在适当的自然条件下，在这个背景下，这样的子群H是非常特殊的，因为它们在J-P定义的意义下是G-不可约的。Serre，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，在我们以前的论文中，用于推导主要结果的许多机械，其中处理了正则幂等元的超群H的特殊情况，可以在这个更一般的背景下使用，以解决关于G-不可约性的相同问题。事实上，G的正则幂等元的约化超群的许多关键性质与其对应的可区别的幂等元的超群是平行的。中心是G-完全可约性和最优抛物子群的概念.然而，我们需要强调的是，这种推广远非仅仅是例行公事，因为Steinberg和Spalstein工作中关于正则幂等元的强大的枢轴定理在这种情况下是不可用的.

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.