Reductive Groups in Positive Characteristic

该项目将研究特征为素数p的一个可能不完美的地面域上的约化线性代数群；这种设置提出了许多重要而有趣的挑战。作者最近利用几何不变理论中的一个结果，给出了一类合适的1阶简单子群的确定存在性和共轭性结果，称为最优，包含任何给定的p阶元素。如果这个元素在一个地面域上是有理的，则该技术产生了在该地面域上定义的最优1阶子群。该项目力求扩大这些成果；在J-P意义上，最优秩1子群是g -完全可约的。我们希望用最优子群来描述任何g -完全可约的秩1子群。此外，该项目将考虑秩大于1的简单子组的这些结果的对立物-例如，希望表征最优子组。在一个稍微不同的方向上，该项目寻求将作者最近关于幂零轨道的结果——在李代数中——扩展到地面场。对于不完全地场，作者证明了对于一类相当一般的约化群，幂零元上有理点群的轨道——即算术幂零轨道——具有良好的性质；作为结果，例如，我们发现在局部地场上有有限多个算术幂零轨道，我们发现幂零轨道积分收敛；Deligne和Ranga Rao在特征0处得到了轨道积分的收敛性。项目将探讨相关问题；例如，它试图利用作者和E. Sommers最近的结果来获得代数曲线上有理函数域上群的幂零轨道的信息。线性代数群的结构和表示——尤其是约化群——对数学的各个部分都很重要。它们的有理点在有限域上的群占有限单群的大多数；它们是许多代数几何问题的自然变换群；它们在局部和全局域上的有理点群的表示包含了深刻的数论信息。该项目将侧重于这些群体的重要方面。上述局部场和全局场在其特征为正的情况下是不完善的；作者现有的关于不完全域上的约化群的结果——以及项目所寻求的结果——对于在这种“数论”设置下研究线性群是必不可少的。解决这些问题的原因有很多：在具有正特征的局部场上的群的算术幂零轨道的结果为研究相关的“调和分析”提供了必要的工具；在地面场上定义的约化子群的结果有助于人们的理解并提供有用的归纳工具。

DMS-0437482George J. McNinch The project will study reductive linear algebraic groups over a, possibly imperfect, ground field whose characteristic is a prime number p; this setting presents many important and interesting challenges. The author recently exploited a result in geometric invariant theory to give definitive existence and conjugacy results for a suitable class of rank 1 simple subgroups, called optimal, containing any given element of order p. If this element is rational over a ground field, the techniques yield an optimal rank one subgroup defined over that ground field. The project seeks to extend these results; an optimal rank 1 subgroup is G-completely reducible in the sense of J-P. Serre, and one hopes to describe any G-completely reducible rank 1 subgroup using the optimal ones. Moreover, the project will consider counterparts of these results for simple subgroups of rank greater than 1 -- e.g. one hopes to characterize the optimal ones. In a slightly different direction, the project seeks to extend recent results of the author on nilpotent orbits -- in the Lie algebra -- over ground fields. For an imperfect ground field, the author has showed for a rather general class of reductive groups that the orbits of the group of rational points on nilpotent elements -- i.e. the arithmetic nilpotent orbits -- have favorable properties; as consequences, for instance, one finds over a local ground field that there are finitely many arithmetic nilpotent orbits, and one finds that nilpotent orbital integrals converge; the convergence of orbital integrals was obtained in characteristic 0 by Deligne and Ranga Rao. The project will explore related issues; for instance, it seeks to exploit recent results of the author and E. Sommers to obtain information about nilpotent orbits for groups over the field of rational functions on an algebraic curve.The structure and representations of linear algebraic groups - and especially the reductive ones -- are important to diverse parts of mathematics. The groups of their rational points over finite fields account for most of the finite simple groups; they are the natural transformation groups of many algebro-geometric questions; representations of the groups of their rational points over local and global fields carry deep number-theoretic information. The project will focus on important aspects of these groups. The local and global fields just mentioned are imperfect when their characteristic is positive; the author's existing results -- and those sought by the project -- on reductive groups over imperfect fields are essential to the study of linear groups in this "number theoretic" setting. Reasons for tackling the proposed problems abound: results on arithmetic nilpotent orbits for a group over a local field of positive characteristic should provide tools needed for the study of the relevant "harmonic analysis"; results on reductive subgroups defined over the ground field contribute to one's understanding and provide useful inductive tools.