Prehomogeneous Spaces for Parabolic Group Actions in Reductive Algebraic Groups

代数群在向量空间和代数簇上的作用在纯数学和应用数学的许多分支中起着重要的作用。在这些群作用中，还原群的作用是最好理解的。下一个自然类由约化群的抛物子群构成。抛物群在其幂幺根的李代数上的作用在某种意义上与约化群的伴随作用一样基本。我们打算研究关于这一作用的特殊几何问题，更具体地说，对于约化群的给定抛物子群P，我们打算分类Lie P的诣零根的所有P-子模，它们允许一个稠密P-轨道。对于一般的线性群，我们希望利用表示论的方法来重新表述这一问题，即对于某些拟线性群，我们希望利用特定类型的模的存在性来解决这一问题。遗传代数。然后，不同的技术应该允许我们从一般群的解推导出其他经典群的问题的解。最后，我们计划通过研究E7和E8型群的开放情形，完成单代数群的所有抛物子群的诣零根的降中心列的成员之间的所有有限轨道模的分类。

Actions of algebraic groups on vector spaces and algebraic varieties play an important role in many branches of pure and applied mathematics. Among these groups actions, the actions of reductive groups are the best understood. The next natural class is formed by parabolic subgroups of reductive groups. The action of a parabolic group on the Lie algebra of its unipotent radical is in some sense as fundamental as the adjoint action of a reductive group. We intend to study particular geometric questions about this action.More specifically, for a given parabolic subgroup P of a reductive group we intend to classify all P-submodules of the nilradical of Lie P which admit a dense P-orbit. Our strategies for obtaining solutions of this problem naturally vary depending on the type of the ambient reductive group.For general linear groups we hope to employ a representation theoretic reformulation of this problem in terms of the existence of particular types of modules for certain quasi-hereditary algebras.Different techniques should then allow us to deduce the solution of our problem for the other classical groups from the solution for the general linear groups.We hope to address this problem for the groups of exceptional type by computational means.Finally, we plan to complete the classification of all finite orbit modules among the members of the descending central series of the nilradicals of all parabolic subgroups of simple algebraic groups by studying the open cases for groups of type E7 and E8.