关于 Vinberg theta-群的 Jordan 类
On Jordan classes for Vinberg's theta-groups
AI总结：
本文局部研究了 Vinberg theta-群作用的 Jordan 类闭包，证明其光滑性及闭包可分解性，并利用小 Weyl 群子群对类与轨道进行参数化，揭示了与对称情形的差异。
摘要：
Popov 最近为 theta-群 (G_0,V) 的作用引入了 Jordan 类（包，或分解类）的类似物，证明了它们是有限个、局部闭、不可约的 G_0-轨道并集，且具有恒定维数并对 V 进行划分。我们对其闭包进行了局部研究，证明 Jordan 类是光滑的，且其闭包是 Jordan 类的并集。我们利用 Vinberg 的小 Weyl 群的子群作用对 Jordan 类及给定类中的 G_0-轨道进行了参数化，并列举了多个例子和反例，以说明与对称情形的差异以及 theta 情形中出现的关键问题。
英文摘要：
Popov has recently introduced an analogue of Jordan classes (packets, or decomposition classes) for the action of a theta-group (G_0,V), showing that they are finitely-many, locally-closed, irreducible unions of G_0-orbits of constant dimension partitioning V. We carry out a local study of their closures showing that Jordan classes are smooth and that their closure is a union of Jordan classes. We parametrize Jordan classes and G_0-orbits in a given class in terms of the action of subgroups of Vinberg's little Weyl group, and include several examples and counterexamples underlying the differences with the symmetric case and the critical issues arising in the theta-situation.
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