代数群中双陪集集的无限性
Infiniteness of Double Coset Collections in Algebraic Groups
摘要：
设$G$是定义在代数闭域上的线性代数群.本文讨论的双陪集问题是：给定闭子群$X$和$P$，其双陪集集合$X\backslash G/P$是有限的还是无限的？我们限制自己的情况下，$X$是最大的秩和还原和$P$抛物线。本文给出了一个只涉及半单元中心化子维数的无穷大判别准则。然后将该结果应用于完成球形$X$的分类。最后，排除$F_4$中的一个例子，我们证明了如果$X\backslash G/P$是有限的，那么$X$是球面的，或者$P$的Levi因子是球面的。这意味着$X\backslash G/P$很少是有限的。主要的证明方法是下降到有限群水平上的计算，然后使用初等特征标理论。
英文摘要：
Let $G$ be a linear algebraic group defined over an algebraically closed field. The double coset question addressed in this paper is the following: Given closed subgroups $X$ and $P$, is the double coset collection $X\backslash G/P$ finite or infinite? We limit ourselves to the case where $X$ is maximal rank and reductive and $P$ parabolic. This paper presents a criterion for infiniteness which involves only dimensions of centralizers of semisimple elements. This result is then applied to finish the classification of those $X$ which are spherical. Finally, excluding a case in $F_4$, we show that if $X\backslash G/P$ is finite then $X$ is spherical or the Levi factor of $P$ is spherical. This implies that it is rare for $X\backslash G/P$ to be finite. The primary method of proof is to descend to calculations at the finite group level and then to use elementary character theory.
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