论文标题
还原式代数组亚组的半密度
Semisimplification for Subgroups of Reductive Algebraic Groups
论文作者
论文摘要
让$ g $为一个还原的代数组 - 可能是非连接的---在$ k $上，让$ h $是$ g $的子组。如果$ g = gl_n $，那么从$ h $获得$ g $的$ h $ a $ h'$ h'$ h'的变性过程；从适当的意义上讲，沿着$ g $的共同限制了$ h $的限制。我们使用$ g $ complete降低的概念将这个想法概括为任意还原$ g $，以及由于作者和疱疹而引起的非偏格闭合字段的几何不变理论的结果。我们的建筑产生了$ g $ - $ g $的$ h $ h'$ h'$ h'$ h'$ h'$ h'$ h'$ g $，唯一最高$ g（k）$ - 共轭，我们称之为$ k $ -smisimimplifate $ h $。这提供了一个单一的统一结构，该结构扩展了文献中的各种特殊情况（特别是，它与$ g = gl_n $的通常概念一致，以及Serre的“ $ g $ -Analogue”的$ g（k）$的子组的“ $ g $ -Analogue”。我们还表明，在一些额外的假设下，可以使用山雀中心的猜想来选择$ h'$，用于球形建筑物和/或Hesselink，Kempf和Rousseau引入的最佳不稳定的Cocharacter的理论。
Let $G$ be a reductive algebraic group---possibly non-connected---over a field $k$ and let $H$ be a subgroup of $G$. If $G= GL_n$ then there is a degeneration process for obtaining from $H$ a completely reducible subgroup $H'$ of $G$; one takes a limit of $H$ along a cocharacter of $G$ in an appropriate sense. We generalise this idea to arbitrary reductive $G$ using the notion of $G$-complete reducibility and results from geometric invariant theory over non-algebraically closed fields due to the authors and Herpel. Our construction produces a $G$-completely reducible subgroup $H'$ of $G$, unique up to $G(k)$-conjugacy, which we call a $k$-semisimplification of $H$. This gives a single unifying construction which extends various special cases in the literature (in particular, it agrees with the usual notion for $G= GL_n$ and with Serre's "$G$-analogue" of semisimplification for subgroups of $G(k)$). We also show that under some extra hypotheses, one can pick $H'$ in a more canonical way using the Tits Centre Conjecture for spherical buildings and/or the theory of optimal destabilising cocharacters introduced by Hesselink, Kempf and Rousseau.
