In this note, we unify and extend various concepts in the area of G-complete reducibility, where G is a reductive algebraic group. By results of Serre and Bate–Martin–Röhrle, the usual notion of G-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of G. We show that other variations of this notion, such as relative complete reducibility and \(\sigma \)-complete reducibility, can also be viewed as special cases of this building-theoretic definition, and hence a number of results from these areas are special cases of more general properties.

中文翻译：

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完全可还原性：塞尔主题的变体

在这篇笔记中，我们统一并扩展了G -完全可约性领域的各种概念，其中G是一个约简代数群。根据 Serre 和 Bate-Martin-Röhrle 的结果，可以将G完全可约性的通常概念重新定义为群在G的恒等分量的球形构建上的动作的属性。我们表明这个概念的其他变体，例如相对完全可约性和\(\sigma \) -完全可约性，也可以被视为这个构建理论定义的特例，因此这些领域的许多结果是特殊的更一般性质的情况。