We study a relative variant of Serre’s notion of $G$ -complete reducibility for a reductive algebraic group $G$ . We let $K$ be a reductive subgroup of $G$ , and consider subgroups of $G$ that normalize the identity component $K^{\circ }$ . We show that such a subgroup is relatively $G$ -completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ }$ is completely reducible. This allows us to generalize a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of $G$ , as well as ‘rational’ versions over nonalgebraically closed fields.

中文翻译：

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相对完全可约性和归一化子群

我们研究了 Serre 概念的一个相对变体 $G$ - 还原代数群的完全可约性 $G$ . 我们让 $K$ 是一个约简子群 $G$ , 并考虑 $G$ 规范化身份组件 $K^{\circ}$ . 我们证明了这样一个子群是相对的 $G$ - 完全可约 $K$ 当且仅当它的图像在自同构群中 $K^{\circ}$ 是完全可还原的。这使我们能够将一些基本结果从绝对设置推广到相对设置。我们还为李代数的李子代数推导出了类似的结果 $G$ ，以及非代数闭域上的“有理”版本。