AI 中文总结
该论文利用$K$-理论版本的等变形式性，得到迷向作用等变形式性的两种刻画，并据此对单李群相关情形给出等变形式性的完整分类。
AI 中文摘要
设$G$为紧连通李群，$K$为其连通李子群。利用文献[F2]中发展的等变形式性的$K$-理论版本（涉及对$G/K$上向量丛的分析），我们得到了$K$在$G/K$上的迷向作用的等变形式性的两种刻画：第一种将等变形式性与$G$的表示环到$K$的表示环的限制映射的“光滑性”条件等同；第二种刻画断言等变形式性等价于两个特定特殊$K$-理论类的等变$K$-理论指标配对在某种意义下非平凡。通过应用这些刻画，我们能够给出圆周子群的迷向作用的等变形式性的表示论判据，以及当$K$是秩比$G$小1的环面时的不变量论判据，最终在进一步假设$G$为单群的情形下，给出了等变形式性的完整分类。
英文摘要
Let $G$ be a compact connected Lie group and $K$ its connected Lie subgroup. Using the $K$-theoretic version of equivariant formality developed in [F2] which involves analysis of vector bundles over $G/K$, we find two characterizations of equivariant formality of the isotropy action of $K$ on $G/K$. The first one equates equivariant formality with a "smoothness" condition of the restriction map of the representation ring of $G$ to that of $K$. The second characterization asserts that equivariant formality amounts to the equivariant $K$-theoretic index pairing of two certain special $K$-theory classes being nontrivial in some sense. By applying these characterizations, we are able to give a representation theoretic criterion for equivariant formality of the isotropy action by a circle subgroup, as well as an invariant theory criterion for the case where $K$ is a torus of dimension one less than the rank of $G$, culminating in a complete classification of equivariant formality where $G$ is further assumed to be simple.
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