In this paper, we obtain a localization formula in differential K-theory for $S^1$-action. Then by combining an extension of Goette's result on the comparison of two types of equivariant $\eta$-invariants, we establish a version of localization formula for equivariant $\eta$-invariants. An important step of our approach is to construct a pre-$\lambda$-ring structure in differential K-theory which is interesting in its own right.

中文翻译：

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$$\eta $$-invariants 的微分 K 理论和定位公式

在本文中，我们在微分K-理论中获得了$S^1$-action 的定位公式。然后结合Goette关于两类等变$\eta$-不变量的比较结果的扩展，我们建立了一个版本的等变$\eta$-不变量的定位公式。我们方法的一个重要步骤是在微分 K 理论中构建一个 pre-$\lambda$-ring 结构，这本身就很有趣。