论文标题
适当动作的均等atiyah-patodi-singer索引定理ii：$ k $ - 理论指数
An equivariant Atiyah-Patodi-Singer index theorem for proper actions II: the $K$-theoretic index
论文作者
论文摘要
考虑一个在Riemannian歧管$ M $带边界的本地紧凑型组$ g $的适当的等轴测操作，因此$ m/g $是紧凑的。然后，在合适的边界条件下，$ m $上的dirac型运算符$ d $具有均值索引$ \ permatatorName {index} _g（d）$ $ k $ - 还原$ c^*$ c^*$ - algebra $ c^*algebra $ c^*rg $ g $的$ k $。这是Baum-Connes分析组装图和（均等）Atiyah-Patodi-Singer指数的常见概括。在本系列的第一部分中，根据$ d $的参数和与$ g $相关的跟踪，为元素$ g \ in g $ in G $中的元素$ g \定义了数值索引$ \ operatotorName {index} _g（d）$。为此指数获得了Atiyah-Patodi-Singer类型指数公式。在本文中，我们表明，在某些条件下，对于trace $τ_g$，$τ_g（\ operatorName {index} _g（d））= \ propatatorName {index} _g（d）$，对于trace $τ_g$定义了由Orbital积分定义的$ G $。这意味着第一部分中的索引定理得出有关$ K $ - 理论索引$ \ operatatorName {index} _g（d）$的信息。它还表明$ \ operatatorName {index} _g（d）$是同型不变的数量。
Consider a proper, isometric action by a unimodular locally compact group $G$ on a Riemannian manifold $M$ with boundary, such that $M/G$ is compact. Then an equivariant Dirac-type operator $D$ on $M$ under a suitable boundary condition has an equivariant index $\operatorname{index}_G(D)$ in the $K$-theory of the reduced group $C^*$-algebra $C^*_rG$ of $G$. This is a common generalisation of the Baum-Connes analytic assembly map and the (equivariant) Atiyah-Patodi-Singer index. In part I of this series, a numerical index $\operatorname{index}_g(D)$ was defined for an element $g \in G$, in terms of a parametrix of $D$ and a trace associated to $g$. An Atiyah-Patodi-Singer type index formula was obtained for this index. In this paper, we show that, under certain conditions, $τ_g(\operatorname{index}_G(D)) = \operatorname{index}_g(D)$, for a trace $τ_g$ defined by the orbital integral over the conjugacy class of $g$. This implies that the index theorem from part I yields information about the $K$-theoretic index $\operatorname{index}_G(D)$. It also shows that $\operatorname{index}_g(D)$ is a homotopy-invariant quantity.
