论文标题
Fredholm条件和指数，以限制不变的假差异操作员对同型组件的限制
Fredholm conditions and index for restrictions of invariant pseudodifferential operators to isotypical components
论文作者
论文摘要
令$γ$成为一个紧凑的团体，在光滑，紧凑的歧管$ m $上，让$ p \ inψ^m（m; e_0，e_1）$是$γ$ -INVARIANT，经典的，古典的伪差操作员在两个等等的vector vector vector $ e_i $ $ $ $ $，$ $，$ y，$ i = 0 0. 0,1 $ i = 0. 0. = 0 0. 0. 0. = 0. 0. 0. 1 c.集团$γ$。然后，$ p $诱导了$π_α（p）：h^s（m; e_0）_α\ to h^{s-m}（m; e_1）_α$之间的$α$ - 异型型组件之间相应的部分Sobolev Sobolev空间的组件。当$γ$有限时，我们明确表征了$π_α（p）$的运算符$ p $，就$ p $的主要符号而言，$π_α（p）$是弗雷德·霍尔姆（Fredholm），以及对向量捆绑$ e_i $的$γ$的操作。当$γ= \ {1 \} $时，也就是说，当没有组时，我们的结果扩展了Fredholm（Pseudo）差异操作员在紧凑型歧管上的经典表征。证明是基于对符号$ c^*$ - 代数的仔细研究以及其原始理想频谱的拓扑。我们还获得了关于不变伪差算子代数的规范封闭的结构及其与诱导表示的关系的结果。每当我们的结果也适用于非差异群体时，我们就会在更大的一般性中证明它们。为了说明结果的普遍性，我们为霍奇理论和奇异商空间的索引理论提供了一些应用。
Let $Γ$ be a compact group acting on a smooth, compact manifold $M$, let $P \in ψ^m(M; E_0, E_1)$ be a $Γ$-invariant, classical pseudodifferential operator acting between sections of two equivariant vector bundles $E_i \to M$, $i = 0,1$, and let $α$ be an irreducible representation of the group $Γ$. Then $P$ induces a map $π_α(P) : H^s(M; E_0)_α\to H^{s-m}(M; E_1)_α$ between the $α$-isotypical components of the corresponding Sobolev spaces of sections. When $Γ$ is finite, we explicitly characterize the operators $P$ for which the map $π_α(P)$ is Fredholm in terms of the principal symbol of $P$ and the action of $Γ$ on the vector bundles $E_i$. When $Γ= \{1\}$, that is, when there is no group, our result extends the classical characterization of Fredholm (pseudo)differential operators on compact manifolds. The proof is based on a careful study of the symbol $C^*$-algebra and of the topology of its primitive ideal spectrum. We also obtain several results on the structure of the norm closure of the algebra of invariant pseudodifferential operators and their relation to induced representations. Whenever our results also hold for non-discrete groups, we prove them in this greater generality. As an illustration of the generality of our results, we provide some applications to Hodge theory and to index theory of singular quotient spaces.
