We develop a local index theory for a class of operators associated with non-proper and non-isometric actions of Lie groupoids on smooth submersions. Such actions imply the existence of a short exact sequence of algebras, relating these operators to their non-commutative symbol. We then compute the connecting map induced by this extension on periodic cyclic cohomology. When cyclic cohomology is localized at appropriate isotropic submanifolds of the groupoid in question, we find that the connecting map is expressed in terms of an explicit Wodzicki-type residue formula, which involves the jets of non-commutative symbols at the fixed-point set of the action.

中文翻译：

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与李群动作相关的算子的局部指数理论

我们为与李群在平滑浸没上的非适当和非等距作用相关的一类算子开发了局部指数理论。这样的动作意味着存在一个简短的精确代数序列，将这些运算符与它们的非交换符号相关联。然后我们计算由这个扩展在周期性循环上同调上引起的连接映射。当循环上同调位于所讨论群的适当各向同性子流形上时，我们发现连接映射用明确的 Wodzicki 型剩余公式表示，该公式涉及在定点集上的非交换符号射流那个行动。