极作用的辛性质

摘要：

称等距紧群作用$G \times (M,g) \rightarrow (M,g)$为极群，如果存在一个闭的嵌入子流形$Σ\subseteq M$，它与所有轨道正交相交.设$Π$是相应的广义Weyl群.研究余切丛$T^*M$上提升作用$G$的性质.特别地，我们证明了限制映射$(C^{\infty}(T^*M))^G \rightarrow (C^{\infty}(T^* Σ))^Π$是Poisson代数的一个满同态。作为推论，奇异辛约化$T^*M // G $和$T^* Σ// Π$同构于分层辛空间，部分回答了Lerman，Montgomery和Sjamaar的一个猜想.

英文摘要：

An isometric compact group action $G \times (M,g) \rightarrow (M,g)$ is called polar if there exists a closed embedded submanifold $Σ\subseteq M$ which meets all orbits orthogonally. Let $Π$ be the associated generalized Weyl group. We study the properties of the lifting action $G$ on the cotangent bundle $T^*M$. In particular, we show that the restriction map $(C^{\infty}(T^*M))^G \rightarrow (C^{\infty}(T^* Σ))^Π$ is a surjective homomorphism of Poisson algebras. As a corollary, the singular symplectic reductions $T^*M // G $ and $T^* Σ// Π$ are isomorphic as stratified symplectic spaces, which gives a partial answer to a conjecture of Lerman, Montgomery and Sjamaar.

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