离散级数轨道的Poisson几何与非紧群作用的动量凸性
Poisson Geometry of Discrete Series Orbits, and Momentum Convexity for Noncompact Group Actions
摘要：
本文的主要结果是非紧半单李群的某些哈密顿作用的动量映射的凸性定理。要求图像落在李代数（的对偶）的某个开子集D内，并且要求动量映射本身作为到D的映射是适当的。集合D大致对应，通过轨道方法，以离散系列的代表性的团体，大部分的文件是致力于研究D本身，其中包括的李代数元素有紧凑的中心化子。当群为Sp（2n）时，这些元素在线性哈密顿动力系统理论中被称为“强稳定”的元素，我们的结果可以看作是某些理论对任意半单李群的推广.作为应用，我们证明了具有指定频率的正定Hamilton和的频率集的一个新的凸性定理。
英文摘要：
The main result of this paper is a convexity theorem for momentum mappings of certain hamiltonian actions of noncompact semisimple Lie groups. The image is required to fall within a certain open subset D of the (dual of the) Lie algebra, and the momentum map itself is required to be proper as a map to D. The set D corresponds roughly, via the orbit method, to the discrete series of representations of the group, Much of the paper is devoted to the study of D itself, which consists of the Lie algebra elements which have compact centralizer. When the group is Sp(2n), these elements are the ones which are called "strongly stable" in the theory of linear hamiltonian dynamical systems, and our results may be seen as a generalization of some of that theory to arbitrary semisimple Lie groups. As an application, we prove a new convexity theorem for the frequency sets of sums of positive definite hamiltonians with prescribed frequencies.
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