论文标题
在共同连接轨道上的动作角度坐标以及部分热带化的无数性空间
Action-angle coordinates on coadjoint orbits and multiplicity free spaces from partial tropicalization
论文作者
论文摘要
紧凑型谎言组的共同连接轨道和多样性空间是汉密尔顿群体作用的符号歧管的重要例子。在这些空间上构建动作角度变量是一项艰巨的任务。该领域的基本结果是为$ k = u（n）组的Gelfand-Zeitlin可集成系统的Guillemin-sternberg构建，SO（n）$。将这些结果扩展到其他类型的组是本文的目标之一。 部分热带化是泊松空间，其持续的泊松支架是使用泊松理论技术和贝伦斯坦·卡兹丹（Berenstein-Kazhdan）的几何晶体构建的。它们提供了谎言代数$ {\ rm lie}（k）^*$与线性泊松支架和多面体锥之间的桥梁，该圆锥将$ g = k^\ k^\ mathbb {c} $的不可减至模块的规范底座进行参数。 我们概括了部分热带化的构建以允许任意群集图，并将其应用于符号几何形状中的问题。对于紧凑型组$ k $的每个常规coadexhexhexhight轨道，我们通过复式域的符号嵌入来构建疲惫。作为产品，我们到达了一个猜想的公式，用于常规旋转轨道的gromov宽度。我们证明了无数$ k $ - 空格的相似结果。
Coadjoint orbits and multiplicity free spaces of compact Lie groups are important examples of symplectic manifolds with Hamiltonian groups actions. Constructing action-angle variables on these spaces is a challenging task. A fundamental result in the field is the Guillemin-Sternberg construction of Gelfand-Zeitlin integrable systems for the groups $K=U(n), SO(n)$. Extending these results to groups of other types is one of the goals of this paper. Partial tropicalizations are Poisson spaces with constant Poisson bracket built using techniques of Poisson-Lie theory and the geometric crystals of Berenstein-Kazhdan. They provide a bridge between dual spaces of Lie algebras ${\rm Lie}(K)^*$ with linear Poisson brackets and polyhedral cones which parametrize the canonical bases of irreducible modules of $G=K^\mathbb{C}$. We generalize the construction of partial tropicalizations to allow for arbitrary cluster charts, and apply it to questions in symplectic geometry. For each regular coadjoint orbit of a compact group $K$, we construct an exhaustion by symplectic embeddings of toric domains. As a by product we arrive at a conjectured formula for Gromov width of regular coadjoint orbits. We prove similar results for multiplicity free $K$-spaces.
