李群中的椭圆域
Elliptic domains in Lie groups
AI总结：
该研究刻画了李群中稳定椭圆元素的性质，通过不动点代数、Weyl群作用等描述其集合连通分支，在单埃尔米特李群中关联其与不变锥的关系，还证明基本连通分支的序区间紧性与整体双曲性。
摘要：
李群中的元素$g$若包含于椭圆元素集合$G^e$的内部，则被称为稳定椭圆元素，这类元素的特征性质是$\text{Ad}(g)$生成一个相对紧子群。稳定椭圆元素自然出现在因果对称空间的几何学与表示论中。我们通过$\text{Ad}(g)$的不动点代数刻画了稳定椭圆元素，并证明稳定椭圆元素集合$G^{se}$的连通分支可通过紧嵌入Cartan子代数上的Weyl群作用来描述。在单埃尔米特李群的情形下，我们将稳定椭圆元素与最大不变锥及相关子半群联系起来。特别地，我们证明了基本连通分支$G^{se}(0)$可通过序区间的紧性来刻画，且$G^{se}(0)$关于诱导的双不变因果结构是整体双曲的。
英文摘要：
An element $g$ of a Lie group is called stably elliptic if it is contained in the interior of the set $G^e$ of elliptic elements, characterized by the property that $\mathrm{Ad}(g)$ generates a relatively compact subgroup. Stably elliptic elements appear naturally in the geometry of causal symmetric spaces and in representation theory. We characterize stably elliptic elements in terms of the fixed point algebra of $\mathrm{Ad}(g)$ and show that the connected components of the set $G^{se}$ of stably elliptic elements can be described in terms of the Weyl group action on a compactly embedded Cartan subalgebra. In the case of simple hermitian Lie groups we relate stably elliptic elements to maximal invariant cones and the associated subsemigroups. In particular we show that the basic connected component $G^{se}(0)$ can be characterized in terms of the compactness of order intervals and that $G^{se}(0)$ is globally hyperbolic with respect to the induced biinvariant causal structure.
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