In this article, we study compactifications of homogeneous spaces coming from equivariant, open embeddings into a generalized flag manifold $G/P$. The key to this approach is that in each case $G/P$ is the homogeneous model for a parabolic geometry; the theory of such geometries provides a large supply of geometric tools and invariant differential operators that can be used for this study. A classical theorem of Wolf shows that any involutive automorphism of a semisimple Lie group $G$ with fixed point group $H$ gives rise to a large family of such compactifications of homogeneous spaces of $H$. Most examples of (classical) Riemannian symmetric spaces as well as many non-symmetric examples arise in this way. A specific feature of the approach is that any compactification of that type comes with the notion of ‘curved analog’ to which the tools we develop also apply. The model example of this is a general Poincaré–Einstein manifold forming the curved analog of the conformal compactification of hyperbolic space. In the first part of the article, we derive general tools for the analysis of such compactifications. In the second part, we analyze two families of examples in detail, which in particular contain compactifications of the symmetric spaces $\mathit{SL}(n,\mathbb{R})/\mathit{SO}(p,n-p)$ and $\mathit{SO}(n,\mathbb{C})/\mathit{SO}(n)$. We describe the decomposition of the compactification into orbits, show how orbit closures can be described as the zero sets of smooth solutions to certain invariant differential operators and prove a local slice theorem around each orbit in these examples.

中文翻译：

---

均匀空间的抛物线紧缩

在本文中，我们研究了来自等变开放嵌入的齐次空间到广义标志流形的紧缩$G/P$. 这种方法的关键在于，在每种情况下$G/P$是抛物线几何的齐次模型；这种几何学理论提供了大量可用于本研究的几何工具和不变微分算子。Wolf 的一个经典定理表明，一个半单李群的任何对合自同构$G$带不动点群$H$产生了一大类这样的齐次空间的紧化$H$. 大多数（经典）黎曼对称空间的例子以及许多非对称例子都是以这种方式出现的。该方法的一个特定特征是，任何这种类型的紧凑化都带有“弯曲模拟”的概念，我们开发的工具也适用于此。这个模型的例子是一个一般的 Poincaré-Einstein 流形，形成双曲空间的保形紧化的弯曲模拟。在文章的第一部分，我们推导出分析这种紧化的通用工具。在第二部分，我们详细分析了两个例子族，特别是包含对称空间的紧化$\mathit{SL}(n,\mathbb{R})/\mathit{SO}(p,np)$和$\mathit{SO}(n,\mathbb{C})/\mathit{SO}(n)$. 我们描述了将紧化分解为轨道，展示了如何将轨道闭合描述为某些不变微分算子的平滑解的零集，并在这些示例中证明了每个轨道周围的局部切片定理。