Symmetric k-varieties are a natural generalization of symmetric spaces to general fields k. We study the action of minimal parabolic k-subgroups on symmetric k-varieties and define a map that embeds these orbits within the orbits corresponding to algebraically closed fields. We develop a condition for the surjectivity of this map in the case of k-split groups that depends only on the dimension of a maximal k-split torus contained within the fixed point group of the involution defining the symmetric k-variety.

中文翻译：

---

作用于对应于 k 分裂群的对称 k 变量的最小抛物线 k 子群

对称的ķ-varieties 是对称空间对一般领域的自然概括ķ. 我们研究最小抛物线的作用ķ-对称上的子群ķ-varieties 并定义一个映射，将这些轨道嵌入对应于代数封闭域的轨道内。在以下情况下，我们为这张地图的超射性开发了一个条件ķ- 只依赖于最大维度的分组ķ-分裂环包含在定义对称的对合的不动点群中ķ-种类。