Minimal Parabolic $k$-subgroups acting on Symmetric $k$-varieties Corresponding to $k$-split Groups

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.