Orbit embedding for double flag varieties and Steinberg map

In the first half of this article, we review the Steinberg theory for double flag varieties for symmetric pairs. For a special case of the symmetric space of type AIII, we will consider $X = GL_{2n}/P_{(n,n)} \times GL_n / B_n^+ \times GL_n / B_n^-$ on which $K = GL_n \times GL_n$ acts diagonally. We give a classification of $K$-orbits in $X$, and explicit combinatorial description of the Steinberg maps. In the latter half, we develop the theory of embedding of a double flag variety into a larger one. This embedding is a powerful tool to study different types of double flag varieties in terms of the known ones. We prove an embedding theorem of orbits in full generality and give an example of type CI which is embedded into type AIII.