I am trying to learn a bit about spherical geometry and wonderful compactifications. Please correct any misconceptions. If I've understood http://www.springerlink.com/content/x62342v721707828/ properly, wonderful compactifications of rank one homogeneous spaces have more familiar descriptions. For example the wonderful compactification of $SO_n/SO_{n-1}$ is a quadric. I was wondering if the same might be true for some rank 2 homogeneous spaces such as:

$SL_n/S(GL_2 \times GL_{n-2})$, where n is at least 4,

Does the wonderful compactification of this variety show up outside of wonderful geometry at least for low n? If not is there a relatively simple algorithm to at least compute its cohomology?