# More familiar description of wonderful compactification of SL_n/S(GL_2 \times GL_n-2)

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?

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For smooth projective G-varieties, you can always find a $\mathbb{G}_m \subset G$ which acts with isolated fixed points, thus giving you a Bialynicki-Birula filtration (that you can compute by hand for low dimension) and thus a stable motivic cell structure which should make it possible to compute the (Betti) cohomology. For the case of $SL_n/S(L_2\times L_{n-2})$ you could also use a homotopy equivalence to $SL_n/P$ where $P$ is a parabolic whose Levi subgroup is $S(L_2\times L_{n-2})$, if I'm not mistaken. Then look at the Schubert cells of $SL_n/P$ for cohomology. –  Konrad Voelkel Jun 11 at 15:00