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I am trying to learn a bit about spherical geometry and wonderful compactifications. Please correct any misconceptions. If I've understood 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 '14 at 15:00

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