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A horospherical variety is a normal algebraic variety where a connected reductive algebraic group acts with an open orbit isomorphic to a torus bundle over a flag variety. In this article we study the cohomology of line bundles on complete horospherical varieties.

In this paper, we will look at the algebra of global differential operators D X on wonderful compactifications X of symmetric spaces G H of type A 1 and A 2 . We will first construct a global differential operator on these varieties that does not come from the infinitesimal action of g. We will then focus on type A 2 , where we will show that D X is an algebra of finite type, and that for any invertible sheaf L on X, H 0 (X, L) is either 0 or a simple left D X,L -module. Finally, we will show with the help of local cohomology that this is still true for higher cohomology groups H i (X, L).

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