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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 onthese varieties that does not come from the infinitesimal action of$\mathfrak{g}$. 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 ${\cal L}$ on$X$, $H^{0}(X,{\cal L})$ is either 0 or a simple left $D_{X,{\cal L}}$-module.Finally, we will show with the help of local cohomology that this is still truefor higher cohomology groups $H^{i}(X,{\cal L})$.

On differential operators on complete symmetric varieties of type $A_1$ and $A_2$