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

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 $\mathfrak{g}$. 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 true for higher cohomology groups $H^{i}(X,{\cal L})$.