Invariant distributions supported on the nilpotent cone of a semisimple Lie algebra

Let $\mathfrak{g}$ be a semisimple complex Lie algebra with adjoint group $G$ and $\mathcal{D}(\mathfrak{g})$ be the algebra of differential operators with polynomial coefficients on $\mathfrak{g}$. If $\mathfrak{g}_0$ is a real form of $\mathfrak{g}$, we give the decomposition of the semisimple $\mathcal{D}(\mathfrak{g})^G$-module of invariant distributions on $\mathfrak{g}_0$ supported on the nilpotent cone.