  Let $G$ be a connected reductive subgroup of a complex connected reductive
group $\hat{G}$. Fix maximal tori and Borel subgroups of $G$ and $\hat{G}$.
Consider the cone $LR^\circ(\hat{G},G)$ generated by the pairs
$(\nu,\hat{\nu})$ of strictly dominant characters such that $V_\nu$ is a
submodule of $V_{\hat\nu}$. The main result of this article is a bijective
parametrisation of the faces of $LR^\circ(\hat G,G)$. We also explain when such
a face is contained in another one. In way, we obtain results about the faces
of the Dolgachev-Hu's $G$-ample cone. We also apply our results to reprove
known results about the moment polytopes.
