  Let $H$ be a connected reductive subgroup of a complex connected reductive
group $G$. Fix maximal tori and Borel subgroups of $H$ and $G$. Consider the
pairs $(V,V')$ of irreducible representations of $H$ and $G$ such that $V$ is a
submodule of $V'$. We are interested in the cone $LR(G,H)$ generated by the
pairs of dominant weights of such a pair of representations. Our main result
gives a minimal set of inequalities describing $LR(G,H)$ as a part of the
dominant chamber. In way, we obtain results about the faces of the
Dolgachev-Hu's $G$-ample cone and variations of this cone.
