Geometric Invariant Theory and Generalized Eigenvalue Problem

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 $(V,V^{\mathrm{\prime }})$ of irreducible representations of $H$ and $G$ such that $V$ is a submodule of $V^{\mathrm{\prime }}$. 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.