Geometric Invariant Theory and Generalized Eigenvalue Problem II

Abstract

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 $L{R}^{\circ }(\hat{G},G)$ 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 $L{R}^{\circ }(\hat{G},G)$. 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.

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