Reductions for branching coefficients

Let $G$ be a connected reductive subgroup of a complex connected reductive group $\hat{G}$. We are interested in the branching problem. Fix maximal tori and Borel subgroups of $G$ and $\hat G$. Consider the cone $lr(G,\hat G)$ generated by the pairs $(ν,\hat nu)$ of dominant characters such that $V_ν^*$ is a submodule of $V_{\hat nu}$. It is known that $lr(G,\hat G)$ is a closed convex polyhedral cone. In this work, we show that every regular face of $lr(G,\hat G)$ gives rise to a {\it reduction rule} for multiplicities. More precisely, we prove that for $(ν,\hat nu)$ on such a face, the multiplicity of $V_ν^*$ in $V_{\hat nu}$ equal to a similar multiplicity for representations of Levi subgroups of $G$ and $\hat G$. This generalizes, by different methods, results obtained by Brion, Derksen-Weyman, Roth...

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