Journal of Lie Theory

\newcommand\hG{{\widehat G}} \newcommand\hnu{{\hat\nu}} \newcommand\LR{\operatorname{LR}} \newcommand\lr{{\mathcal{LR}}} Let $G$ be a connected reductive subgroup of a complex connected reductive group $\hG$. The branching problem consists in decomposing irreducible $\hG$-representations as sums of irreducible $G$-representations. The appearing multiplicities are parameterized by the pairs $(\nu,\hnu)$ of dominant weights for $G$ and $\hG$ respectively. The support $\LR(G,\hG)$ of these decompositions is a finitely generated semigroup of such pairs of weights. The cone $\lr(G,\hG)$ generated by $\LR(G,\hG)$ is convex polyhedral and the explicit list of inequalities characterizing it is known. There are the inequalities stating that $\nu$ and $\hnu$ are dominant and those giving faces containing regular weights (called regular faces), that are parameterized by cohomological conditions.\\ In this paper, we describe the multiplicities corresponding to the pairs $(\nu,\hnu)$ belonging to any regular face of $\lr(G,\hG)$. More precisely, we prove that such a multiplicity is equal to a similar multiplicity for strict Levi subgroups of $G$ and $\hG$. This generalizes, unifies and simplifies, by different methods, results obtained by Brion, Derksen-Weyman, Roth, and others.