
Journal of Lie Theory 31 (2021), No. 3, 885896 Copyright Heldermann Verlag 2021 Reductions for Branching Coefficients Nicolas Ressayre Institute de Mathématiques et de Modélisation, Université Montpellier 2, 34095 Montpellier, France ressayre@math.univmontp2.fr [Abstractpdf] \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, DerksenWeyman, Roth, and others. Keywords: Branching rules, eigencone. MSC: 20G05, 20G20. [ Fulltextpdf (142 KB)] for subscribers only. 