分支系数的约化
Reductions for branching coefficients
摘要：
设$G$是复连通约化群$\hat{G}$的连通约化子群.我们感兴趣的是分支问题。固定$G$和$\hat G$的极大环面和博雷尔子群。考虑由显性性状对$(ν,\hat nu)$产生的锥$lr(G,\hat G)$，使得$V_ν^*$是$V_{\hat nu}$的子模块。已知$lr(G,\hat G)$是封闭的凸多面锥。在这项工作中，我们表明，每一个正规面的$lr(G,\hat G)$产生一个{ \it减少规则}的多重性。更准确地说，我们证明了，对于$(ν,\hat nu)$在这样的一个面，在$V_{\hat nu}$的$V_ν^*$的多重性等于一个类似的多重性表示的Levi子群的$G$和$\hat G$。这概括了，通过不同的方法，结果得到的布里昂，德克森韦曼，罗斯.
英文摘要：
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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