Consider a simple Lie algebra $\mathfrak{g}$ and $\overline{\mathfrak{g}}%\subset \mathfrak{g}$ a Levi subalgebra. Two irreducible $\overline{%\mathfrak{g}}$-modules yield isomorphic inductions to $\mathfrak{g}$ when theirhighest weights coincide up to conjugation by an element of the Weyl group $W$of $\mathfrak{g}$ which is also a Dynkin diagram automorphism of $%\overline{\mathfrak{g}}$. In this paper we study the converse problem: giventwo irreducible $\overline{\mathfrak{g}}$-modules of highest weight $\mu $ and$\nu $ whose inductions to $\mathfrak{g}$ are isomorphic, can we conclude that$\mu $ and $\nu $ are conjugate under the action of an element of $W$ which isalso a Dynkin diagram automorphism of $\overline{\mathfrak{g}% }$ ? Weconjecture this is true in general. We prove this conjecture in type $A$ and,for the other root systems, in various situations providing $\mu $ and $\nu $satisfy additional hypotheses. Our result can be interpreted as an analogue forbranching coefficient of the main result of \cite{Raj} on tensor productmultiplicities.
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机译：考虑一个简单的李代数$ \ mathfrak {g} $和$ \ overline {\ mathfrak {g}}％\ subset \ mathfrak {g} $ Levi子代数。两个不可约的$ \ overline {％\ mathfrak {g}} $-模块在它们的最高权重与$ \ mathfrak {g的Weyl组元素$ W $发生共轭时，会产生$ \ mathfrak {g} $同构感应。 } $也是$％\ overline {\ mathfrak {g}} $的Dynkin图自同构。在本文中，我们研究相反的问题：给定两个具有不可重约的$ \ overline {\ mathfrak {g}} $-模块，它们的权重最高，分别为\\ mu $和$ \ nu $，它们对$ \ mathfrak {g} $的诱导是同构的，可以我们得出结论，$ \ mu $和$ \ nu $在$ W $元素的作用下是共轭的，这也是$ \ overline {\ mathfrak {g}％} $的Dynkin图自同构。我们推测这通常是正确的。我们在$ A $类型中证明了这个猜想，对于其他根系统，在各种情况下也提供了$ \ mu $和$ \ nu $ s满足其他假设。我们的结果可以解释为\ cite {Raj}在张量积乘数上的主要结果的模拟分支分支系数。
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