  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 their
highest 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: given
two 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 is
also a Dynkin diagram automorphism of $\overline{\mathfrak{g}% }$ ? We
conjecture 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 for
branching coefficient of the main result of \cite{Raj} on tensor product
multiplicities.
