Exercise 21, § 7

Let $\lambda , \mu\in P_{++}$. Let $E,F,G$ be simple $\mathfrak{g}$-modules of highest weights $\lambda , \mu, \lambda +\mu$, and let $n$ be an integer $\geq 1$. If $\omega$ is a weight of E of multiplicity $n$, then $\omega +\mu$ is a weight of G of multiplicity $\geq n$. (We have $G^{\omega+\mu}\subset \bigoplus_{\nu+\sigma=\omega+\mu}E^{\nu}\otimes F^{\sigma}$. If dim $G^{\omega+\mu}< n$, the projection of $G^{\omega+\mu}$ onto

$E^{\omega}\otimes F^\mu$ is of the form $E'\otimes F^\mu$, with $E'$ strictly contained in $E^{\omega}$. Derive a contradiction from this by choosing adapted bases of E and F and imitating the proof of Prop. 10.)