Exercise 11, § 9

Assume that $\mathfrak{g}$ is not a product of algebras of rank 1. Let $n\in \mathbf{N}$. Show that there exists a simple $\mathfrak{g}$-module one of whose weights is of multiplicity $\geq n$. (Suppose not. Let $E_{\lambda}$ be the simple $\mathfrak{g}$-module of highest weight $\lambda \in P_{++}$. Compare dim $E_{\lambda}$ and the number of distinct weights of $E_{\lambda}$ when $(\lambda |\lambda )\rightarrow \infty$ (with the notations of Th. 2).)