Exercise 3, § 8

Let $u$ be an irreducible representation of G, and let $\lambda$ be its highest weight. For $x\in \mathfrak{g}$, denote by $\overline{x}$ the unique element of $\overline{\mathbf{C}}$ that is conjugate to $x$ under Ad(G). Prove the equality $\|u(x)\|_{\infty}=|\delta (\lambda )(\overline{x})|$.