Exercise 9, § 8

The notations are those of no. 5. Let $\lambda \in \mathfrak{h}^*, z\in Z,z'$ the image of $z$ under the principal anti-automorphism of $U(\mathfrak{g})$, and $w_0$ the element of W that transforms B into $-B$. Show that $\chi_{\lambda}(z) =\chi_{-w_0\lambda}(z')$.

(It suffices to prove this for $\lambda \in P_{++}$. Consider the operation of $z$ on $E(\lambda )$ and $E(\lambda )^*$; use Prop. 11 of §7.)