Exercise 2, § 9

Show that there exists a unique polynomial function $d$ on $\mathfrak{h}^*$ such that $d(\lambda ) =$ dim $E(\lambda )$ for all $\lambda \in P_{++}$; its degree is Card(R$_+$). We have

$d(w\lambda -\rho ) =\varepsilon (w)d(\lambda -\rho )$ if $w\in W, \lambda \in \mathfrak{h}^*$.

In particular, the function $\lambda \rightarrow d(\lambda -\rho )^2$ is invariant under W. Deduce that there exists a unique element $u$ of the centre of $U(\mathfrak{g})$ such that $\chi_{\lambda}(u) =d(\lambda )^2$ for all $\lambda \in \mathfrak{h}^*$ (apply Th. 2 of §8, no. 5). When $\mathfrak{g}=\mathfrak{s}\mathfrak{l}(2, k)$, we have $u= C+1$, where C is the element defined in Exerc. 1 of §1.