Exercise 8, § 9

Let $\psi$ be a polynomial function on $\mathfrak{h}^*$ of degree $r$. Show that there exists a unique polynomial function $\Psi$ on $\mathfrak{h}^*$ that is invariant under W, of degree $\leq r$, and such that

$\sum_{\alpha\in P}\psi (\mu)$ dim $E^\mu=\Psi (\lambda +\rho )$ dim E

for any simple $\mathfrak{g}$-module E of highest weight $\lambda$.

(Treat first the case in which $\psi (\mu) = (\mu|\nu )^r$, where $\nu \in P$ is not orthogonal to any root; for this use the homomorphism $f_{\nu}$ in the proof of Th. 2 and Chap. V, §5, no. 4, Prop. 5 (i).)