 Solution

In the shifted normalization of part (a), the Harish-Chandra isomorphism is the algebra isomorphismThus the centre is identified with the Weyl-invariant polynomials on ${\mathfrak{h}}^{\ast }$. Central characters satisfy ${\theta }_{\chi }={\theta }_{\xi }$ exactly when $\chi$ and $\xi$ belong to the same Weyl orbit. The $\rho$ shift is already incorporated in $M^{\chi }$ having highest weight $\chi -\rho$; without that shift the corresponding Weyl action is the dot action.