= Solution
In the shifted normalization of part (a), the <Harish-Chandra isomorphism> is the algebra isomorphism
$$
\boxed{Z(U(\mathfrak g))\overset\sim\longrightarrow S(\mathfrak h)^W,\qquad z\longmapsto j_z.}
$$
Thus the centre is identified with the Weyl-invariant polynomials on $\mathfrak h^*$. 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.
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