= Solution
For every $\chi$ on a hyperplane $\langle\chi,\alpha^\vee\rangle=k$, with $k\in\mathbb Z_{\ge0}$, part (b) embeds one <Verma module> into the other. A central element acts on the nonzero image by both scalars, so $j_z(\chi)=j_z(s_\alpha\chi)$ there.
The difference of these two expressions is a <polynomial> on $\mathfrak h^*$. Fix all coordinates parallel to those hyperplanes and vary the remaining coordinate. It vanishes at every nonnegative integer, so as a <polynomial> in that coordinate it is identically zero. Its coefficient polynomials therefore vanish as well. Consequently the equality holds for all $\chi$. Since the simple reflections generate the <Weyl group>,
$$
\boxed{j_z(w\chi)=j_z(\chi)\qquad(w\in W).}
$$
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