Past exam of the representation theory of semisimple Lie algebras course of the University of Oxford / 2010 / 2 / c / Solution

For every $\chi$ on a hyperplane $⟨\chi ,{\alpha }^{\vee }⟩=k$, with $k\in {\mathbb{Z}}_{\ge 0}$, 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 ${\mathfrak{h}}^{\ast }$. 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,