Proposition 7

Let $\lambda \in \mathfrak{h}^*$, E a $\mathfrak{g}$-module generated by a primitive element of weight $\lambda ,\chi$ the central character of E, and $\varphi$ the Harish-Chandra homomorphism from $U^0$ to V. Then, $\chi (z) = (\varphi (z))(\lambda )$ for all $z$ in the centre of U.

Let $v$ be a primitive element of E of weight $\lambda$, and $z$ an element of the centre of U. There exist $u_1, . . . , u_p\in U$ and $n_1, . . . , n_p\in \mathfrak{n}_+$ such that $z=\varphi (z) +u_1n_1+\cdots +u_pn_p$. Then