Proposition 6

Let V be a $\mathfrak{g}$-module, $\lambda$ an element of $\mathfrak{h}^*$ and $v$ a primitive element of V of weight $\lambda$.

(i) There exists a unique homomorphism of $\mathfrak{g}$-modules $\psi : Z(\lambda )\rightarrow V$ such that $\psi (e) =v$.

(ii) Assume that $v$ generates V. Then $\psi$ is surjective. Moreover, $\psi$ is bijective if and only if, for every non-zero element $u$ of $U(\mathfrak{n}_-),u_V$ is injective.

(iii) The map $u \rightarrow u\otimes 1$ from $U(\mathfrak{n}_-)$ to $Z(\lambda )$ is bijective.

Let K be the kernel of the representation of $U(\mathfrak{b}_+)$ on $L_{\lambda}$; it is of codimension 1 in $U(\mathfrak{b}_+)$. Let $J = U(\mathfrak{g})K$ be the left ideal of $U(\mathfrak{g})$ generated by K; then $L_{\lambda}$ can be identified with $U(\mathfrak{b}_+)/K$ as a left $U(\mathfrak{b}_+$)-module, and $Z(\lambda )$ can be identified with $U(\mathfrak{g})/J$ as a left $U(\mathfrak{g}$)-module. We have $K.v= 0$, so $J.v= 0$, which proves (i).

Now assume that $v$ generates V. It is clear that $\psi$ is surjective.

By the Poincaré-Birkhoff-Witt theorem (Chap. I, §2, no. 7, Cor. 6 of Th. 1), a basis of $U(\mathfrak{n}_-)$ over $k$ is also a basis of $U(\mathfrak{g})$ as a right $U(\mathfrak{b}_+)$-module. Hence the map $\varphi :u \rightarrow u\otimes 1$ from $U(\mathfrak{n}_-)$ to $U(\mathfrak{g})\otimes_{U(\mathfrak{b}_+)}L_{\lambda}$ is bijective. Let $u\in U(\mathfrak{n}_-)$. Then $\varphi^{-1}\circ u_{Z(\lambda)}\circ \varphi$ is left multiplication by $u$ on $U(\mathfrak{n}_-)$. In view of Chap. I, §2, no. 7, Cor. 7 of Th. $1,u_{Z(\lambda)}$ is injective if $u\not= 0$. Consequently, if $\psi$ is bijective, then $u_V$ is injective for non-zero $u$ in $U(\mathfrak{n}_-)$.

Assume that $\psi$ is not injective. There exists $u\in U(\mathfrak{n}_-)$ such that $u\not= 0$ and $\psi (\varphi (u)) = 0$. Then