Proposition 5

(i) The element $e$ of $Z(\lambda )$ is primitive of weight $\lambda$ and generates the $\mathfrak{g}$-module $Z(\lambda )$.

(ii) Let $Z^+(\lambda ) =\sum_{\lambda\not=\mu}Z(\lambda )^\mu$. Every submodule of $Z(\lambda )$ distinct from $Z(\lambda )$

is contained in $Z^+(\lambda )$.

(iii) There exists a largest submodule $F_{\lambda}$ of $Z(\lambda )$ distinct from $Z(\lambda )$. The quotient module $Z(\lambda )/F_{\lambda}$ is simple and has highest weight $\lambda$.

It is clear that $e$ generates the $\mathfrak{g}$-module $Z(\lambda )$. If $x\in \mathfrak{b}_+$,

hence (i).

The $\mathfrak{h}$-module $Z(\lambda )$ is semi-simple (Prop. 1). If G is a $\mathfrak{g}$-submodule of $Z(\lambda )$, then $G =\sum_{\mu\in\mathfrak{h}^*}(G\cap Z(\lambda )^\mu)$. The hypothesis $G\cap Z(\lambda )^{\lambda}\not= 0$ implies

that $G = Z(\lambda )$, since dim $Z(\lambda )^{\lambda}= 1$ and $e$ generates the $\mathfrak{g}$-module $Z(\lambda )$. If $G\not= Z(\lambda )$, then $G =\sum_{\mu\not=\lambda}G\cap Z(\lambda )^\mu\subset Z^+(\lambda )$.

Let $F_{\lambda}$ be the sum of the $\mathfrak{g}$-submodules of $Z(\lambda )$ distinct from $Z(\lambda )$. By (ii), $F_{\lambda}\subset Z^+(\lambda )$. Hence $F_{\lambda}$ is the largest submodule of $Z(\lambda )$ distinct from $Z(\lambda )$. It is clear that $Z(\lambda )/F_{\lambda}$ is simple and that the canonical image of $e$ in $Z(\lambda )/F_{\lambda}$ is primitive of weight $\lambda$.

In the remainder of this chapter, the $\mathfrak{g}$-module $Z(\lambda )/F_{\lambda}$ of Prop. 5 will be denoted by $E(\lambda )$.