Proposition 2

Let V be a $\mathfrak{g}$-module generated by a primitive element $e$ of weight $\omega$, and X a semi-simple $\mathfrak{g}$-module. Let $\Phi$ be the set of homomorphisms from the $\mathfrak{g}$-module V to the $\mathfrak{g}$-module X. Then $\varphi \rightarrow \varphi (e)$ is an isomorphism from the vector space $\Phi$ to the vector space $X^{\omega}_{\pi}$.

It is clear that $\varphi (e)\in X^{\omega}_{\pi}$ for all $\varphi \in \Phi$. If $\varphi \in \Phi$ and $\varphi (e) = 0$, then $\varphi = 0$ since $e$ generates the $\mathfrak{g}$-module V. We show that, if $f$ is a non-zero element of $X^{\omega}_{\pi}$, there exists $\varphi \in \Phi$ such that $\varphi (e) =f$. Let $X'$ be the submodule of X generated by $f$. By Prop. $1, X'$ is indecomposable, hence simple since X is semi-simple. The element $(e, f)$ is primitive in the $\mathfrak{g}$-module $V\times X$. Let N be the submodule of $V\times X$ generated by $(e, f)$. Then $N\cap X\subset$ pr$_2(N) = X'$, so $N\cap X = 0$ or $X'$; if $N\cap X = X', N$ contains the linearly independent elements $(e, f)$ and $(0, f)$ which are primitive of weight $\omega$; this is absurd (Prop. 1), so $N\cap X = 0$. Thus pr$_1|N$ is an injective map $h$ from N to V; this map is surjective since its image contains $e$. Thus $\varphi =$ pr$_2\circ h^{-1}$ is a homomorphism from the $\mathfrak{g}$-module V to the $\mathfrak{g}$-module X such that $\varphi (e) =f$.