Proposition 4

Let E be a finite dimensional $\mathfrak{s}\mathfrak{l}(2, k)$-module, $m$ an integer $\geq 0$, $P_m$ the set of primitive elements of weight $m$. Let L be the vector space of homomorphisms from the $\mathfrak{s}\mathfrak{l}(2, k)$-module $V(m)$ to the $\mathfrak{s}\mathfrak{l}(2, k)$-module E. The map $f \rightarrow f(u^m)$ from L to E is linear, injective, and its image is $P_m\cup \{0\}$.

This map is clearly linear, and it is injective because $u^m$ generates the $\mathfrak{s}\mathfrak{l}(2, k$)-module $V(m)$. If $f\in L$,

so $f(u^m)\in P_m\cup \{0\}$. Let $e\in P_m$, and V the submodule of E generated by $e$. By Prop. 1, there exists an isomorphism from the module $V(m)$ to the module V that takes $u^m$ to $e$. Then $L(u^m) = P_m\cup \{0\}$.