Definition 1

Let E be an $\mathfrak{s}\mathfrak{l}(2, k)$-module. An element of E is said to be primitive if it is a non-zero eigenvector of $H_E$ and belongs to the kernel of $X_{+E}$.

A non-zero element $e$ of E is primitive if and only if $ke$ is stable under the operation of $kH+kX_+$; this follows for example from Lemma 3.

Examples The element $X_+$ is primitive of weight 2 for the adjoint representation of $\mathfrak{s}\mathfrak{l}(2, k)$. The element $(1,0)$ of $k^2$ is primitive of weight 1 for the identity representation of $\mathfrak{s}\mathfrak{l}(2, k)$ on $k^2$.