Remark 4

Let $(\mathfrak{g},\mathfrak{h})$ be a split reductive Lie algebra, $\mathfrak{c}$ the centre of $\mathfrak{g},\mathfrak{g}'=\mathscr{D}\mathfrak{g}$, $\mathfrak{h}'=\mathfrak{h}\cap \mathfrak{g}'$. Then $\mathfrak{h}=\mathfrak{c}\times \mathfrak{h}'$, and we identify $\mathfrak{h}^{'*}$ with a vector subspace of $\mathfrak{h}^*$. For any $\lambda \in \mathfrak{h}^*$ such that $\lambda \not= 0$, the primary subspace $\mathfrak{g}^{\lambda}$ relative to $\lambda$ is equal to ${\mathfrak{g}'}^{\lambda|\mathfrak{h}'}$. A non-zero weight of $\mathfrak{h}$ on $\mathfrak{g}$ is called a root of $(\mathfrak{g},\mathfrak{h})$; every root vanishes on $\mathfrak{c}$. Denote by $R(\mathfrak{g},\mathfrak{h})$ the set of roots of $(\mathfrak{g},\mathfrak{h})$; it can be identified canonically with $R(\mathfrak{g}',\mathfrak{h}')$. Let $\alpha \in R(\mathfrak{g},\mathfrak{h})$. We define $h_{\alpha},H_{\alpha}$, $\mathfrak{s}_{\alpha}$, the isomorphisms $\mathfrak{s}\mathfrak{l}(2, k)\rightarrow \mathfrak{s}_{\alpha}$, and the representations of $\mathfrak{s}\mathfrak{l}(2, k)$ on $\mathfrak{g}$ associated to $\alpha$, as in the semi-simple case.