Particular on the structure of a weight $L$-module $M$, with $L$ semisimple Lie-algebra.

Question

Let be $L$ a semisimple Lie-algebra with its root system $R \subset H^*$ Cartan subalgebra and root decomposition \begin{gather} L=H \oplus \bigoplus_{\alpha \in R}^n L_{\alpha} \end{gather} Let $M$ be an $L$-module and $M_\lambda$ a non-zero weight space w.r.t. the weight $\lambda \in H^*$ and $x \in L_\alpha$. Then we have $x.M_\lambda \subseteq M_{\lambda + \alpha}$, is this inclusion actually an equality for all $\alpha$?

Answer 1

No. Consider the adjoint rep of sl3: the outer weightspaces are 1-d and the zero weightspace is 2d so there is no way the map induced by the action of $e_\alpha$ from $M_{-\alpha}$ to the zero weight space can be onto. – mt_ 5 hours ago