Let $\mathfrak{g}$ be a classical simple Lie algebra. Let $V \in \mathcal{O}$ (Category O)and let $\mu \in \mathfrak{h}^*$ (Dual of the Cartan sub-algebra).
Let $\mu + \bigwedge _\text{root}$ be the coset for $\mu$ for the $\mathbb{Z}$-submodule $\bigwedge _\text{root}$ of $\mathfrak{h}^*$. Here $\bigwedge_\text{root}$ denotes the root lattice $\mathbb{Z}\Delta$.
Define $$V^{\mu + \bigwedge_\text{root}} = \bigoplus_{ \lambda \in \mu + \bigwedge_\text{root}} V_\lambda$$ where $V_\lambda$ is the weight space of $V$ of weight $\lambda$.
Question: Why is $V^{\mu + \bigwedge_\text{root}}$ a $\mathfrak{g}$ sub-module of $V$ and that $V$ is a direct sum of finitely many such submodules.
If $x\in \mathfrak{g}_\alpha$ ($\alpha\in \Delta$) and $v\in V_\lambda$ for some $\lambda\in\mu+\bigwedge_{\mathrm{root}}$, then $x.v\in V_{\lambda+\alpha}$ and $\lambda+\alpha\in\mu+\bigwedge_{\mathrm{root}}$ again.
It follows that $U(\mathfrak{g}).V^{\mu+\bigwedge_{\mathrm{root}}}\subset V^{\mu+\bigwedge_{\mathrm{root}}}$.