I have a problem as follows.
Let $E_{i,j}\in M_2(\mathbb{C})$ be the elementary matrix and $\mathfrak{g}:=\mathfrak{sl}(2) = \{ A \in M_2({\mathbb{C}})| tr(A) =0 \}$ the special linear Lie algebra. Fix $\mathfrak{h}=\mathbb{C}(E_{11} - E_{22})$ the standard Cartan sub-algebra of $\mathfrak{g}$, and $\{\alpha\}$ the only simple root of $\mathfrak{g}$ . Recall that there is a standard bilinear form $<,>$ on $\mathfrak{h}^*$. Let $V$ be a weight $\mathfrak{g}$-module (i.e. a $\mathfrak{g}$-module which is semi-simple over $\mathfrak{h}$.)
$\bf My$ $\bf Question:$ Is it true that if every weight $\mu$ of $V$ satisfies that $<\mu, \alpha'> \notin \mathbb{Z}$. ($\alpha':=2\alpha/<\alpha,\alpha>$ the co-root) then $E_{21}$ acts as one-to-one linear operator on $V$? Thank you very much!
Here is an example that shows how this fails outside category $\mathcal{O}$ (I will give a general example that may be good to keep in mind for other Lie algebras as well).
Let $\mathfrak{g} = \mathfrak{n}^-\oplus \mathfrak{h}\oplus\mathfrak{n}^+$ be the usual decomposition and set $\mathfrak{b^-} = \mathfrak{n}^-\oplus\mathfrak{h}$ be the opposite of the usual Borel subalgebra.
For $\lambda\in \mathfrak{h}^*$, we define an "opposite" Verma module (not to be confused with a dual Verma module) by $$W(\lambda) = U(\mathfrak{g})\otimes_{U(\mathfrak{b}^-)}\lambda$$ where $\lambda$ is a $\mathfrak{b}^-$-module by letting $\mathfrak{h}$ act via $\lambda$ and $\mathfrak{n}^-$ act as $0$.
An easy argument shows that $\mathfrak{n}^-$ acts locally nilpotently on $W(\lambda)$.
It is now easy to check that in the case of $\mathfrak{sl}(2)$, if $\lambda$ is not integral then the same holds for all weights of $W(\lambda)$ (in fact, the weights of $W(\lambda)$ are precisely those of the form $\lambda + 2k$ for some $k\geq 0$). In particular, we see that $W(\lambda)$ satisfies the conditions, but since $E_{21}$ is in $\mathfrak{n}^-$ it acts locally nilpotently and thus it cannot act as an injective operator.
The "easy arguments" mentioned above are simply the same as the ones for the usual Verma modules, with the roles of $\mathfrak{n}^+$ and $\mathfrak{n}^-$ switched.