Reducibility of an sl2 Verma module
= Reducibility of an sl2 Verma module
{c}
{title2=$M_\lambda$ reducible iff $\lambda\in\mathbb Z_{\geq0}$}
The Verma module $M_\lambda$ has basis $(f^rv_\lambda)_{r\geq0}$ and
$$
e f^rv_\lambda=r(\lambda-r+1)f^{r-1}v_\lambda.
$$
It is reducible exactly when $\lambda=m\in\mathbb Z_{\geq0}$; then its unique proper nonzero submodule is generated by $f^{m+1}v_\lambda$ and is isomorphic to $M_{-m-2}$.