Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 102 / 5 / a / ii / Solution

For ${\mathfrak{sl}}_2$, the Verma module has basisby the Poincare-Birkhoff-Witt theorem. If $hv=\lambda v$, thenFor $\lambda =-d$ with $d>0$, the coefficient isfor every $r\ge 1$. Thus no $f^{r}v$ with $r>0$ is a singular vector.

Any nonzero submodule contains a weight vector $f^{r}v$ because the $h$-weights are distinct. Applying $e^r$ gives a nonzero multiple of $v$, after which applying powers of $f$ generates all of $M(-d)$. HenceThis is also the negative-highest-weight case of Reducibility of an sl2 Verma module.