Verma modules need not be finite-dimensional

The classes of the monomials $f^k$, $k\ge 0$, are linearly independent in $M$: let $\pi :U({\mathfrak{s}\mathfrak{l}}_2)\to U(\mathbb{C}f)$ be the linear map sending a PBW monomial $f^a{h}^b{e}^c$ to $f^a$ when $b=0$ and $c=0$ and to $0$ otherwise, so that $\pi (u)=u$ for $u\in U(\mathbb{C}f)$; then $\pi$ vanishes on $J=U({\mathfrak{s}\mathfrak{l}}_2)e+U({\mathfrak{s}\mathfrak{l}}_2)h$, because every element of $U({\mathfrak{s}\mathfrak{l}}_2)e$ has zero component along $f^a$ in the ordered PBW basis of [L1], and for $uh$ with $u={\sum }_A{f}^A{u}_A$, $u_A\in U(\mathfrak{h}\oplus \mathbb{C}e)$, the component of $f^A{u}_{A}h$ along $f^A$ equals the augmentation $\epsilon (u_{A}h)=\epsilon (u_A)\epsilon (h)=0$; hence ${\sum }_k{c}_k{f}^k\in J$ forces ${\sum }_k{c}_k{f}^k=\pi ({\sum }_k{c}_k{f}^k)=0$ and all $c_k=0$.

Therefore the vectors $f^k\cdot v$, $k\ge 0$, form an infinite linearly independent family in $M$; in particular $M$ is infinite-dimensional.

Since $M$ is a Verma module (it is the induced module $U({\mathfrak{s}\mathfrak{l}}_2){\otimes }_{U(\mathfrak{b})}{\mathbb{C}}_0$ in the realisation above) and its dimension is infinite, the universal statement of the Statement section is false.