For a while I've been reading J.E.Humphreys's book "Representations of semisimple Lie algebras in the BGG category $\mathcal O$" under the impression that any [module in $\mathcal O$](http://en.wikipedia.org/wiki/Category_O) has a finite generating set composed of highest weight vectors. Now I've realised that I'm lacking a proof. So what would serve as a counterexample? Or has my assumption been correct for some reason?


I apologize if the answer can be found further on in the book itself!



Module in category O not generated by a finite set of HWVs.
This is false. Consider the contragradient dual on a Verma module $V\_\lambda$. This is a module given by linear functions on $V\_\lambda$ which kill all but finitely many weight spaces. This module has no highest weight vectors of weight other than $\lambda$; any non-zero vector $\xi$ of weight $< \lambda$ is non-zero on $F\_i v$ for some $v$ (since $V\_\lambda$ is generated in degree $\lambda$), and $E\_i\xi\neq 0$ since it has non-zero value on $v$. 


Thus, this module is generated by weight vectors if and only if it is generated by vectors of weight $\lambda$, but this is only possible if the Verma module is irreducible (otherwise, the dual of the simple quotient of the Verma is the proper submodule generated by the vectors of degree $\lambda$). Thus, this is a counter-example.


