Well, the question is probably a rather basic one but I haven't been able to find the answer in literature or come up with it myself so here we go.


Do there exist irreducible representations of the Lie algebra $\mathfrak g$ which aren't a highest weight module with respect to some Borel subalgebra in the case when


a) $\mathfrak g$ is a complex finite dimensional semisimple algebra,


b) $\mathfrak g=\mathfrak{gl}\_n(\mathbb C)$?



Irreps which aren't highest-weight modules.
Here's a more specific reference for the rank 1 simple Lie algebra; the "classification" by Block shows clearly how sparse the highest weight representations are among all irreducible representations [here](http://www.ams.org/journals/bull/1979-01-01/S0273-0979-1979-14573-7/home.html). Block was basically responding to the remarks of Dixmier and others to the effect that no "classification" would be possible, but of course his treatment is not very concrete. 


Also, the close connection between irreducible representations of a general linear Lie algebra and its derived (special linear) algebra make it obvious how close the answers to (a) and (b) are. 


