TITLE: Irreps which aren't highest-weight modules.
QUESTION [7 upvotes]: 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) $\mathfrak g=\mathfrak{gl}_n(\mathbb C)$?

REPLY [7 votes]: I think it's very far from true, and misleading to state too informally, that every representation is highest weight. For example if we think of representations (with a fixed infinitesimal character) geometrically on the flag manifold (on $P^1$ for $SL_2$), then highest weight modules with respect to a unipotent subgroup $N$ are locally constant (or rather have some central curvature) along the corresponding Schubert stratification (ie for SL2 they are attached geometrically to the stratification of $P^1$ by $A^1$ and a point), while a representation can have arbitrary "shape" - ie we can fix any stratification we want. (Note the finite dimensional representations, which as was stated are all highest weight, are attached to the stratification of the flag variety by just one piece - by Borel-Weil they arise from line bundles, which definitely satisfy this "shape requirement".) In particular the representations that people care most about tend to be Harish Chandra modules, which can be realized geometrically as stratified with respect to the (finitely many) orbits of a symmetric subgroup of $G$ - for instance, the admissible representations of $SL2R$ correspond to the stratification of $P^1$ by the two poles and $C^\times$, and most of these are evidently not highest weight.
What is true, and very useful and important (some might say the most fundamental statement about g-modules, but that's not a debate I want to start), is that representations DO live on the flag manifold - in other words, they are made out of "points" of the flag manifold, which correspond to representations that are highest weight (Verma modules). So the sense in which representations relate to highest weight modules is just that they can be realized geometrically on the flag manifold ($P^1$).
Technically this is captured by a version of Casselman's submodule theorem (reps of real groups live inside principal series) due to Beilinson-Bernstein: for every representation
there IS an N so that the representation has nonvanishing N-coinvariants (ie not highest weight vectors but a form of highest weight covectors). Such N give the support of the corresponding picture of the representation on the flag manifold.