Are all irreducible representations of solvable Lie algebras 1-dimensional?

Question

Let $\mathfrak{g}$ be a solvable Lie algebra. By Lie's theorem, it is easy to see that any finite dimensional irreducible representation is 1 dimensional. Is it possible to remove the condition that the representation be finite dimensional?

Answer 1

The link given by Qiaochu Yuan contains a counterexample if you remove finite dimensional. Consider the 3-dimensional Lie algebra (over a field $k$) generated by $x$, $d/dx$ and $1$. This is a nilpotent Lie algebra (hence solvable), with $1$ central and $$[d/dx,x]=1.$$ This Lie algebra acts on the polynomial ring $k[x]$ without eigenvalues. In fact, the enveloping algebra of this Lie algebra is the Heisenberg algebra, $H$, and $k[x]$ is a faithful irreducible $H$-module.