Let $\mathfrak{g}$ be a Kac-Moody Lie algebra (actually, I'd already be happy with an answer addressing the case where $\mathfrak{g}$ is a simple Lie algebra over $\mathbb{C}$).

**1st ?**: I'm wondering what results are known about the characters of $L(\lambda)$ (the irreducible module of highest weight $\lambda$) when $\lambda$ is not dominant. I know very little Lie theory (just the basics really, like in Kac's book) and am not at all up to date on the current literature.

I do know that in principle one could use KL polynomials to calculate characters, but that's not the kind of answer I'd like. Rather, I'd be interested in closed formulas like the Weyl-Kac character formula or explicit combinatorial descriptions of the weight space decomposition with respect to a Cartan. I'm sure no general results are known; I'll be happy with formulas for particular examples or special classes.

**2nd ?**: Is there a reason we shouldn't hope for such formulas for non-integrable modules in general? (Other than that current techniques don't provide them)

`$\mathcal{O}$`

as in my 2008 AMS book). But here no closed character formula like Weyl's can be expected in general, given the lack of Weyl group symmetry and complexity of KL polynomial values. For Kac-Moody algebras in general there is also lots of theory, but it gets even more complicated. Many people have worked on this, such as Kashiwara, Tanisaki, Kumar, .... $\endgroup$