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)



Character formulas for non-integrable modules?
As one special case you may consider the Kac-Wakimoto admissible representations for affine Kac-Moody algebras. These representations are a bit more general than the integrable ones and satisfy modular invariance properties.


Kac-Wakimoto gave a character formula for them in <http://www.pnas.org/content/85/14/4956.short>


A highest weight module $L(\lambda)$ for an affine Lie algebra $\mathfrak{g}$ is *admissible* when the weight lattice of integral weights with respect to $\lambda$ has the same dimension as the original weight lattice of $\mathfrak{g}$, and $\lambda$ satisfies a natural nonnegativity constraint with respect to the root lattice.


