Let $G$ be a complex reductive Lie group, $B$ a Borel subgroup, with which to define "dominant weight". Let $\lambda$ be an integral weight, not necessarily dominant, but nonetheless giving a one-dimensional $B$-rep $\mathbb C_\lambda$.

Then we can define the Verma module $U{\mathfrak g} \otimes_{U\mathfrak b} \mathbb C_\lambda$, which has compatible actions of $\mathfrak g$ and $B$ (it's a "$(\mathfrak g,B)$-module"), and a unique irreducible quotient $V_\lambda$, again a $(\mathfrak g,B)$-module.

If $\lambda$ is dominant, then $V_\lambda$ is actually a (finite-dimensional) $G$-irrep, so we know how to compute its weight multiplicities in manifestly positive ways, e.g. counting Littelmann paths. For $\lambda$ not dominant, $V_\lambda$ is infinite-dimensional, but its weight multiplicities are still finite (since they're bounded by those of the Verma module).

Are there combinatorial formulae for the weight multiplicities of $V_\lambda$, when $\lambda$ is not dominant?

If so, references please!