Weight multiplicity formulae for $(\mathfrak g,B)$-irreps 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!
 A: Maybe it should be emphasized that the question here concerns only Lie algebra representations, so the initial group language (about Lie groups or perhaps algebraic groups) is unnecessary.   The framework is the 1976 BGG category $\mathcal{O}$ for a semisimple Lie algebra over $\mathbb{C}$ or other algebraically closed field of characteristic 0.    In general, each linear functional $\lambda \in \mathfrak{h}^*$ (for a fixed Cartan subalgebra $\mathfrak{h}$ of the given Lie algebra $\mathfrak{g}$) defines a Verma module and its unique simple quotient, while other possible highest weight modules occupy intermediate positions.    
Here the weights are assumed to be integral.   Such weights form an $\ell$-dimensional lattice $\Lambda \subset \mathfrak{h}^*$ (where $\ell = \dim \mathfrak{h}$), which can be viewed also as the abstract weight lattice of the underlying root system or as the character group $X(T)$ of a maximal torus $T$ having Lie algebra $\mathfrak{h}$ in a simply connected algebraic group with Lie algebra $\mathfrak{g}$.   (But since $G$ usually doesn't act analytically or rationally on modules in category $\mathcal{O}$, the potential rational $T$-action doesn't add anything to the given action of $\mathfrak{h}$.)
As Victor observes, there are "trivial" cases in which there exists a combinatorial formula for the weight multiplicities of a given (integral) weight $\mu$ in a simple highest weight module of highest weight $\lambda$.   A Verma module is in fact simple if and only if its highest weight is antidominant (see section 4.4 in my book on category $\mathcal{O}$).   For a parabolic (= generalized) Verma module, Jantzen developed a rather complicated criterion (see sections 9.12-9.13).    But even in such "trivial" cases, it's necessary to compute Kostant's partition function.
Other than this case, and the narrow case in type $A$ which Victor cites, I'm unaware of any infinite dimensional simple modules admitting such combinatorial formulas.   Naturally it's impossible to prove that such formulas can't exist elsewhere.   To approach this experimentally would be extremely challenging, since the combination of Kostant's partition function (for weight multiplicities in Verma modules) and the Kazhdan-Lusztig polynomial values at 1 will lead to lengthy calculations involving many cancellations even in fairly small ranks.    The resulting raw data might or might not suggest any patterns, so a clever a priori theoretical approach would probably be needed to get new formulas.    While the formulation and proof of the Kazhdan-Lusztig conjecture was a major breakthrough in terms of theoretical understanding of the infinite dimensional modules in question, the applications (to unitary representations of Lie groups in particular) continue to be quite indirect. 
A: The "trivial" case is when the simple highest weight module is a generalized Verma module (i.e. the module induced from a character of a parabolic). The multiplicity is given by the variant of Kostant's partition function; this can be reformulated in terms of lattice points in polyhedra, etc.
There is also Mathieu-Papadopoulo character formula (in the general linear case). The proof is somewhat involved; I worked out a simpler proof in a special case based on the theory of standard monomials.
