Let $\mathfrak{g}$ be a simple Lie algebra (e.g. $\mathfrak{sl}_n$), and let $M_\lambda$ be the Verma module with highest weight $\lambda$. Is there a simple formula for $\dim (M_\lambda)_\mu$, where $(M_\lambda)_\mu$ denotes the weight space with weight $\mu$? Of course Weyl character formula gives me the generating function, but I am looking for a closed form formula for $\dim (M_\lambda)_\mu$.
Edit : Apparently this is called the Kostant partition function.