Simple formula for the dimension of weight spaces of Verma module?

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.

• Which formulation of Weyl's character formula do you mean? I recall it usually being given directly, rather than via a generating function. – Tobias Kildetoft Mar 18 at 17:01
• @TobiasKildetoft For instance, if you expand $1/\Delta(z)$ into a power series where $\Delta(z)$ is a Weyl denominator, the coefficients of this series are the certain dimensions that I want. – Henry Mar 19 at 19:35
• As you write in your edit, it's the Kostant partition function: the number of ways to write $\lambda-\mu$ as a non-negative integer linear combination of positive roots. You won't get a better formula than that in general. – Stephen Mar 20 at 22:54
• @Stephen But isn't Kostant partition function piecewise polynomial? If an explicit form of the piecewise polynomial is known for, say, $A_n$, then that would be helpful. – Henry Mar 20 at 23:18