How to find the multiplicity of weight in a Verma module? In particular, let $\mathfrak g$ be the semisimple Lie algebra of type $A_{2}$ et let $\alpha,\beta$ be its simple roots.
How can the multiplicity of weight $-2\alpha -3\beta$ be calculated in the Verma module $M(\alpha+2\beta)$?
As noted in the Wikipedia article on Verma modules, their definition relies on a stack of relatively dense notation. On the other hand, I have searched for useful Verma module formalisms and example solutions, and it seems that such information is scarce.
 A: Disclaimer: I think the below computation is correct but it's very late and I might have made a mistake out of tiredness. I will recheck my answer tomorrow to make sure I haven't misled you somewhere.
Computing weights in Verma modules is actually not bad and just uses the PBW theorem. Suppose $\mathfrak{n}^{-}, \mathfrak{n}^{+}$ are the subalgebra of negative and positive root spaces respectly and let $\mathfrak{h}$ be the Cartan subalgebra. Then, the PBW theorem tells you that we have an isomorphism of vector spaces 
$$U(\mathfrak{n}^{-}) \otimes U(\mathfrak{h}) \otimes U(\mathfrak{n}^{+}).$$
In particular, if $V_{\lambda}$ is your one dimensional module over the borel $\mathfrak{b} = \mathfrak{h} + \mathfrak{n}^{+}$, then by definition, the Verma module is 
$$M_{\lambda} = U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_{\lambda}$$
which by the above isomorphism is, as a vector space
$$U(\mathfrak{n}^{-1}) \otimes V_{\lambda}.$$
With the knowledge that applying an element $g_{\theta}$ in the negative root space $\mathfrak{g}_{\theta}$ simply lowers the corresponding weight by $\theta$, you can see that the multiplicity of a weight $\omega$ is the number of distinct ways you can write
$$\omega - \lambda$$
as a sum of negative roots.
