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.
In particular, the following is the definition in my reference: $M\left(\lambda\right)=\mathcal{U}\left(\widetilde{\mathfrak{g}}\right)/J\left(\lambda\right)$, where
$$
J\left(\lambda\right)=\mathcal{U}\left(\widetilde{\mathfrak{g}}\right)\hat{n}^{+}+\sum_{h\in\widetilde{\mathfrak{h}}^{*}}\mathcal{U}\left(\widetilde{\mathfrak{g}}\right)\left(\lambda-\lambda\left(h\right)\right)\subset\mathcal{U}\left(\widetilde{\mathfrak{g}}\right)
$$
On the other hand, I have searched for useful Verma module formalisms and example solutions, and it seems that such information is scarce.
 A: Information about Verma modules is not so scarce.   In any case, the multiplicities of weights are (in principle) quite easy to calculate, which led BGG in their 1971 paper to an elegant proof of the Weyl and Kostant formulas.    (My 2008 book Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$ published by AMS has an extensive treatment of these matters.   See for example 1.16.)   
In general, given a weight $\lambda$ (in the dual of a Cartan subalgebra) for an arbitrary semisimple Lie algebra, the Verma module $M(\lambda)$ with this highest weight has the weight $\mu \leq \lambda$ with multiplicity equal to $\mathcal{P}(\lambda - \mu)$.   Here $\mathcal{P}$ is the Kostant partition function, which counts the number of ways of writing a given linear functional as a sum of positive roots.  
In your particular example, you are asking for $\mathcal{P}(3 \alpha + 5 \beta)$ with $\alpha, \beta$ the simple roots in the Lie algebra of type $A_2$.   If my arithmetic is correct, the answer is 4.    (Of course, computing the partition function can get arbitrarily long to carry out in practice, but the theory is quite simple.) 
A: There is an easy formula for the character of the Verma module $M_\lambda$ with  highest weight $\lambda$, namely 
    $ch(M_{\lambda})=\frac{e^{\lambda}}{\prod_{\alpha>0}(1-e^{-\alpha})}$ (see for instance the book by Fuchs and Schweigert,  Symmetries, Lie algebras and Representation). From this you get easily the multiplicity of any weight.
