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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.

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    $\begingroup$ The question is legitimate but too elementary to be considered "research-level". Did you try math.stackexchange.com? In any case, there's no need to go back to the definition of a Verma module, given the large amount of coverage in the literature over the past 50 years. $\endgroup$ – Jim Humphreys Nov 30 '14 at 14:40
  • $\begingroup$ @JimHumphreys Yes, I tried, but got no answers. In my opinion, this question is "elementary research-level". Unfortunately, this kind of question is often unanswered in math.se, but voted to close on this site. $\endgroup$ – Jake Nov 30 '14 at 15:07
  • $\begingroup$ @Jake: what exactly is your reference for the definition? this notation looks so unnecessarily heavy... $\endgroup$ – Vladimir Dotsenko Nov 30 '14 at 15:22
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    $\begingroup$ @VladimirDotsenko The course is in fact on affine Kac-Moody algebras, but this particular question within the context of the course is somehow about $A_{2}$. $\endgroup$ – Jake Nov 30 '14 at 15:34
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    $\begingroup$ @VladimirDotsenko Yes, I see. The course is in fact quite terse and often unnecessary challenging, for reasons exactly like this. I looked at the Wikipedia entry; however, in the definition it includes material not present in this or previous courses I took. I see that I should go beyond the course scope to have a better understanding of the subject matter. $\endgroup$ – Jake Nov 30 '14 at 15:55
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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.)

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    $\begingroup$ Jim, it might help the OP to remark that positive roots for $sl_3$ are $\alpha$, $\beta$, and $\alpha+\beta$, so in general a linear combination representing $m\alpha+n\beta$ is completely defined by the coefficient of $\alpha+\beta$, which prompts the answer $\min(m,n)+1$ (of course agreeing with the answer $4$ that you give). $\endgroup$ – Vladimir Dotsenko Nov 30 '14 at 15:24
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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.

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