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.

  • $\begingroup$ If you look at the definition of the module, you''ll notice thatit is the number of ways of writing the weight you want as a sum of simple roots. $\endgroup$ – Mariano Suárez-Álvarez Nov 28 '14 at 4:45
  • $\begingroup$ @MarianoSuárez-Alvarez I have looked at the definition of a Verma module (something like quotient of a quotient of an enveloping algebra), and it appears to be so complicated, I have zero intuition about it. Perhaps you are referring to a different, but equivalent definition? $\endgroup$ – Jake Nov 30 '14 at 10:16

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.


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