Verma module

Let $𝔤$ be a semi-simple Lie algebra, $𝔥$ a Cartan subalgebra, and$𝔟$ a Borel subalgebra. We work over a field $F$. Given a weight $\lambda \in {𝔥}^{*}$, let $F_{\lambda }$ be the 1-d dimensional $𝔟$-module on which $𝔥$ acts by multiplication by $\lambda$, and the positive root spaces act trivially. Now, the Verma module $M_{\lambda }$ of the weight $\lambda$ is the $𝔤$-module induced from $F_{\lambda }$, i.e.

$$M_{\lambda }=𝒰(𝔤){\otimes }_{𝒰(𝔟)}{F}_{\lambda }.$$

Using the Poincaré-Birkhoff-Witt theorem we see that as a vector space $M_{\lambda }$ is isomorphic to $𝒰(\overline{𝔫})$, where $\overline{𝔫}$ is the sum of the negative weight spaces (so $𝔤=𝔟\oplus \overline{𝔫})$. In particular $M_{\lambda }$ is infinite dimensional.

We say a $𝔤$-module $V$ is a highest weight module if it has a weight $\mu \in {𝔥}^{*}$ and a non-zero vector $v\in V_{\mu }$ with $Xv=0$ for any $X$ in a positive root space and such that $V$ is generated as a $𝔤$-module by $v$. The Verma module $M_{\lambda }$ is a highest weight module and we fix a generator $1\otimes 1$.

The most important property of Verma modules is that they are universal amongst highest weight modules, in the following sense. If $V$ is a highest weight module generated by $v$ which has weight $\lambda$ then there is a unique surjective homomorphism $M_{\lambda }\to V$ which sends $1\otimes 1$ to $v$. That is, all highest weight modules with highest weight $\lambda$ are quotients of $M_{\lambda }$. Also, $M_{\lambda }$ has a unique maximal submodule, so there is a unique irreducible representation with highest weight $\lambda$. If $\lambda$ is dominant and integral then this module is finite dimensional.