I’m now aiming to get to the major character formulas for the (finite-dimensional) simple quotients ${L(\lambda)}$ for ${\lambda}$ dominant integral. They will follow from formal manipulations with character symbols and a bit of reasoning with the Weyl group. First, however, it is necessary to express ${\mathrm{ch} L(\lambda)}$ as a sum of characters of Verma modules. We will do this by considering any highest weight module of weight ${\lambda}$ for ${\lambda}$ integral but not necessarily dominant, and considering a filtration on it whose quotients are simple modules ${L(\mu)}$ where there are only finitely many possibilities for ${\mu}$. Applying this to the Verma module, we will then get an expression for ${\mathrm{ch} V(\lambda)}$ in terms of ${\mathrm{ch} L(\lambda)}$, which we can then invert.

First, it is necessary to study the action of the Casimir element (w.r.t. the Killing form). Recall that this is defined as follows: consider a basis ${B}$ for the semisimple Lie algebra ${\mathfrak{g}}$ and its dual basis ${B'}$ under the Killing form isomorphism ${\mathfrak{g} \rightarrow \mathfrak{g}^{\vee}}$. Then the Casimir element is

$\displaystyle \sum_{b \in B} b b^{\vee} \in U \mathfrak{g}$

for ${b^{\vee} \in B'}$ dual to ${b}$. As we saw, this is a central element. I claim now that the Casimir acts by a constant factor on any highest weight module, and that the constant factor is determined by the weight in such a sense as to give information about the preceding filtration (this will become clear shortly).

Central characters

Let ${D \in Z(\mathfrak{g}) := \mathrm{cent} \ U \mathfrak{g}}$ and let ${v_+ \in V(\lambda)}$ be the Verma module. Then ${Dv_+}$ is also a vector with weight ${v_+}$, so it is a constant multiple of ${v_+}$. Since ${v_+}$ generates ${V(\lambda)}$ and ${D}$ is central, it follows that ${D}$ acts on ${V(\lambda)}$ by a scalar ${\mathrm{ch}i_{\lambda}(D)}$. Then ${\mathrm{ch}i_{\lambda}}$ becomes a character ${ Z(\mathfrak{g}) \rightarrow \mathbb{C}}$, i.e. an algebra-homomorphism. There is in fact a theorem of Harish-Chandra that states that ${\mathrm{ch}i_{\lambda}}$ determines the weight ${\lambda}$ up to “linkage” (i.e. up to orbits of the dot action of the Weyl group: ${w \dot \lambda := w(\lambda + \rho) - \rho}$), though I shall not prove this here. (more…)

As we saw in the first post, a representation of a finite group ${G}$ can be thought of simply as a module over a certain ring: the group ring. The analog for Lie algebras is the enveloping algebra. That’s the topic of this post.
The basic idea is as follows. Just as a representation of a finite group ${G}$ was a group-homomorphism ${G \rightarrow Aut(V)}$ for a vector space, a representation of a Lie algebra ${\mathfrak{g}}$ is a Lie-algebra homomorphism ${\mathfrak{g} \rightarrow \mathfrak{g}l(V)}$. Now, ${\mathfrak{g}l(V)}$ is the Lie algebra constructed from an associative algebra, ${End(V)}$—just as ${Aut(V)}$ is the group constructed from ${End(V)}$ taking invertible elements.