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…)

The purpose of this post is to show that the category of finite-dimensional representations of a semismple Lie algebra is  a semisimple category; there is thus an analogy with Maschke’s theorem, except in this case the proofs are more complicated.  They can be simplified somewhat if one uses the cohomology of Lie algebras (i.e., appropriate Ext groups), which I may talk more about, but most likely only later.  Here we will give the proofs based on linear algebra.

The first step is to construct certain central elements in the enveloping algebra.

Casimir elements

Let ${B}$ be a nondegenerate invariant bilinear form on the Lie algebra ${\mathfrak{g}}$. (E.g. ${\mathfrak{g}}$ could be semisimple and ${B}$ the Killing form.) Given a basis ${e_i \in \mathfrak{g}}$, we can consider the dual basis ${f_j}$ with respect to it, i.e. such that ${B(e_i, f_j) = \delta_{ij}}$. Consider the Casimir element

$\displaystyle C := \sum e_i f_i \in U \mathfrak{g}.$

I claim that ${C}$ is independent of the choice of ${e_i}$ and is in the center of the enveloping algebra. First off, consider the isomorphisms of ${\mathfrak{g}}$-modules,

$\displaystyle \hom_k( \mathfrak{g}, \mathfrak{g}) \simeq \mathfrak{g} \otimes \mathfrak{g}^{\vee} \simeq \mathfrak{g} \otimes \mathfrak{g} .$

The last one is given by the form ${B}$. (more…)