Weyl’s character formula (to be proved shortly) gives an expression for the character of  a finite-dimensional simple quotient of a Verma module. In here, we will express the character of the simple quotient using Verma module characters.  Next time, we will calculate the coefficients involved.

Filtration on highest weight modules

Let ${W(\lambda)}$ be any highest weight module with highest weight ${\lambda}$. Then ${W(\lambda)}$ is a quotient of ${V(\lambda)}$, so the Casimir ${C}$ acts on ${W(\lambda)}$ by scalar multiplication by ${(\lambda + \rho, \lambda+\rho) - (\rho, \rho)}$.

Suppose we have a composition series

$\displaystyle 0 \subset W^0 \subset W^1 \subset \dots \subset W^t = W(\lambda)$

with successive quotients simple module ${L(\mu)}$. Then ${C}$ acts on the successive quotients by scalars that we compute in two different ways, whence by yesterday’s formula:

$\displaystyle \boxed{ (\mu + \rho, \mu + \rho) = (\lambda + \rho, \lambda+ \rho).}$

In fact, such a filtration exists:

Proposition 1 ${W(\lambda)}$ has a finite filtration whose quotients are isomorphic to ${L(\mu)}$, where ${\mu \in \lambda - \sum_{\delta \in \Delta} \mathbb{Z}_{\geq 0} \delta}$ (which we write as ${\mu \leq \lambda}$) and ${\mu}$ satisfies the boxed formula.

In general, this follows simply because every element in ${\mathcal{O}}$ has finite length, and the ${L(\mu)}$ are the only candidates for simple modules!

Theorem 2 The category ${\mathcal{O}}$ is artinian.

The only proofs I can find of this use Harish-Chandra’s theorem on characters though, so I’ll follow Sternberg in proving the proposition directly. (I hope later I’ll come back to it.) (more…)