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…)

Let ${X}$ be a representation of a semisimple Lie algebra ${\mathfrak{g}}$, a Cartan subalgebra ${\mathfrak{h}}$, and some choice of splitting ${\Phi = \Phi^+ \cup \Phi^-}$ on the roots.

Recall from the representation theory of finite groups that to each representation of a finite group ${G}$ one can associate a character function ${\chi}$, and that the ring generated by the characters is the Grothendieck ring of the semisimple tensor category ${Rep(G)}$. There is something similar to be said for semisimple Lie algebras. So, assume ${\mathfrak{h}}$ acts semisimply on ${X}$ and that the weight spaces are finite-dimensional, and set formally

$\displaystyle \mathrm{ch}(X) := \sum_{\lambda} \dim X_{\lambda} e(\lambda).$

In other words, we define the character so as to include all the information on the size of the weight spaces at once.

It is necessary, however, to define what ${e(\lambda)}$ for ${\lambda \in \mathfrak{h}^{\vee}}$. Basically, it is just a formal symbol; ${\mathrm{ch}(X)}$ can more rigorously be thought of as a function ${\mathfrak{h}^{\vee} \rightarrow \mathbb{Z}_{\geq 0}}$. Nevertheless, we want to think of ${e(\lambda)}$ as a formal exponential in a sense; we want to have ${e(\lambda) e(\lambda') = e(\lambda + \lambda')}$. The reason is that we can tensor two representations, and we want to talk about multiplying to characters.

I now claim that the above condition on ${X}$ makes sense for ${X \in \mathcal{O}}$, the BGG category. This will follow because it is true for highest weight modules and we have:

Proposition 1 If ${M \in \mathcal{O}}$, then there is a finite filtration on ${M}$ whose quotients are highest weight modules. (more…)

So, let’s suppose that we have a splitting of the roots ${\Phi = \Phi^+ \cup \Phi^-}$, as before, associated to a semisimple Lie algebra ${\mathfrak{g}}$ and a Cartan subalgebra ${\mathfrak{h}}$. Recall that a vector ${v \in V}$ for a representation ${V}$ of ${\mathfrak{g}}$ (not necessarily finite-dimensional!) is called a highest weight vector if ${v}$ is annihilated by the nilpotent algebra ${\mathfrak{n} = \bigoplus_{\alpha \in \Phi^+}}$.

Let ${V}$ be a highest weight module, generated by a highest weight vector ${v}$. We proved before, using a PBW basis for ${U\mathfrak{g}}$, that ${V}$ is the direct sum of its finite-dimensional weight spaces—in particular, ${\mathfrak{h}}$ acts semisimply, which is not a priori obvious since ${V}$ is finite-dimensional—and so is any subrepresentation. The highest weight space is one-dimensional.  Now I am actually going to talk about them in a bit more detail.

Proposition 1 ${V}$ is indecomposable and has a unique maximal submodule and unique simple quotient.

Indeed, let ${W,W' \subset V}$ be any proper submodules; we will prove ${W + W' \neq V}$. If either contains ${v}$, then it is all of ${V}$. So we may assume both don’t contain ${v}$; by the above fact that ${W,W'}$ decompose into weight spaces, they have no vectors of weight the same as ${v}$. So neither does ${W + W'}$, which means that ${W+W' \neq V}$.

We can actually take the sum of all proper submodules of ${V}$; the above argument shows that this sum does not contain ${v}$ (and has no vectors with nonzero ${v}$-component). The rest of the proposition is now clear.

There is an important category, the BGG category ${\mathcal{O}}$, defined as follows: ${X \in \mathcal{O}}$ if ${X}$ is a representation of ${\mathfrak{g}}$ on which ${\mathfrak{n}}$ acts locally nilpotently (i.e., each ${x \in X}$ is annihilated by some power of ${\mathfrak{n}}$ in ${U\mathfrak{g}}$), ${\mathfrak{h}}$ acts semisimply, and ${X}$ is finitely generated over the enveloping algebra ${U\mathfrak{g}}$. I’m hoping to say a few things about category ${\mathcal{O}}$ in the future, but for now, what we’ve seen is that highest weight modules belong to it. It is in fact a theorem that any object in ${\mathcal{O}}$ has a filtration whose quotients are highest weight modules.

Proposition 2 Any simple highest weight modules of the same weight are isomorphic. (more…)