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…)

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…)

Recall that in the representation theory of ${\mathfrak{sl}_2}$, one considered an element ${H}$ and its action on a representation ${V}$. We looked for its largest eigenvalue and the corresponding highest weight vector.

There is something along the same lines to be done here for arbitrary semisimple Lie algebras, though it is much more complicated (and interesting).   I’m only going to scratch the surface today.

Let ${\mathfrak{g}}$ be a semisimple Lie algebra and ${\mathfrak{h}}$ a Cartan subalgebra. Then ${\mathfrak{h}}$ is to play the role of ${H}$ in ${\mathfrak{sl}_2}$; the ${X,Y}$ matrices in ${\mathfrak{sl}_2}$ are now replaced by the root space decomposition

$\displaystyle \mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_{\alpha}.$

We know that ${\mathfrak{h}}$ acts on a representation ${V}$ of ${\mathfrak{g}}$ by commuting semisimple transformations, so we can write

$\displaystyle \mathfrak{h} = \bigoplus_{\beta \in \mathfrak{h}^{\vee}} V_{\beta}$

where ${V_{\beta} := \{ v \in V: hv = \beta(h) v \ \forall h \in \mathfrak{h} \}}$. These are called the weight spaces, and the ${\beta}$ are called weights.

Now

$\displaystyle g_{\alpha} V_{\beta} \subset V_{\alpha + \beta }$

by an analog of the “fundamental calculation,” proved as follows. Let ${h \in \mathfrak{h}, x \in \mathfrak{g}_{\alpha}, v \in V_{\beta}}$. Then

$\displaystyle h (x v) =xh(v) + [h,x] v = x (\alpha(h)) v + \beta(h) x v = (\alpha + \beta)(h) xv.$ (more…)

This post is the second in the series on ${\mathfrak{sl}_2}$ and the third in the series on Lie algebras. I’m going to start where we left off yesterday on ${\mathfrak{sl}_2}$, and go straight from there to classification.  Basically, it’s linear algebra.

Classification

We’ve covered all the preliminaries now and we can classify the ${\mathfrak{sl}_2}$-representations, the really interesting material here. By Weyl’s theorem, we can restrict ourselves to irreducible representations. Fix an irreducible ${V}$.

So, we know that ${H}$ acts diagonalizably on ${V}$, which means we can write

$\displaystyle V = \bigoplus_\lambda V_\lambda$

where ${Hv_\lambda = \lambda v_{\lambda}}$ for each ${\lambda}$, i.e. ${V_\lambda}$ is the ${H}$-eigenspace.