Now choose a dominant integral weight ${\lambda}$. By yesterday, we have:

$\displaystyle \mathrm{ch} L(\lambda) = \sum_{\mu < \lambda} b(\lambda, \mu) \mathrm{ch} V(\mu).$

Our first aim is to prove

Proposition 1 ${b(\lambda, w \cdot \lambda) = (-1)^w}$ for ${w \in W}$, the Weyl group, and ${\cdot}$ the dot action. For ${\mu \notin W\lambda}$, we have ${b(\lambda, \mu)=0}$.

After this, it will be relatively easy to obtain WCF using a few formal manipulations. To prove it, though, we use a few such formal manipulations already.

Manipulations in the group ring

I will now define something that is close to an “inverse” of the Verma module character ${p \ast e(\lambda)}$ for ${p(\lambda)}$ the Kostant partition function evaluated at ${-\lambda}$ (inverse meaning in the group ring ${\mathbb{Z}[L]}$, where ${L}$ is the weight lattice of ${\beta}$ with ${<\beta, \delta> \in \mathbb{Z} \ \forall \delta \in \Delta}$). Define ${q}$ by

$\displaystyle q = \prod_{\alpha \in \Phi^+} \left( e(\alpha/2) - e(-\alpha/2) \right).$

I claim that

$\displaystyle q = e(\rho) \prod_{\alpha \in \Phi^+} (1 - e(-\alpha)), \ \ wp = (-1)^w p, \quad \forall w \in W.$

(Note that since ${w}$ acts on the weight lattice ${L}$, it clearly acts on the group ring. Here, as usual, ${\rho = \frac{1}{2} \sum_{\gamma \in \Phi^+} \gamma}$.)

The first claim is obvious. The second follows because the minimal expression of ${w}$ as a product of reflections has precisely as many terms as the number of positive roots that get sent into negative roots by ${w}$, and a reflection has determinant ${-1}$. (more…)

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…)

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…)

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…)

Today’s is going to be a long post, but an important one.  It tells us precisely what weights are allowed to occur as highest weights in finite-dimensional representations of a semisimple Lie algebra.

Dominant integral weights

Let ${V}$ be a finite-dimensional simple representation of a semisimple Lie algebra ${\mathfrak{g}}$, with Cartan subalgebra ${\mathfrak{h}}$, and root space decomposition ${\mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_{\alpha}}$. Suppose given a base ${\Delta}$ and a corresponding division ${\Phi = \Phi^+ \cup \Phi^-}$.

For each ${\alpha \in \Phi^+}$, choose ${X_{\alpha} \in \mathfrak{g}_{\alpha}, Y_{\alpha} \in \mathfrak{g}_{-\alpha}}$ such that ${[X_{\alpha},Y_{\alpha}] = H_{\alpha}}$ and ${X_{\alpha}, Y_{\alpha}, H_{\alpha}}$ generate a subalgebra ${\mathfrak{s}_{\alpha}}$ isomorphic to ${\mathfrak{sl}_2}$.

Consider the weight space decomposition

$\displaystyle V = \bigoplus_{\beta \in \Pi} V_{\beta}$

where ${\Pi}$ denotes the set of weights of ${V}$. Then if ${\beta \in \Pi}$ is the weight associated to a highest weight vector, ${\beta(H_{\alpha})}$ is necessarily a nonnegative integer by the representation theory of ${\mathfrak{sl}_2}$. In other words,

$\displaystyle <\beta, \alpha> := 2 \frac{ (\beta, \alpha)}{(\alpha, \alpha)} \in \mathbb{Z}_{\geq 0}.$

Any weight ${\beta}$ satisfying that identity for all ${\alpha \in \Phi^+}$ is called dominant integral. We have shown that the highest weight of a finite-dimensional simple ${\mathfrak{g}}$-representation is necessarily dominant integral. In fact, given a dominant integral weight, we can actually construct such a finite-dimensional simple module.

The set of merely integral weights—those ${\beta}$ with ${<\beta, \alpha> \in \mathbb{Z}}$ for ${\alpha \in \Phi}$—form a lattice, spanned by vectors ${\lambda_i}$ such that ${<\lambda_i, \delta_j> = \delta_{ij}}$, where the last ${\delta_{ij}}$ is the Kronecker delta.

Theorem 1 The unique simple quotient of the Verma module ${V(\beta)}$ is finite-dimensional if and only if ${\beta}$ is dominant integral. (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…)

I’m keeping the same notation as all the previous posts here on semisimple Lie algebras.

Consider the real vector space

$\displaystyle E = \sum_{\alpha \in \Phi} \mathbb{R} \alpha \subset \mathfrak{h}^{\vee}.$

I claim that the form ${(\cdot, \cdot)}$ (obtained by the isomorphism ${\mathfrak{h}^{\vee} \rightarrow \mathfrak{h}}$ induced by the Killing form and the Killing form itself) is actually an inner product making ${E}$ into a euclidean space. To see this, we will check that ${(\alpha, \alpha) > 0}$ for all ${\alpha}$. Indeed:

$\displaystyle (\alpha, \alpha) = B(T_{\alpha}, T_{\alpha})$

where ${B}$ is the Killing form, by definition.

Now

$\displaystyle B(T_{\alpha}, T_{\alpha}) = \mathrm{Tr}_{\mathfrak{g}} ( \mathrm{ad} T_{\alpha}^2) = \sum_{\beta \in \Phi} \mathrm{Tr}_{\mathfrak{g}_{\beta}} ( \mathrm{ad} T_{\alpha}^2) .$

Now ${T_{\alpha}}$ acts by the scalar ${\beta(T_{\alpha}) = (\beta, \alpha)}$ on ${\mathfrak{g}_{\beta}}$, so after dividing by ${(\alpha, \alpha)^2}$, this becomes

$\displaystyle (\alpha, \alpha)^{-1} = \sum_{\beta \in \Phi} \left( \frac{ (\beta, \alpha)}{(\alpha, \alpha ) } \right)^2.$

But as we showed yesterday, ${\frac{ (\beta, \alpha)}{(\alpha, \alpha )} \in \mathbb{Q}}$, so the sum in question is actually positive. This proves one half of:

Proposition 1 ${E}$ is a euclidean space and ${\mathfrak{h}^{\vee} = E \oplus iE}$. (more…)

OK, now we’ve gotten some of the basic facts about the root space decomposition down. So, as usual ${\mathfrak{g}}$ is a semisimple Lie algebra and ${\mathfrak{h}}$ a Cartan subalgebra; we have the decomposition ${\mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_{\alpha}}$, where ${\Phi \subset \mathfrak{h}^{\vee}}$ is the root system. For each ${\alpha \in \Phi}$, we can choose a pair of vectors ${X_{\alpha} \in \mathfrak{g}_{\alpha}< Y_{\alpha} \in \mathfrak{g}_{-\alpha}, H_{\alpha} \in \mathfrak{h}}$. Then ${X_{\alpha}, Y_{\alpha}, H_{\alpha}}$ generate a subalgebra ${\mathfrak{s}_{\alpha} \subset \mathfrak{g}}$ which is isomorphic to ${\mathfrak{sl}_2}$. Here ${\alpha(H_{\alpha})=2}$ and ${H_{\alpha}}$ is a multiple of ${T_{\alpha}}$, which in turn is the dual to ${\alpha}$ under the Killing form that identifies ${\mathfrak{h} \simeq \mathfrak{h}^{\vee}}$.

That was a lightning review of where we are; if you’ve missed something, check back at this post.

The notation ${\mathfrak{s}_{\alpha}}$ suggests that the algebra should only depend on ${\alpha}$ and not on the particular choice of ${X_{\alpha}, Y_{\alpha}}$ (but ${H_{\alpha}}$ is uniquely determined from ${\alpha(H_{\alpha})=2}$ and ${H_{\alpha} \in \mathbb{C} T_{\alpha}}$). Indeed, this is the case, and it follows from

Proposition 1 When ${\alpha \in \Phi}$, ${\mathfrak{g}_{\alpha}}$ is one-dimensional.

Choose any ${\mathfrak{s}_{\alpha}}$ coming from suitable ${X_{\alpha}, Y_{\alpha}}$ and ${H_{\alpha}}$. We have a representation of ${\mathfrak{s}_{\alpha}}$ on

$\displaystyle V := \bigoplus_{\mathbb{Z} \alpha} \mathfrak{g}_{\alpha}$

(recall ${\mathfrak{g}_0 = \mathfrak{h}}$) and we can apply the representation theory of ${\mathfrak{sl}_2}$ to it. (more…)

I talked about the Lie algebra ${\mathfrak{sl}_2}$ a while back.  Now I’m going to do it more properly, and using the tools developed.  This is going to feature prominently in some of the proofs in the sequel.

Now, let’s see how all this works for the familiar case of ${\mathfrak{sl}_2}$, with its usual generators ${H,X,Y}$. This is a simple Lie algebra in fact. To see this, let’s consider the ideal ${I}$ of ${\mathfrak{sl}_2}$ generated by some nonzero vector ${aX + bH + cY}$; I claim it is all of ${\mathfrak{sl}_2}$.

Consider the three cases ${a \neq 0, b \neq 0, c \neq 0}$:

First, assume ${a}$ or ${c}$ is nonzero. Bracketing with ${H}$, and again, gives

$\displaystyle -2aX + 2 c Y \in I , \ (-2)^2 a X + 2^2 cY \in I, \ (-2)^3 a X + 2^3 cY \in I.$

Using a vanderMonde invertibility of this system of linear equations, we find that either ${X}$ or ${Y}$ belongs to ${I}$. Say ${X}$ does, for definiteness; then ${H = [X,Y] \in I}$ too; from this, ${Y = -\frac{1}{2} [H,Y] \in I}$ as well. Thus ${I = \mathfrak{sl}_2}$.

If ${a=c=0}$, then from ${b \neq 0}$, we find ${H \in I}$, which implies ${X = \frac{1}{2}[H,X] \in I}$ and similarly for ${Y}$. Thus ${I= \mathfrak{sl}_2}$.

I claim now that the algebra ${\mathbb{C} H}$ is in fact a Cartan subalgebra. Indeed, it is easily checked to be maximal abelian. Moreover, since ${H}$ acts by a diagonalizable operator on the faithful representation on ${\mathbb{C}^2}$, it follows that ${H \in \mathfrak{sl}_2}$ is (abstractly) semisimple. (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…)