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…)

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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…)