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.

The goal is to compute {\mathrm{ch}i_{\lambda}(C)} for {C} the Casimir element as above. We shall first find a general expression.

By the PBW basis theorem, if we choose a basis {h_1 \dots h_l } of the Cartan subalgebra {\mathfrak{h}} and a {\mathfrak{sl}_2}-type basis {f_1 \dots f_m, e_1 \dots e_m} of {\bigoplus_{\alpha \in \Phi} \mathfrak{g}_{\alpha}} (with {f_i} being paired with {e_i} and spanning a subalgebra isomorphic to {\mathfrak{sl}_2} with {[f_i, e_i]}, etc.), then we can find a basis for {U\mathfrak{g}} via

\displaystyle f_1^{i_1} \dots f_m^{i_m} h_1^{j_1} \dots h_l^{j_l} e_1^{k_1} \dots e_m^{k_m} \quad (*).

There is a vector subspace {U \mathfrak{h} \subset U\mathfrak{g}}, and any weight {\lambda} extends to a character {U \mathfrak{h} \rightarrow \mathbb{C}}, still denoted {\lambda}. With respect to the above basis, there is a projection {pr: U \mathfrak{g} \rightarrow U \mathfrak{h}}.

Proposition 1 \displaystyle \mathrm{ch}i_{\lambda}(z) = \lambda(pr(z)), \quad \forall z \in Z(\mathfrak{g}).

This computes the character {\mathrm{ch}i}.

Write {z} as a sum of terms as in (*). Those where all the {i,k} are zero contribute precisely {\lambda(pr(z))}; it must be seen that the others contribute nothing. Any term with some {k_r>0} must annihilate the vector {v_+} because these correspond to positive weights. I claim now that no term of the form

\displaystyle f_1^{i_1} \dots f_m^{i_m} h_1^{j_1} \dots h_l^{j_l}

can even appear in the expansion for {z} in this basis. Then bracketing with any {h \in \mathfrak{h}} is an operator on {U \mathfrak{g}} diagonalizable with respect to this basis. Moreover:

\displaystyle [h, f_1^{i_1} \dots f_m^{i_m} h_1^{j_1} \dots h_l^{j_l}] = \left( \sum_q i_q \alpha_q(h) \right) f_1^{i_1} \dots f_m^{i_m} h_1^{j_1} \dots h_l^{j_l}

if {\alpha_q} ranges over the weights corresponding to {f_q}. If we choose {h} so that the thing in brackets is nonzero, we find that this is impossible, because anything in the kernel of bracketing with {h} (e.g. in the center of {U \mathfrak{g}}) cannot have any term in this basis with nonzero eigenvalue.

This proves the proposition.

Action of the Casimir

It is now necessary to compute {\lambda(pr(C))} for {C} the Casimir.

First, choose a basis {h_i} for {\mathfrak{h}} and let {k_j} be the dual basis. Next, choose for each root {\alpha \in \Phi} some {u_{\alpha} \in \mathfrak{g}_{\alpha}} and a dual {v_{\alpha} \in \mathfrak{g}_{-\alpha}}; then {B(u_{\alpha}, v_{\beta}) = \delta_{\alpha \beta}}, of course. We can arrange things such that {u_{\alpha} = v_{-\alpha}, v_{\alpha} = u_{-\alpha}}.

We have

\displaystyle C = \sum h_i k_i + \sum u_{\alpha} v_{\alpha} = \sum h_i k_i + \sum_{\alpha \in \Phi^+} u_{\alpha} v_{\alpha} + v_{\alpha} u_{\alpha}

by the way we arranged things. However, the term {u_{\alpha} v_{\alpha}} is not in the right form for the basis. So we write this as

\displaystyle \sum h_i k_i + 2 \sum_{\alpha \in \Phi^+} v_{\alpha} u_{\alpha} + \sum_{\alpha \in \Phi^+} [ u_{\alpha}, v_{\alpha}].

Now, in the projection {pr}, the middle term disappears, so we find that

\displaystyle \mathrm{ch}i_{\lambda}(C) = \sum \lambda(h_i) \lambda(k_i) + \lambda\left( \sum_{\alpha \in \Phi^+} t_{\alpha} \right)

where {t_{\alpha}} is the dual to {\alpha} under the identification {\mathfrak{h} \simeq \mathfrak{h}^{\vee}}. (This is a general fact about brackets between {\mathfrak{g}^{\alpha}, \mathfrak{g}^{-\alpha}}; I proved it here.)

Consider the bilinear form {(., .)} on {\mathfrak{h}^{\vee}} obtained by duality, which makes the dual bases of {h_i, k_i} in {\mathfrak{h}^{\vee}} biorthogonal; then

\displaystyle (\lambda, \lambda) = \sum \lambda(h_i) \lambda(k_i)

because this is checked for those dual bases in {\mathfrak{h}^{\vee}}. Moreover, by assumption {(\lambda, \alpha) = (t_{\lambda}, t_{\alpha}) = \lambda(t_{\alpha})}, so in total we find:

Proposition 2 \displaystyle \mathrm{ch}i_{\lambda}(C) = (\lambda, \lambda) + 2(\lambda, \rho) = (\lambda + \rho, \lambda+\rho) - (\rho, \rho)

for {\rho} one-half the sum of the positive roots. 


Next: to show that the character of simple quotients is a sum of Verma module characters, for integral weights anyway.