Weyl character formula

Remarks

The Weyl denominator formula says $\sum_{w \in W} \varepsilon(w) e^{w\rho} = \prod_{\alpha \in \Phi^+} (e^{\alpha/2} - e^{-\alpha/2})$, the case $\lambda = 0$. One algebraic proof writes $\operatorname{ch} L(\lambda)$ as an alternating sum of characters of Verma modules $M(w(\lambda + \rho) - \rho)$. The Harish-Chandra isomorphism shows that only these Verma modules can occur, because they share the central character of $L(\lambda)$. Each Verma character equals $e^{\mu} \prod_{\alpha \in \Phi^+} (1 - e^{-\alpha})^{-1}$, which gives the formula. Example: for $\mathfrak{sl}_2(\C)$, write $e^k$ for the weight with $\lambda(h) = k$; then $\rho$ is $1$ and $W = \set{\pm 1}$. The formula reads $(e^{1} - e^{-1}) \operatorname{ch} L(n) = e^{n+1} - e^{-n-1}$, so $\operatorname{ch} L(n) = e^n + e^{n-2} + \dots + e^{-n}$. For compact Lie groups the same formula, evaluated on a maximal torus, is the Weyl character formula for compact groups.

Going deeper

For a student

The formula has many consequences: the Weyl dimension formula, Kostant's multiplicity formula, and the decomposition of tensor products. The Freudenthal multiplicity formula computes the same multiplicities by a recursion that is easier to run by hand. The Bernstein–Gelfand–Gelfand resolution lifts the alternating sum to an exact sequence of Verma modules. For affine Lie algebras the analogue is the Kac–Weyl character formula.