## The Weyl character formula for an affine Lie algebra

Last update: 27 July 2013

## The Weyl character formula for an affine Lie algebra

For $\lambda \in {𝔥}^{*},$ the theta function ${\Theta }_{\lambda }$ corresponding to $\lambda$ is given by

$e|λ|22mδ Θλ=∑β∈𝔥∘ℤ etβ(λ), (1.22)$

the sum over the ${\stackrel{\circ }{𝔥}}_{ℤ}\text{-orbit}$ of $\lambda$ (see [Kac1104219, (12.7.2)]). The Weyl numerator (see [Mac1983, (5.1)] and [Kac1104219, Theorem 10.4]) is

$Aλ+ρ = Aaδ+λ‾+mΛ0+ρ‾+h∨Λ0 =∑w∈Wdet(w) ew(λ+ρ)= ∑w∈W0det(w) ∑β∈𝔥∘ℤ etβ(w(λ+ρ)) = e12(m+h∨)(λ+ρ|λ+ρ)δ ∑w∈W0det(w) Θw(λ‾+ρ‾)+(m+h∨)Λ0, (1.23)$

for $\lambda =a\delta +\stackrel{‾}{\lambda }+m{\Lambda }_{0}\text{.}$ Using (1.20), $\left(\lambda +\rho |\lambda +\rho \right)=a\left(m+{h}^{\vee }\right)+\left(\stackrel{‾}{\lambda }|\stackrel{‾}{\lambda }\right)$ so that all dependence on $a$ is in the initial exponential factor of ${A}_{\lambda +\rho }\text{.}$ Let

$q=e-δ. (1.24)$

By (1.11) and (1.4) the Weyl denominator formula is

$Aρ = eρ‾+h∨Λ0 ∏α∈R+ (1-e-α) dim(𝔤α) Aρ = eρ‾+h∨Λ0 ∏n=1∞ (1-qn)ℓ ∏α∈R+ (1-qn-1e-α) (1-qneα). (1.25)$

(see [Kac1104219, (10.4.4) and (12.7.4)])

The Weyl character is (see [Kac1104219, (12.7.11)])

$charL(λ‾+mΛ0) =Aλ+ρAρ= e12(m+g)⟨λ+ρ,λ+ρ⟩δ ∑w∈W0det(w) Θw(λ‾+ρ‾)+(m+g)Λ0 e12g⟨ρ,ρ⟩δ ∑w∈W0det(w) Θwρ‾+gΛ0 . (1.26)$

The string functions (see [Kac1104219, (12.7.13) and (13.8.1)]) are the coefficients ${K}_{\stackrel{‾}{\lambda }+m{\Lambda }_{0},\stackrel{‾}{\mu }+m{\Lambda }_{0}}$ (which are functions of $q\text{)}$ in the expansion

$charL(λ)= e12(m+g)⟨λ+ρ,λ+ρ⟩δ e12g⟨ρ,ρ⟩δ ∑μ∈P mod m𝔥∘ℤ Kλ‾+mΛ0,μ‾+mΛ0 Θμ‾+mΛ0,$

where $P=\left\{\lambda \in {𝔥}^{*} | ⟨\lambda ,{\alpha }_{i}^{\vee }⟩\in ℤ, \text{for} i=1,\dots ,n\right\}\text{.}$