The Weyl character formula for an affine Lie algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 27 July 2013

The Weyl character formula for an affine Lie algebra

For λ𝔥*, the theta function Θλ corresponding to λ is given by

e|λ|22mδ Θλ=β𝔥 etβ(λ), (1.22)

the sum over the 𝔥-orbit of λ (see [Kac1104219, (12.7.2)]). The Weyl numerator (see [Mac1983, (5.1)] and [Kac1104219, Theorem 10.4]) is

Aλ+ρ = Aaδ+λ+mΛ0+ρ+hΛ0 =wWdet(w) ew(λ+ρ)= wW0det(w) β𝔥 etβ(w(λ+ρ)) = e12(m+h)(λ+ρ|λ+ρ)δ wW0det(w) Θw(λ+ρ)+(m+h)Λ0, (1.23)

for λ=aδ+λ+mΛ0. Using (1.20), (λ+ρ|λ+ρ)= a(m+h)+ (λ|λ) so that all dependence on a is in the initial exponential factor of Aλ+ρ. Let

q=e-δ. (1.24)

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

Aρ = eρ+hΛ0 αR+ (1-e-α) dim(𝔤α) Aρ = eρ+hΛ0 n=1 (1-qn) αR+ (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+ρδ wW0det(w) Θw(λ+ρ)+(m+g)Λ0 e12gρ,ρδ wW0det(w) Θwρ+gΛ0 . (1.26)

The string functions (see [Kac1104219, (12.7.13) and (13.8.1)]) are the coefficients Kλ+mΛ0,μ+mΛ0 (which are functions of q) in the expansion

charL(λ)= e12(m+g)λ+ρ,λ+ρδ e12gρ,ρδ μPmodm𝔥 Kλ+mΛ0,μ+mΛ0 Θμ+mΛ0,

where P= { λ𝔥*| λ,αi ,fori=1,,n } .

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