The Weyl character formula for an affine Lie algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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

$$\begin{array}{cc}{e}^{\frac{{\left|\lambda \right|}^2}{2m}\delta }{\Theta }_{\lambda }=\sum _{\beta \in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}}{e}^{t_{\beta }\left(\lambda \right)},& \text{(1.22)}\end{array}$$

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

$$\begin{array}{cc}\begin{array}{ccc}{A}_{\lambda +\rho }& =& A_{a\delta +\bar{\lambda }+m{\Lambda }_0+\bar{\rho }+h^{\vee }{\Lambda }_0}=\sum _{w\in W}\text{det}\left(w\right)e^{w(\lambda +\rho )}=\sum _{w\in W_0}\text{det}\left(w\right)\sum _{\beta \in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}}{e}^{t_{\beta }\left(w(\lambda +\rho )\right)}\\ & =& e^{\frac{1}{2(m+h^{\vee })}(\lambda +\rho |\lambda +\rho )\delta }\sum _{w\in W_0}\text{det}\left(w\right){\Theta }_{w(\bar{\lambda }+\bar{\rho })+(m+h^{\vee }){\Lambda }_0},\end{array}& \text{(1.23)}\end{array}$$

for $\lambda =a\delta +\bar{\lambda }+m{\Lambda }_0\text{.}$ Using (1.20), $(\lambda +\rho |\lambda +\rho )=a(m+h^{\vee })+\left(\bar{\lambda }\right|\bar{\lambda })$ so that all dependence on $a$ is in the initial exponential factor of $A_{\lambda +\rho }\text{.}$ Let

$$\begin{array}{cc}q=e^{-\delta }\text{.}& \text{(1.24)}\end{array}$$

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

$$\begin{array}{cc}\begin{array}{ccc}{A}_{\rho }& =& e^{\bar{\rho }+h^{\vee }{\Lambda }_0}\prod _{\alpha \in R^{+}}{(1-e^{-\alpha })}^{\text{dim}\left({𝔤}_{\alpha }\right)}\\ A_{\rho }& =& e^{\bar{\rho }+h^{\vee }{\Lambda }_0}\prod _{n=1}^{\infty }{(1-q^n)}^{\ell }\prod _{\alpha \in R^{+}}(1-q^{n-1}{e}^{-\alpha })(1-q^n{e}^{\alpha })\text{.}\end{array}& \text{(1.25)}\end{array}$$

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

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

$$\begin{array}{cc}{\text{char}}_{L(\bar{\lambda }+m{\Lambda }_0)}=\frac{A_{\lambda +\rho }}{A_{\rho }}=\frac{e^{\frac{1}{2(m+g)}⟨\lambda +\rho ,\lambda +\rho ⟩\delta }{\sum }_{w\in W_0}\text{det}\left(w\right){\Theta }_{w(\bar{\lambda }+\bar{\rho })+(m+g){\Lambda }_0}}{e^{\frac{1}{2g}⟨\rho ,\rho ⟩\delta }{\sum }_{w\in W_0}\text{det}\left(w\right){\Theta }_{w\bar{\rho }+g{\Lambda }_0}}\text{.}& \text{(1.26)}\end{array}$$

The string functions (see [Kac1104219, (12.7.13) and (13.8.1)]) are the coefficients $K_{\bar{\lambda }+m{\Lambda }_0,\bar{\mu }+m{\Lambda }_0}$ (which are functions of $q\text{)}$ in the expansion

$${\text{char}}_{L\left(\lambda \right)}=\frac{e^{\frac{1}{2(m+g)}⟨\lambda +\rho ,\lambda +\rho ⟩\delta }}{e^{\frac{1}{2g}⟨\rho ,\rho ⟩\delta }}\sum _{\mu \in P\hspace{0.17em}\text{mod}\hspace{0.17em}m{\stackrel{\circ }{𝔥}}_{\mathbb{Z}}}{K}_{\bar{\lambda }+m{\Lambda }_0,\bar{\mu }+m{\Lambda }_0}{\Theta }_{\bar{\mu }+m{\Lambda }_0},$$

where $P=\{\lambda \in {𝔥}^{*}\hspace{0.17em}|\hspace{0.17em}⟨\lambda ,{\alpha }_i^{\vee }⟩\in \mathbb{Z},\hspace{0.17em}\text{for}\hspace{0.17em}i=1,\dots ,n\}\text{.}$

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