Theta functions in affine Kac-Moody algebras

introduction

notation

• let $k\in \mathbb{Z}_{\geq 1}$ be the level
• Let $M=Q^{\vee}$. This is also the $\mathbb{Z}$-span of $W\theta$ where $\theta$ is the highest root
• for $\gamma\in M$, define $t_{\gamma} : \mathfrak{h}^{*}\to \mathfrak{h}^{*}$ by

$$t_{\gamma}(\lambda)=\lambda+\lambda(c)\gamma-\left(\frac{1}{2}\lambda(c)|\gamma|^2+(\gamma,\lambda)\right)\delta $$

• note $\lambda=\bar{\lambda}+k\Lambda_0$
• note that $|\lambda|^2=|\bar{\lambda}|^2$

definition

The theta function is defined by $$ \begin{align} \Theta_{k,\lambda} &=e^{-\frac{|\bar{\lambda}|^2\delta}{2k}}\sum_{\gamma \in M}e^{t_{\gamma}(\lambda)} \\ &=e^{-\frac{|\bar{\lambda}|^2\delta}{2k}}e^{\lambda}\sum_{\gamma \in M}e^{k\gamma}q^{\frac{k(\gamma,\gamma)}{2}+(\gamma,\lambda)}\\ &=e^{\lambda}\sum_{\gamma \in M}e^{k\gamma}q^{\frac{1}{2k}(k\gamma+\lambda,k\gamma+\lambda)} \end{align} $$ We also have $$ \begin{align} \Theta_{k,\lambda} &=e^{-\frac{|\bar{\lambda}|^2\delta}{2k}}\sum_{\gamma \in M}e^{t_{\gamma}(\lambda)} \\ &=e^{k\Lambda_0}\sum_{\mu\in Q^{\vee}+\frac{\bar{\lambda}}{k}}e^{k\mu-\frac{1}{2}k( \mu,\mu ) \delta}\\ &=e^{k\Lambda_0}\sum_{\mu\in Q^{\vee}+\frac{\bar{\lambda}}{k}}e^{k\mu}q^{\frac{k}{2}( \mu,\mu )} \end{align} $$

$A_1$ example

• level k=1, $\lambda=0$
• let $z=e^{-\alpha_1}$

$$ \Theta_{1,0}=1 + q (1/z + z) + q^4 (1/z^2 + z^2) + q^9 (1/z^3 + z^3)+\cdots $$

related items

• Affine Weyl group
• Theta functions
• Cube root of the j-invariant and E8

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxbW9iUTgtaThCM2s/edit