Theta functions in affine Kac-Moody algebras

introduction

• let $k\in \mathbb{Z}_{\geq 1}$ be the level
• definition

$$ \begin{align} \theta_{k,\lambda} &=\sum_{\mu\in Q^{\vee}+\frac{\lambda}{k}}e^{k(\mu-\frac{1}{2}\langle \mu,\mu \rangle \delta)}\\ &=\sum_{\mu\in Q^{\vee}+\frac{\lambda}{k}}e^{k\mu}q^{\frac{k}{2}\langle \mu,\mu \rangle} \end{align} $$

$A_1$ example

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

$$ \theta_{1,0}=1 + q (1 + 1/z + z) + q^2 (2 + 1/z + z) + q^3 (3 + 2 (1/z + z)) + q^4 (5 + 1/z^2 + z^2 + 3 (1/z + z))+\cdots $$

computational resource

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