# Theta functions in affine Kac-Moody algebras

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## introduction

### notation

• 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

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

definition

Let $$k\in \mathbb{Z}_{\geq 1}$$ be the level of $$\lambda$$. Note $$\lambda=\bar{\lambda}+k\Lambda_0+\xi\delta$$. and $$|\lambda|^2=|\bar{\lambda}|^2+2k\xi$$. The theta function is defined by \begin{align} \Theta_{k,\lambda} &=e^{-\frac{|\lambda|^2\delta}{2k}}\sum_{\gamma \in M}e^{t_{\gamma}(\lambda)} \\ &=e^{-\frac{|\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{|\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}

### analytic function

• FMS 605p, (14.318)
• let $$(\zeta;\tau;t)=\zeta+ \tau \Lambda_0+ t\delta\in \mathfrak{h}^{*}$$
• once we evaluate them at $$\xi=-2\pi i (\zeta;\tau;t)$$, we get

$\Theta_{\lambda}^{(k)}(\zeta,\tau,t)=e^{-2\pi i k t}\sum_{\alpha^{\vee}\in Q^{\vee}}e^{-2\pi i k (\alpha^{\vee}+\lambda/k,\zeta)}e^{i\pi k \tau |\alpha^{\vee}+\lambda/k|^2}$

thm

$\Theta_{\lambda}^{(k)}(\frac{\zeta}{\tau},-\frac{1}{\tau},t+\frac{|\zeta|^2}{2\tau})=(-\frac{i \tau}{k})^{r/2}\frac{1}{\operatorname{vol}(Q^{\vee})}\sum_{\mu\in P/k Q^{\vee}}e^{-2\pi i (\mu,\nu)/k} \Theta_{\mu}^{(k)}(\zeta,\tau,t)$

## $$A_1$$ example

• let $$z=e^{-\alpha_1}$$

### level k=1

• $$\lambda=0$$

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