Theta functions in affine Kac-Moody algebras
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
- \(\lambda=0\)
\[ \Theta_{1,0}=1 + q (1/z + z) + q^4 (1/z^2 + z^2) + q^9 (1/z^3 + z^3)+\cdots \]