"Theta functions in affine Kac-Moody algebras"의 두 판 사이의 차이

2015년 5월 25일 (월) 05:01 판

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 $\lambda=\bar{\lambda}+k\Lambda_0$. and $|\lambda|^2=|\bar{\lambda}|^2$. 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 $$ \begin{align} \Theta_{k,\lambda} &=e^{-\frac{|\bar{\lambda}|^2\delta}{2k}}\sum_{\gamma \in M}e^{t_{\gamma}(\lambda)} \\ &=e^{k\Lambda_0}\sum_{\mu\in Q^{\vee}+\frac{\bar{\lambda}}{k}}e^{k\mu-\frac{1}{2}k( \mu,\mu ) \delta}\\ &=e^{k\Lambda_0}\sum_{\mu\in Q^{\vee}+\frac{\bar{\lambda}}{k}}e^{k\mu}q^{\frac{k}{2}( \mu,\mu )} \end{align} $$

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

$\lambda=0$

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

related items

computational resource

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

@@ 23번째 줄: / 23번째 줄: @@
 ===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
 $$