Jacobi's theta function from a representation theoretic viewpoint

abstract

• title: Jacobi's theta function from a representation theoretic viewpoint
• Jacobi introduced his theta functions to develop the theory of elliptic functions. Weil's approach to theta functions opened up the way to study them from a representation theoretic point of view. This involves the Heisenberg group, the Stone-von Neumann theorem and the Weil representation of the metaplectic group. I will give an introduction to this topic focusing on the classical transformation properties of theta functions.

questions

• why consider conjugate linear functionals?
• precise definition of metapletic group

overview

• $g\in \mathbb{Z}$, $g\geq 1$
• Heisenberg group $H(2g, \mathbb{R})$ and its Lie algebra
• Schrodinger representation of $H(2g, \mathbb{R})$ on $\mathcal{H}=L^2(\mathbb{R}^g)$
• Stone-von Neumann theorem induces an action of $Sp(2g,\mathbb{R})$ on $\mathcal{H}$
  • but this is only a projective representation
• we can turn it into a genuine representation of the metaplectic group and we call it the Weil representation
• a smooth vector $f_{\Omega}\in \mathcal{H}_{\infty}$, Schwartz space
• a functional $\mu_{\mathbb{Z}}\in \mathcal{H}_{-\infty}$, where $\mathcal{H}_{-\infty}$ is the space conjugate linear continuous maps from $\mathcal{H}_{\infty}$ to $\mathbb{C}$
• let $\mathbf{x}=(x_1,x_2)$ and $\underline{\mathbf{x}}=\Omega x_1+x_2$
• $\Theta(\underline{\mathbf{x}},\Omega)$ appears as pairing

$$ \langle U_{(1,x)}f_{\Omega}, \mu_{\mathbb{Z}}\rangle=c\exp(\pi i ^tx_1 \mathbf{x})\Theta(\underline{\mathbf{x}},\Omega) $$

• modular transformation properties follows from the action of $Mp(2g,\mathbb{R})$ on $\mathfrak{H}_g$ and $H(2g,\mathbb{R})$

theta function

Jacobi theta function

• $\theta:\mathbb{C}\times \mathbb{H}_1\to \mathbb{C}$

$$ \theta (z,\tau)= \sum_{n \in \mathbb{Z}} e^{\pi i n^2 \tau} \, \E^{2 \pi i n z},\, \tau\in \mathbb{H}_1,z\in \mathbb{C} $$

• $\theta (z+a\tau +b,\tau)=\exp(-\pi i a^2 \tau -2\pi i az)\theta(z,\tau)$
• for $\gamma=\left(

\begin{array}{cc} a & b \\ c & d \\ \end{array} \right)\in SL_2(\mathbb{Z})$ and $ac,bd$ even, we have $$ \theta\left(\frac{z}{c\tau+d},\frac{a\tau+b}{c\tau+d}\right) = \zeta_{\gamma}(c\tau+d)^{1/2}\exp(\frac{\pi i cz^2}{c\tau+d})\theta(z,\tau) $$

지겔 모듈라 군

• 지겔 모듈라 군 $\Gamma_g:=\operatorname{Sp}(2g,\R)\cap \operatorname{GL}(2g,\mathbb{Z})$
• 행렬 $\gamma=\begin{pmatrix}A & B \\ C & D \\\end{pmatrix}\in \Gamma_g$는 다음의 조건을 만족해야 한다

$$ \begin{align} A^tC=C^tA \\ B^tD=D^tB \\ A^tD-C^tB= I_g \end{align} $$

• 지겔 상반 공간 $\mathbb{H}_g$

$$ \mathbb{H}_g=\left\{\Omega \in \operatorname{Mat}_{g \times g}(\mathbb{C}) \ \big| \ \Omega^t=\Omega, \Im \Omega>0 \right\} $$

• 사교군 $\Gamma_g$ 은 $\mathbb{H}_g$에 다음과 같이 작용

$$ \Omega\mapsto \gamma(\Omega)=(A\Omega +B)(C\Omega + D)^{-1} $$

• $C\Omega + D$는 가역이고, $\Im{\gamma(\Omega)}>0 $임을 확인

이구사 부분군

• 이구사 부분군 $\Gamma_{1,2}:=\{\gamma\in \Gamma_g|Q(\gamma \mathbf{x})=Q(\mathbf{x}) \pmod 2\}$, 여기서 $\mathbf{x}=(\mathbf(x_1),\mathbf(x_2))\in \mathbb{Z}^g\times \mathbb{Z}^g=\mathbb{Z}^{2g}$, $Q(\mathbf{x})=\mathbf(x_1)^t \cdot\mathbf(x_2)$
• $\gamma=\begin{pmatrix}A & B \\ C & D \\ \end{pmatrix}\in \Gamma_{1,2}$는 $A^tC, B^tD$의 대각성분이 짝수라는 사실과 동치

Riemann theta function

• Siegel upper-half plane $\mathbb{H}_g=\left\{\Omega \in M_{g \times g}(\mathbb{C}) \ \big| \ \Omega^t=\Omega, \operatorname{Im}(\Omega) \text{ positive definite} \right\}$
• $\Theta:\mathbb{C}^g\times \mathbb{H}_g\to \mathbb{C}$

$$ \Theta(\mathbf{z},\Omega):=\sum_{{\mathbf{n}\in{\mathbb Z}^g}}e^{{2\pi i\left(\frac{1}{2}\mathbf{n}\cdot\boldsymbol{\Omega}\cdot\mathbf{n}+\mathbf{n}\cdot\mathbf{z}\right)}} ,\, \Omega\in \mathbb{H}_g,\mathbb{z}\in \mathbb{C}^g $$

• for $\Omega\in \mathbb{H}_g$, define a lattice $\Lambda_{\Omega}=\mathbb{Z}^g+\Omega \mathbb{Z}^g\subset \mathbb{C}^g$
• $\Theta(\mathbf{z},\Omega)$

thm (quasi-periodicity)

Let $\mathbf{a},\mathbf{b}\in \mathbb{Z}^g,\mathbf{z}\in \mathbb{C}^g,\Omega\in \mathbb{H}_g$. We have $$ \Theta (\mathbf{z}+\Omega \mathbf{a}+\mathbf{b},\Omega)=\exp(-\pi i\cdot \mathbf{a}^t \Omega a-2\pi i \mathbf{a}^t\mathbf{z})\Theta(\mathbf{z},\Omega) $$

thm (modularity)

Let $\gamma=\begin{pmatrix}A & B \\ C & D \\ \end{pmatrix}\in \Gamma_{1,2}$. We have $$ \Theta \left(((C\Omega + D)^{-1})^t \mathbf{z}, (A\Omega+B)(C\Omega + D)^{-1}\right)=\zeta_{\gamma}\det(C\Omega+D)^{1/2}\exp(\pi i\cdot ^t\mathbf{z}(C\Omega+D)^{-1}C\mathbf{z})\Theta(\mathbf{z},\Omega),\,\mathbf{z}\in \mathbb{C}^g,\Omega\in \mathbb{H}_g $$ 여기서 $\zeta_\gamma$는 $\gamma$에 의존하는 적당한 8-th root of unity

Heisenberg group

• Heisenberg group and Heisenberg algebra
• Let $V$ be a $2g$-dimensional symplectic space
• we can set $V=(\mathbb{R}^{2g},A)$, where $A$ is the form $A(x,y)=^tx_1y_2-^tx_2y_1$
• Heisenberg group $H(2g, \mathbb{R})$ : central extension of $V$ by $S^1$
• note that $\psi(x,y)=\exp(\pi i A(x,y))$ is a 2-cocycle
• the following operation defines a group

$$ (\lambda,x)\cdot (\mu, y)=(\lambda \mu \psi(x,y),x+y) $$ \[ 1 \rightarrow S^1~\rightarrow~H(2g, \mathbb{R})~\rightarrow~V \rightarrow 0\]

thm (Stone-von Neumann)

There exists a uniqu irreducible unitary representation $$ U:H(2g,\mathbb{R})\to Aut(\mathcal{H}) $$ such that $U_{\lambda}=\lambda \operatorname{id}_{\mathcal{H}}$ for all $\lambda \in S^1$

realization

• let $\mathcal{H}=L^2(\mathbb{R}^g)$
• for $(\lambda,y_1,y_2)\in H(2g, \mathbb{R})$, $x_1\in \mathbb{R}^g$ and $\varphi\in \mathcal{H}$, define

$$ U_{(\lambda,y_1,y_2)}\varphi(x_1):=\lambda \exp(2\pi i (^tx_1y_2+^ty_1y_2/2))\varphi(x_1+y_1) $$

Heisenberg algebra

• the Lie algebra $\mathfrak{g}$ of $H(2g,\mathbb{R})$ has a basis : $A_1,\cdots,A_g, B_1,\cdots,B_g,C$ with

$$ [A_i, B_j] = \delta_{ij}C, [A_i, C] =[B_j, C] = 0 $$

• $A_i=p_i,B_i=q_i$ in usual notation for Heisenberg algebra
• we want to get a reprsentation of $\mathfrak{g}$ on a certain dense set of $\mathcal{H}$
• $A_i$ acts as $\frac{\partial f}{\partial x_i}$
• $B_i$ acts as $2\pi i x_i f(x)$
• $C$ acts as $2\pi i f(x)$

theta as matrix coefficients

prop

There is a unique $f_{\Omega}\in \mathcal{H}_{\infty}$, unique up to scalars, such that blah blah

• Let $L=\mathbb{Z}^g$ and let $\sigma:L\to H(2g, \mathbb{R})$ defined by

$$ \sigma(n)=(e_{*}(n/2),n) $$ where $e_{*}(x)=\exp(4\pi i ^tx_1 x_2)$.

• note that $e_{*}(n/2)\in \{\pm 1\}$ for any $n\in L$

prop

There is a unique $\mu_{\mathbb{Z}}\in \mathcal{H}_{-\infty}$, unique up to scalars, which is invariant under $U_x,\, x\in \sigma(L)$

thm

$$ \langle U_{(1,x)}f_{\Omega}, \mu_{\mathbb{Z}}\rangle=c\exp(\pi i ^tx_1 \mathbf{x})\Theta(\underline{\mathbf{x}},\Omega) $$

quasi-periodicity

• for $n=(n_1,n_2)\in \mathbb{Z}^{2g}$,

$$ \begin{aligned} \exp(\pi i ^tx_1 \mathbf{x})\Theta(\underline{\mathbf{x}},\Omega)&=\langle U_{(1,x)}f_{\Omega}, \mu_{\mathbb{Z}}\rangle \\ &=\langle U_{\sigma(n)}U_{(1,x)}f_{\Omega}, U_{\sigma(n)} \mu_{\mathbb{Z}}\rangle \\ &=\langle U_{e_{*}(n/2)e(n/2,x),x+n}f_{\Omega},\mu_{\mathbb{Z}}\rangle \\ &=e_{*}(n/2)e(n/2,x)\exp(\pi i ^t(x_1+n_1)(\underline{x+n}))\Theta(\underline{\mathbf{x+n}},\Omega) \end{aligned} $$

metaplectic group

• coverting of the symplectic group

\[ 1 \rightarrow S^1~\rightarrow~\widetilde{Mp}(2g,\mathbb{R})~\overset{\rho}{\rightarrow}~Sp(2g,\mathbb{R}) \rightarrow 1\]

• assume $c=1$ so that

$$ \langle U_{(1,x)}f_{\Omega}, \mu_{\mathbb{Z}}\rangle=\exp(\pi i ^tx_1 \mathbf{x})\Theta(\underline{\mathbf{x}},\Omega) $$

• Let $\gamma\in Sp(2g,\mathbb{R})$ and $P\in \widetilde{Mp}(2g,\mathbb{R})$ such that $\rho(P)=\gamma$. Then

$$ \begin{aligned} \exp(\pi i ^tx_1 \mathbf{x})\Theta(\underline{\mathbf{x}},\Omega)&=\langle PU_{(1,x)}f_{\Omega}, P\mu_{\mathbb{Z}}\rangle\\ &=\langle U_{(1,\gamma x)}P f_{\Omega}, P\mu_{\mathbb{Z}}\rangle \end{aligned} $$

• once we compute $P f_{\Omega}, P\mu_{\mathbb{Z}}$, the functional equation of $\Theta$ will fall out

computing $P f_{\Omega}$

thm

Let $P\in \widetilde{Mp}(2g,\mathbb{R})$, $\rho(P)=\gamma$. We choose $f_{\Omega}(x)=\exp(\pi i ^tx \Omega x)$ for $\Omega\in \mathbb{H}_{g}$. Then $$ Pf_{\Omega}=C(P,\Omega)f_{\gamma*\Omega}, $$ where $C(P,\Omega)$ is, up to a scalar of absoulte value one, a branch of $\det(-B\Omega+C)^{-1/2}$ on $\mathbb{H}_{g}$

computing $P\mu_{\mathbb{Z}}$

• Recall that $\mu_{\mathbb{Z}}\in \mathcal{H}_{-\infty}$ is killed by $U_x-1$ for any $x\in \sigma(\mathbb{Z}^g)$.
• for $\tilde{\gamma}\in Mp(2g,\mathbb{R})$ with $\rho(\tilde{\gamma})=\gamma\in \Gamma_{1,2}$, $\tilde{\gamma}\mu_{\mathbb{Z}}$ is killed by $U_{T_{\gamma}x}-1$ for $x\in \sigma(\mathbb{Z}^g)$.
• from the uniqueness of $\mu_{\mathbb{Z}}$, we get

$$ \tilde{\gamma}\mu_{\mathbb{Z}}=\eta(\tilde{\gamma})\mu_{\mathbb{Z}} $$ where $\eta(\tilde{\gamma})\in \mathbb{C}^{\times}$.

• $\eta:\rho^{-1}(\Gamma_{1,2})\cap Mp(2g,\mathbb{R})\to \mathbb{C}^{\times}$ is a character

lemma

1. $\eta$ surjects on the 8-th root of unity
2. Consider $\eta^2$ as a character on $\Gamma_{1,2}$. If $\operatorname{ker} \eta^2=\Delta$, then $\Delta$ contains $\Gamma_4=\{\gamma\in Sp_{2g}(\mathbb{Z}):\gamma=I_g \mod 4\}$

functional equation

• for $x \in \mathbb{R}^{2g}$ and $\Omega\in \mathbb{H}_g$, let

$$ \Theta[x](\Omega):=\exp(\pi i ^tx_1 \mathbf{x})\Theta(\underline{\mathbf{x}},\Omega) $$

thm

For $x\in \mathbb{x}\in \mathbb{R}^{2g}, \Omega\in \mathbb{H}_g$ and $\tilde{\gamma}\in Mp(2g,\mathbb{R})$ with $\rho(\tilde{\gamma})=\gamma=\begin{pmatrix}A & B \\ C & D \\ \end{pmatrix}\in \Gamma_{1,2}$, we have $$ \Theta[x](\Omega)=\overline{\eta(\tilde{\gamma})} \det(-B\Omega+A)^{1/2}\Theta[\gamma x]\left((D\Omega-C)(-B\Omega+A)^{-1}\right) $$

related items

• Segal-Shale-Weil representation
• Howe duality
• Heisenberg group and Heisenberg algebra

computational resource

• https://drive.google.com/drive/u/0/folders/0B8XXo8Tve1cxTkRfOElvdjRuSFE