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.
• Mumford, David, M. Nori, and P. Norman. Tata Lectures on Theta III. Boston: Birkhäuser, 2006.

questions

• semi-direct product and 2-cocycle
• Hilbert space
• unitary operator
• statement of the Stone-von Neumann theorem
• $C\Omega + D$는 가역이고, $\Im{\gamma(\Omega)}>0 $임을 확인
• why consider conjugate linear functionals?
  • a given sesquilinear form $\langle \cdot, \cdot \rangle$ determines an isomorphism of $V$ with the complex conjugate of the dual space
• equivariant action on $\mathcal{H}_{\infty}$ and $\mathcal{H}_{-\infty}$

overview

• $g\in \mathbb{Z}$, $g\geq 1$
• $V=(\mathbb{R}^{2g},A)$, where $A$ is the form $A(x,y)=^tx_1y_2-^tx_2y_1$, $2g$-dimensional symplectic space
• symplectic group, isometry of $V$
• $Sp_{2g}(\mathbb{R})=\{M\in \operatorname{GL}_{2g}(\mathbb{R})|M^T J_{n} M = J_{n}\}$

where $$ J_{n} =\begin{pmatrix}0 & I_n \\-I_n & 0 \\\end{pmatrix} $$

• Heisenberg group $H(2g, \mathbb{R})$ and its Lie algebra
• 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
• transformation properties of theta function follows from the action of $Mp(2g,\mathbb{R})$ and $H(2g,\mathbb{R})$ on $\mathcal{H}$

theta functions

Jacobi theta function

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

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

• for $a,b\in \mathbb{Z}$,

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

• for $\gamma=\left(

\begin{array}{cc} a & b \\ c & d \\ \end{array} \right)\in SL_2(\mathbb{Z})$ and $ac,bd$ even, we have $$ \theta\left(\frac{z}{c\tau+d},\frac{a\tau+b}{c\tau+d}\right) = \zeta_{\gamma}(c\tau+d)^{1/2}\exp(\frac{\pi i cz^2}{c\tau+d})\theta(z,\tau) $$ where $\zeta_\gamma$ is an 8-th root of unity depending in $\gamma$

Riemann theta function

• Siegel modular group $\Gamma_g:=\operatorname{Sp}_{2g}(\R)\cap \operatorname{GL}_{2g}(\mathbb{Z})$
• Siegel upper-half space $\mathbb{H}_g=\left\{\Omega \in \operatorname{Mat}_{g \times g}(\mathbb{C}) \ \big| \ \Omega^t=\Omega, \Im \Omega>0 \right\}$
• $\Gamma_g$ acts on $\mathbb{H}_g$ by

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

• Igusa subgroup $\Gamma_{1,2}$, $\gamma=\begin{pmatrix}A & B \\ C & D \\ \end{pmatrix}\in \Gamma_{1,2}$ iff diagonals of $^tAC, ^tBD$ are even
• $\Theta:\mathbb{C}^g\times \mathbb{H}_g\to \mathbb{C}$

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

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 $$ \Theta (\mathbf{z}+\Omega \mathbf{a}+\mathbf{b},\Omega)=\exp(-\pi i\cdot ^t\mathbf{a} \Omega \mathbf{a}-2\pi i ^t\mathbf{a}\mathbf{z})\Theta(\mathbf{z},\Omega) $$

thm (modularity)

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

Heisenberg group

• Heisenberg group and Heisenberg algebra
• Heisenberg group $H(2g, \mathbb{R})$ : central extension of $V$ by $S^1=\{z\in \mathbb{C}:|z|=1\}$
• note that $\psi(x,y)=\exp(\pi i A(x,y))$ is a 2-cocycle
• Heisenberg group $H(2g, \mathbb{R}):=\{(\lambda,x)|\lambda\in S^1,x\in V\}$ with

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

thm (Stone-von Neumann)

There exists a unique irreducible unitary representation $$ U:H(2g,\mathbb{R})\to Aut(\mathcal{H}) $$ such that $U_{\lambda}=\lambda \operatorname{id}_{\mathcal{H}}$ for all $\lambda \in S^1$. In other words, if there are two representations $U^{(1)}$ and $U^{(2)}$ on $\mathcal{H}_1$ and $\mathcal{H}_2$, then there exists a unitary isomorphism $A: \mathcal{H}_1 \to \mathcal{H}_2$ such that $$ A\circ U^{(1)}\circ A^{-1}=U^{(2)} \\ \begin{array}{ccc} \mathcal{H}_1 & \overset{A}{\longrightarrow } & \mathcal{H}_2 \\ \downarrow U^{(1)} & \text{} & \downarrow U^{(2)} \\ \mathcal{H}_1 & \overset{A}{\longrightarrow } & \mathcal{H}_2 \end{array} $$

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

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

• called the Schrodinger representation of $H(2g, \mathbb{R})$

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
• want to get a reprsentation $\delta U$ of $\mathfrak{g}$ on a certain dense set $\mathcal{H}_{\infty}$ of $\mathcal{H}$
• for $X\in \mathfrak{g}$, let

$$ \delta U_{X}f:=\lim_{t\to 0}\frac{(U_{\exp_H(tX)}f)-f}{t} $$

• $A_i$ 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

• $\mathcal{H}_{\infty}$, Schwartz space
• $\mathcal{H}_{-\infty}$, the space of conjugate linear continuous maps from $\mathcal{H}_{\infty}$ to $\mathbb{C}$
• let $W_{\Omega}:=\langle \delta U_{A_i}-\sum_{j}\Omega_{ij} \delta U_{B_j},\, i=1,\cdots, g\rangle$, subalgebra of $\mathfrak{g}\otimes \mathbb{C}$

prop

There is a unique $f_{\Omega}\in \mathcal{H}_{\infty}$, unique up to scalars, such that $\delta U_{X} f_{\Omega}=0, \forall X\in W_{\Omega}$

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

$$ \sigma(n):=((-1)^{^tn_1n_2},n),\, n\in L $$

prop

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

• we get a function on $H(2g,\mathbb{R})$ as a matrix coefficient

$$ h\to \langle U_hf_{\Omega},\mu_{\mathbb{Z}} \rangle :=\overline{\mu_{\mathbb{Z}}(U_hf_{\Omega})},\,h\in H(2g,\mathbb{R}) $$

thm

Let $\Omega\in \mathbb{H}_g$ be fixed. Let $\mathcal{H}$ be a representation of $H_{2g}(\mathbb{R})$ and $f_{\Omega},\mu_{\mathbb{Z}}$ as above. For $x\in V=\mathbb{R}^{2g}$, $$ \langle U_{(1,x)}f_{\Omega}, \mu_{\mathbb{Z}}\rangle=c\exp(\pi i ^tx_1 \underline{\mathbf{x}})\Theta(\underline{\mathbf{x}},\Omega) $$ for some $c\in \mathbb{C}^{\times}$

quasi-periodicity

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

$$ \begin{aligned} \exp(\pi i ^tx_1 \underline{\mathbf{x}})\Theta(\underline{\mathbf{x}},\Omega)&=\langle U_{(1,x)}f_{\Omega}, \mu_{\mathbb{Z}}\rangle \\ &=\langle U_{\sigma(n)}U_{(1,x)}f_{\Omega}, U_{\sigma(n)} \mu_{\mathbb{Z}}\rangle \\ &=\langle U_{(-1)^{^tn_1n_2}\psi(n,x),x+n}f_{\Omega},\mu_{\mathbb{Z}}\rangle \\ &=(-1)^{^tn_1n_2}\psi(n,x)\exp(\pi i ^t(x_1+n_1)(\underline{\mathbf{x+n}}))\Theta(\underline{\mathbf{x+n}},\Omega) \end{aligned} $$

metaplectic group

covering of the symplectic group

• let $\gamma\in Sp_{2g}(\mathbb{R})$. As it preserves $A$, it induces an automorphism of $H(2g,\mathbb{R})$ by

$$ (\lambda,x)\mapsto (\lambda, \gamma x) $$

• define a new action $U'$ of $H(2g,\mathbb{R})$ on $\mathcal{H}$ by

$$ U'_{(\lambda,x)}f:=U_{(\lambda,\gamma x)}f $$

• by the Stone-von Neumann theorem, there exists a unitary map $A_{\gamma}:\mathcal{H}\to \mathcal{H}$ intertwining these two
• let $U(\mathcal{H})$ be the group of unitary isomorphisms of $\mathcal{H}$ and define

$$ \widetilde{Mp}(2g,\mathbb{R}):=\{A\in U(\mathcal{H}) : A=A_{\gamma} \text{for some } \gamma \in Sp_{2g}(\mathbb{R})\} $$

• then for $A\in \widetilde{Mp}(2g,\mathbb{R})$, there exists $\gamma \in Sp_{2g}(\mathbb{R})$ such that

$$ AU_{(\lambda,x)}A^{-1}=U_{(\lambda,\gamma x)} $$

lemma

Given $A\in \widetilde{Mp}(2g,\mathbb{R})$, there exists unique $\gamma \in Sp_{2g}(\mathbb{R})$ such that $A=A_{\gamma}$.

• we get an exact sequence

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

• assume $c=1$ so that

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

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

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

• 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 $\det(-B\Omega+A)^{-1/2}$ 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 $x\in \sigma(\mathbb{Z}^{2g})$.
• 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 $x\in \sigma(\mathbb{Z}^{2g})$.
• 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

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

thm

For $\mathbb{x}\in \mathbb{R}^{2g}, \Omega\in \mathbb{H}_g$ 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 $$ \Theta[x](\Omega)=\overline{\eta(\tilde{\gamma})} \det(-B\Omega+A)^{1/2}\Theta[\gamma x]\left((D\Omega-C)(-B\Omega+A)^{-1}\right) $$

memo

• $\gamma=\begin{pmatrix}A & B \\ C & D \\\end{pmatrix}\in \Gamma_g$

$$ \begin{align} ^tAC=^tCA \\ ^tBD=^tDB \\ ^tAD-^tCB= I_g \end{align} $$

• Igusa subgroup $\Gamma_{1,2}:=\{\gamma\in \Gamma_g|Q(\gamma \mathbf{x})=Q(\mathbf{x}) \pmod 2\}$, where $\mathbf{x}=(\mathbf{x_1},\mathbf{x_2})\in \mathbb{Z}^g\times \mathbb{Z}^g=\mathbb{Z}^{2g}$, $Q(\mathbf{x})=^t\mathbf{x_1} \mathbf{x_2}$
• for $\Omega\in \mathbb{H}_g$, define a lattice $\Lambda_{\Omega}=\mathbb{Z}^g+\Omega \mathbb{Z}^g\subset \mathbb{C}^g$
• $\Theta(\mathbf{z},\Omega)$
• a smooth vector $f_{\Omega}\in \mathcal{H}_{\infty}$, (Schwartz space, rapidly decreasing smooth function)
• 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

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

related items

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

computational resource

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