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
• \(C\Omega + D\) is invertible and \(\Im{\gamma(\Omega)}>0 \)
• 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 symplectic form \(A(x,y)=^tx_1y_2-^tx_2y_1\), \(2g\)-dimensional symplectic space
• symplectic group, isometry of \(V\), \(\gamma\) s.t. \(A(\gamma x,\gamma y)=A(x,y)\)
• \(Sp_{2g}(\mathbb{R})=\{M\in \operatorname{GL}_{2g}(\mathbb{R})|M^T J_{n} M = J_{n}\}\) where

\[ J_{n} =\begin{pmatrix}0 & I_n \\-I_n & 0 \\\end{pmatrix} \]

• representation of Heisenberg group \(H(2g, \mathbb{R})\) on a Hilbert space \(\mathcal{H}\)
• Stone-von Neumann theorem -> projective representation of \(Sp_{2g}(\mathbb{R})\) on \(\mathcal{H}\)
• Weil representation of \(Mp(2g,\mathbb{R})\), double cover of the symplectic group
• interpret \(\Theta\) as representation theoretic quantity
• 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
• \(\Theta:\mathbb{C}^g\times \mathbb{H}_g\to \mathbb{C}\)

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

• quasi-periodicity

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

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

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

• central extension of \(V\) by \(S^1\)

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 such representations \(U^{(1)}\) and \(U^{(2)}\) on \(\mathcal{H}_1\) and \(\mathcal{H}_2\), then there exists an isomorphism \(A: \mathcal{H}_1 \to \mathcal{H}_2\) such that \[ A\circ U^{(1)}\circ A^{-1}=U^{(2)} \\ \begin{array}{ccc} \mathcal{H}_1 & \overset{A}{\longrightarrow } & \mathcal{H}_2 \\ \downarrow U^{(1)} & \text{} & \downarrow U^{(2)} \\ \mathcal{H}_1 & \overset{A}{\longrightarrow } & \mathcal{H}_2 \end{array} \]

• \(A\) is an intertwinter between \(U^{(1)}\) and \(U^{(2)}\)
• related to the equivalence of matrix mechanics and wave mechanics in the early days of quantum mechanics

realization

• let \(\mathcal{H}_1:=L^2(\mathbb{R}^g)\)
• 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 \(\mathfrak{h}(2g,\mathbb{R})\) of \(H(2g,\mathbb{R})\) has a basis \[A_1,\cdots,A_g, B_1,\cdots,B_g,C\] with

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

• want to get a reprsentation \(\delta U\) of \(\mathfrak{h}(2g,\mathbb{R})\) on a certain dense subspace \(\mathcal{H}_{\infty}\) of \(\mathcal{H}_1\)
• for \(X\in \mathfrak{h}(2g,\mathbb{R})\), let

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

• on \(\mathcal{H}_1\)
• \(A_i\) 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 \(\mathfrak{h}(2g,\mathbb{R})\otimes \mathbb{C}\)

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 \(\sigma:\mathbb{Z}^{2g}\to H(2g, \mathbb{R})\) defined by

\[ \sigma(n):=((-1)^{^tn_1n_2},n),\, n\in \mathbb{Z}^{2g} \]

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

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

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 representation \(U'\) 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 \(U\) and \(U'\)
• 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

\[ AU_{(\lambda,x)}A^{-1}=U_{(\lambda,\gamma x)} \label{star} \]

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

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

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

\[ \begin{aligned} \exp(\pi i ^tx_1 \underline{\mathbf{x}})\Theta(\underline{\mathbf{x}},\Omega)&=\langle PU_{(1,x)}f_{\Omega}, P\mu_{\mathbb{Z}}\rangle\\ &=\langle U_{(1,\gamma x)}P f_{\Omega}, P\mu_{\mathbb{Z}}\rangle \end{aligned} \] where the second equality follows from \ref{star}

• 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 \(\mathbb{H}_{g}\)

• \(\chi: \widetilde{Mp}(2g,\mathbb{R})\to S^1\), \(\chi(P):=\det(-B\Omega+A)C(P,\Omega)^2\) is a character
• \(\operatorname{ker}(\chi)=Mp(2g,\mathbb{R})\) central ext of \(Sp_{2g}(\mathbb{R})\) by \(\{\pm 1\}\)

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 \(\Lambda_{\Omega}=\mathbb{Z}^g+\Omega \mathbb{Z}^g\subset \mathbb{C}^g\)
• 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

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

• \(A_i=p_i,B_i=x_i\) in usual notation for Heisenberg algebra
• \([X,P] = X P - P X = i \hbar\)

related items

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

computational resource

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