"Jacobi's theta function from a representation theoretic viewpoint"의 두 판 사이의 차이

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==abstract==
 
==abstract==
−
* Jacobi introduced his theta functions to develop the theory of elliptic functions. Jacobi's theta function has two notable properties : quasi-periodicity and modularity. Weil's approach to theta functions opened up the way to understand these classical transformation properties from a representation theoretic viewpoint, which paved the way to the theory of Howe duality. This involves the Heisenberg group, the Stone-Von Neumann theorem and Weil representations of metapletic groups. In this talk, I will explain the basics of these notions.
+
* 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.
  
−
==introduction==
+
==questions==
−
* $g\in \mathbb{Z}$, $g\geq 1$
+
* semi-direct product and 2-cocycle
−
* Heisenberg group $H(2g, \mathbb{R})$ and its Lie algebra
+
* Hilbert space
−
* Schrodinger representation of $H(2g, \mathbb{R})$ on $\mathcal{H}=L^2(\mathbb{R}^g)$
+
* unitary operator
−
* Stone-von Neumann theorem induces an action of $Sp(2g,\mathbb{R})$ on $\mathcal{H}$
+
* statement of the Stone-von Neumann theorem
−
** but this is only a projective representation
+
* <math>C\Omega + D</math> is invertible and <math>\Im{\gamma(\Omega)}>0 </math>
−
* we can turn it into a genuine representation of the metaplectic group and we call it the Weil representation
+
* why consider conjugate linear functionals?
−
* a smooth vector $f_{\Omega}\in \mathcal{H}_{\infty}$, Schwartz space
+
** a given sesquilinear form <math>\langle \cdot, \cdot \rangle</math> determines an isomorphism of <math>V</math> with the complex conjugate of the dual space
−
* a functional $\mu_{\mathbb{Z}}\in \mathcal{H}_{-\infty}$
+
* equivariant action on <math>\mathcal{H}_{\infty}</math> and <math>\mathcal{H}_{-\infty}</math>
−
* let $\mathbf{x}=(x_1,x_2)$ and $\underline{x}=\Omega x_1+x_2$
+
 
−
* $\theta(\underline{x},\Omega)$ appears as pairing
+
==overview==
−
$$
+
* <math>g\in \mathbb{Z}</math>, <math>g\geq 1</math>
−
\langle U_{(1,x)}f_{\Omega}, \mu_{\mathbb{Z}}\rangle=c\exp(\pi i ^tx_1 \mathbf{x})\theta(\underline{x},\Omega)
+
* <math>V=(\mathbb{R}^{2g},A)</math>, where <math>A</math> is the symplectic form <math>A(x,y)=^tx_1y_2-^tx_2y_1</math>, <math>2g</math>-dimensional symplectic space
−
$$
+
* symplectic group, isometry of <math>V</math>, <math>\gamma</math> s.t. <math>A(\gamma x,\gamma y)=A(x,y)</math>
−
* modular transformation properties follows from the action of $Mp(2g,\mathbb{R})$ on $\mathfrak{h}_g$ and $H$
+
* <math>Sp_{2g}(\mathbb{R})=\{M\in \operatorname{GL}_{2g}(\mathbb{R})|M^T J_{n} M = J_{n}\}</math> where
 +
:<math>
 +
J_{n} =\begin{pmatrix}0 & I_n \\-I_n & 0 \\\end{pmatrix}
 +
</math>
 +
* representation of Heisenberg group <math>H(2g, \mathbb{R})</math> on a Hilbert space <math>\mathcal{H}</math>
 +
* Stone-von Neumann theorem -> projective representation of <math>Sp_{2g}(\mathbb{R})</math> on <math>\mathcal{H}</math>
 +
* Weil representation of <math>Mp(2g,\mathbb{R})</math>, double cover of the symplectic group
 +
* interpret <math>\Theta</math> as representation theoretic quantity
 +
* transformation properties of theta function follows from the action of <math>Mp(2g,\mathbb{R})</math> and <math>H(2g,\mathbb{R})</math> on <math>\mathcal{H}</math>
 +
 
 +
==theta functions==
 +
===Jacobi theta function===
 +
* <math>\theta:\mathbb{C}\times \mathbb{H}\to \mathbb{C}</math>
 +
:<math>
 +
\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}
 +
</math>
 +
* for <math>a,b\in \mathbb{Z}</math>,
 +
:<math>\theta (z+a\tau +b,\tau)=\exp(-\pi i a^2 \tau -2\pi i az)\theta(z,\tau)</math>
 +
* for <math>\gamma=\left(
 +
\begin{array}{cc}
 +
a & b \\
 +
c & d \\
 +
\end{array}
 +
\right)\in SL_2(\mathbb{Z})</math> and <math>ac,bd</math> even, we have
 +
:<math>
 +
\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)
 +
</math>
 +
where <math>\zeta_\gamma</math> is an 8-th root of unity depending in <math>\gamma</math>
 +
 
 +
 
 +
===Riemann theta function===
 +
* Siegel modular group <math>\Gamma_g:=\operatorname{Sp}_{2g}(\R)\cap \operatorname{GL}_{2g}(\mathbb{Z})</math>
 +
* Siegel upper-half space <math>\mathbb{H}_g=\left\{\Omega \in \operatorname{Mat}_{g \times g}(\mathbb{C}) \ \big| \ \Omega^t=\Omega, \Im \Omega>0 \right\}</math>
 +
* <math>\Gamma_g</math> acts on <math>\mathbb{H}_g</math> by
 +
:<math>
 +
\Omega\mapsto \gamma(\Omega)=(A\Omega +B)(C\Omega + D)^{-1}
 +
</math>
 +
* Igusa subgroup <math>\Gamma_{1,2}</math>, <math>\gamma=\begin{pmatrix}A & B \\ C & D \\ \end{pmatrix}\in \Gamma_{1,2}</math> iff diagonals of <math>^tAC, ^tBD</math> are even
 +
* <math>\Theta:\mathbb{C}^g\times \mathbb{H}_g\to \mathbb{C}</math>
 +
:<math>
 +
\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
 +
</math>
 +
* quasi-periodicity
 +
Let <math>\mathbf{a},\mathbf{b}\in \mathbb{Z}^g,\mathbf{z}\in \mathbb{C}^g,\Omega\in \mathbb{H}_g</math>. We have
 +
:<math>
 +
\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)
 +
</math>
 +
* modularity
 +
Let <math>\gamma=\begin{pmatrix}A & B \\ C & D \\ \end{pmatrix}\in \Gamma_{1,2}</math>. We have
 +
:<math>
 +
\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
 +
</math>
 +
where <math>\zeta_\gamma</math> is an 8-th root of unity depending in <math>\gamma</math>
 +
 
 +
==Heisenberg group==
 +
* [[Heisenberg group and Heisenberg algebra|Heisenberg group]] <math>H(2g, \mathbb{R})</math> : central extension of <math>V</math> by <math>S^1=\{z\in \mathbb{C}:|z|=1\}</math>
 +
* note that <math>\psi(x,y)=\exp(\pi i A(x,y)),\,x,y\in V</math> is a 2-cocycle
 +
* Heisenberg group <math>H(2g, \mathbb{R}):=\{(\lambda,x)|\lambda\in S^1,x\in V\}</math> with
 +
:<math>
 +
(\lambda,x)\cdot (\mu, y):=(\lambda \mu \psi(x,y),x+y)
 +
</math>
 +
: <math> 1 \rightarrow S^1~\rightarrow~H(2g, \mathbb{R})~\rightarrow~V \rightarrow 0</math>
 +
* central extension of <math>V</math> by <math>S^1</math>
 +
;thm (Stone-von Neumann)
 +
There exists a unique irreducible unitary representation
 +
:<math>
 +
U:H(2g,\mathbb{R})\to Aut(\mathcal{H})
 +
</math>
 +
such that <math>U_{\lambda}=\lambda \operatorname{id}_{\mathcal{H}}</math> for all <math>\lambda \in S^1</math>. In other words, if there are two such representations <math>U^{(1)}</math> and <math>U^{(2)}</math> on <math>\mathcal{H}_1</math> and <math>\mathcal{H}_2</math>, then there exists an isomorphism <math>A: \mathcal{H}_1 \to \mathcal{H}_2</math> such that
 +
:<math>
 +
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}
 +
</math>
 +
* <math>A</math> is an intertwinter between <math>U^{(1)}</math> and <math>U^{(2)}</math>
 +
* related to the equivalence of matrix mechanics and wave mechanics in the early days of quantum mechanics
 +
 
 +
===realization===
 +
* let <math>\mathcal{H}_1:=L^2(\mathbb{R}^g)</math>
 +
* for <math>(\lambda,y_1,y_2)\in H(2g, \mathbb{R})</math>, <math>x_1\in \mathbb{R}^g</math> and <math>\varphi\in \mathcal{H}</math>, define
 +
:<math>
 +
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)
 +
</math>
 +
* called the Schrodinger representation of <math>H(2g, \mathbb{R})</math>
 +
 
 +
===Heisenberg algebra===
 +
* the Lie algebra <math>\mathfrak{h}(2g,\mathbb{R})</math> of <math>H(2g,\mathbb{R})</math> has a basis : <math>A_1,\cdots,A_g, B_1,\cdots,B_g,C</math> with
 +
:<math>
 +
[A_i, B_j] = \delta_{ij}C, [A_i, C] =[B_j, C] = 0
 +
</math>
 +
* want to get a reprsentation <math>\delta U</math> of <math>\mathfrak{h}(2g,\mathbb{R})</math> on a certain dense subspace <math>\mathcal{H}_{\infty}</math> of <math>\mathcal{H}_1</math>
 +
* for <math>X\in \mathfrak{h}(2g,\mathbb{R})</math>, let
 +
:<math>
 +
\delta U_{X}f:=\lim_{t\to 0}\frac{(U_{\exp_H(tX)}f)-f}{t}
 +
</math>
 +
* on <math>\mathcal{H}_1</math>
 +
* <math>A_i</math> acts as <math>\frac{\partial f}{\partial x_i}</math>
 +
* <math>B_i</math> acts as <math>2\pi i x_i f(x)</math>
 +
* <math>C</math> acts as <math>2\pi i f(x)</math>
 +
 
 +
===theta as matrix coefficients===
 +
* <math>\mathcal{H}_{\infty}</math>, Schwartz space
 +
* <math>\mathcal{H}_{-\infty}</math>, the space of conjugate linear continuous maps from <math>\mathcal{H}_{\infty}</math> to <math>\mathbb{C}</math>
 +
* let <math>W_{\Omega}:=\langle \delta U_{A_i}-\sum_{j}\Omega_{ij} \delta U_{B_j},\, i=1,\cdots, g\rangle</math>, subalgebra of <math>\mathfrak{h}(2g,\mathbb{R})\otimes \mathbb{C}</math>
 +
;prop
 +
There is a unique <math>f_{\Omega}\in \mathcal{H}_{\infty}</math>, unique up to scalars, such that <math>\delta U_{X} f_{\Omega}=0, \forall X\in W_{\Omega}</math>
 +
* Let <math>\sigma:\mathbb{Z}^{2g}\to H(2g, \mathbb{R})</math> defined by
 +
:<math>
 +
\sigma(n):=((-1)^{^tn_1n_2},n),\, n\in \mathbb{Z}^{2g}
 +
</math>
 +
;prop
 +
There is a unique <math>\mu_{\mathbb{Z}}\in \mathcal{H}_{-\infty}</math>, unique up to scalars, which is invariant under <math>U_x,\, x\in \sigma(L)</math>
 +
* we get a function on <math>H(2g,\mathbb{R})</math> as a matrix coefficient
 +
:<math>
 +
h\to \langle U_hf_{\Omega},\mu_{\mathbb{Z}} \rangle :=\overline{\mu_{\mathbb{Z}}(U_hf_{\Omega})},\,h\in H(2g,\mathbb{R})
 +
</math>
 +
;thm
 +
Let <math>\Omega\in \mathbb{H}_g</math> be fixed. Let <math>\mathcal{H}</math> be a representation of <math>H(2g,\mathbb{R})</math> and <math>f_{\Omega},\mu_{\mathbb{Z}}</math> as above. For <math>x\in V=\mathbb{R}^{2g}</math>,
 +
:<math>
 +
\langle U_{(1,x)}f_{\Omega}, \mu_{\mathbb{Z}}\rangle=c\exp(\pi i ^tx_1 \underline{\mathbf{x}})\Theta(\underline{\mathbf{x}},\Omega)
 +
</math>
 +
for some <math>c\in \mathbb{C}^{\times}</math>
 +
 
 +
===quasi-periodicity===
 +
* for <math>n=(n_1,n_2)\in \mathbb{Z}^{g}\times \mathbb{Z}^{g}\mathbb{Z}^{2g}</math>,
 +
:<math>
 +
\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}
 +
</math>
 +
 
 +
==metaplectic group==
 +
===covering of the symplectic group===
 +
* let <math>\gamma\in Sp_{2g}(\mathbb{R})</math>. As it preserves <math>A</math>, it induces an automorphism of <math>H(2g,\mathbb{R})</math> by
 +
:<math>
 +
(\lambda,x)\mapsto (\lambda, \gamma x)
 +
</math>
 +
* define a new representation <math>U'</math> of <math>H(2g,\mathbb{R})</math> on <math>\mathcal{H}</math> by
 +
:<math>
 +
U'_{(\lambda,x)}f:=U_{(\lambda,\gamma x)}f
 +
</math>
 +
* by the Stone-von Neumann theorem, there exists a unitary map <math>A_{\gamma}:\mathcal{H}\to \mathcal{H}</math> intertwining <math>U</math> and <math>U'</math>
 +
* let <math>U(\mathcal{H})</math> be the group of unitary isomorphisms of <math>\mathcal{H}</math> and define
 +
:<math>
 +
\widetilde{Mp}(2g,\mathbb{R}):=\{A\in U(\mathcal{H}) : A=A_{\gamma} \text{for some } \gamma \in Sp_{2g}(\mathbb{R})\}
 +
</math>
 +
* then for <math>A\in \widetilde{Mp}(2g,\mathbb{R})</math>, there exists <math>\gamma \in Sp_{2g}(\mathbb{R})</math> such that
 +
:<math>
 +
AU_{(\lambda,x)}A^{-1}=U_{(\lambda,\gamma x)} \label{star}
 +
</math>
 +
;lemma
 +
Given <math>A\in \widetilde{Mp}(2g,\mathbb{R})</math>, there exists unique <math>\gamma \in Sp_{2g}(\mathbb{R})</math> such that <math>A=A_{\gamma}</math>.
 +
* we get an exact sequence
 +
: <math> 1 \rightarrow S^1~\rightarrow~\widetilde{Mp}(2g,\mathbb{R})~\overset{\rho}{\rightarrow}~Sp(2g,\mathbb{R}) \rightarrow 1</math>
 +
* Let <math>\gamma\in Sp(2g,\mathbb{R})</math> and <math>P\in \widetilde{Mp}(2g,\mathbb{R})</math> such that <math>\rho(P)=\gamma</math>. Then
 +
:<math>
 +
\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}
 +
</math>
 +
where the second equality follows from \ref{star}
 +
* once we compute <math>P f_{\Omega}, P\mu_{\mathbb{Z}}</math>, the functional equation of <math>\Theta</math> will fall out
 +
 
 +
===computing <math>P f_{\Omega}</math>===
 +
;thm
 +
Let <math>P\in \widetilde{Mp}(2g,\mathbb{R})</math>, <math>\rho(P)=\gamma</math>. We choose <math>f_{\Omega}(x)=\exp(\pi i ^tx \Omega x)</math> for <math>\Omega\in \mathbb{H}_{g}</math>. Then
 +
:<math>
 +
Pf_{\Omega}=C(P,\Omega)f_{\gamma*\Omega},
 +
</math>
 +
where <math>C(P,\Omega)</math> is, up to a scalar of absoulte value one, a branch of <math>\det(-B\Omega+A)^{-1/2}</math> on <math>\mathbb{H}_{g}</math>
 +
* <math>\chi: \widetilde{Mp}(2g,\mathbb{R})\to S^1</math>, <math>\chi(P):=\det(-B\Omega+A)C(P,\Omega)^2</math> is a character
 +
* <math>\operatorname{ker}(\chi)=Mp(2g,\mathbb{R})</math> central ext of <math>Sp_{2g}(\mathbb{R})</math> by <math>\{\pm 1\}</math>
 +
 
 +
===computing <math>P\mu_{\mathbb{Z}}</math>===
 +
* Recall that <math>\mu_{\mathbb{Z}}\in \mathcal{H}_{-\infty}</math> is killed by <math>U_x-1</math> for any <math>x\in \sigma(\mathbb{Z}^{2g})</math>.
 +
* for <math>\tilde{\gamma}\in Mp(2g,\mathbb{R})</math> with <math>\rho(\tilde{\gamma})=\gamma\in \Gamma_{1,2}</math>, <math>\tilde{\gamma}\mu_{\mathbb{Z}}</math> is killed by <math>U_{T_{\gamma}x}-1</math> for <math>x\in \sigma(\mathbb{Z}^{2g})</math>.
 +
* from the uniqueness of <math>\mu_{\mathbb{Z}}</math>, we get
 +
:<math>
 +
\tilde{\gamma}\mu_{\mathbb{Z}}=\eta(\tilde{\gamma})\mu_{\mathbb{Z}}
 +
</math>
 +
where <math>\eta(\tilde{\gamma})\in \mathbb{C}^{\times}</math>.
 +
* <math>\eta:\rho^{-1}(\Gamma_{1,2})\cap Mp(2g,\mathbb{R})\to \mathbb{C}^{\times}</math> is a character
 +
;lemma
 +
# <math>\eta</math> surjects on the 8-th root of unity
 +
# Consider <math>\eta^2</math> as a character on <math>\Gamma_{1,2}</math>. If <math>\operatorname{ker} \eta^2=\Delta</math>, then <math>\Delta</math> contains <math>\Gamma_4=\{\gamma\in Sp_{2g}(\mathbb{Z}):\gamma=I_g \mod 4\}</math>
 +
 
 +
===functional equation===
 +
* for <math>x \in \mathbb{R}^{2g}</math> and <math>\Omega\in \mathbb{H}_g</math>, let
 +
:<math>
 +
\Theta[x](\Omega):=\exp(\pi i ^tx_1 \underline{\mathbf{x}})\Theta(\underline{\mathbf{x}},\Omega)
 +
</math>
 +
;thm
 +
For <math>\mathbb{x}\in \mathbb{R}^{2g}, \Omega\in \mathbb{H}_g</math> and <math>\tilde{\gamma}\in Mp(2g,\mathbb{R})</math> with <math>\rho(\tilde{\gamma})=\gamma=\begin{pmatrix}A & B \\ C & D \\ \end{pmatrix}\in \Gamma_{1,2}</math>, we have
 +
:<math>
 +
\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)
 +
</math>
 +
 
 +
==memo==
 +
* <math>\gamma=\begin{pmatrix}A & B \\ C & D \\\end{pmatrix}\in \Gamma_g</math>
 +
:<math>
 +
\begin{align}
 +
^tAC=^tCA \\
 +
^tBD=^tDB \\
 +
^tAD-^tCB= I_g
 +
\end{align}
 +
</math>
 +
* Igusa subgroup <math>\Gamma_{1,2}:=\{\gamma\in \Gamma_g|Q(\gamma \mathbf{x})=Q(\mathbf{x}) \pmod 2\}</math>, where <math>\mathbf{x}=(\mathbf{x_1},\mathbf{x_2})\in \mathbb{Z}^g\times \mathbb{Z}^g=\mathbb{Z}^{2g}</math>, <math>Q(\mathbf{x})=^t\mathbf{x_1} \mathbf{x_2}</math>
 +
* for <math>\Omega\in \mathbb{H}_g</math>, define a lattice <math>\Lambda_{\Omega}=\mathbb{Z}^g+\Omega \mathbb{Z}^g\subset \mathbb{C}^g</math>
 +
* a smooth vector <math>f_{\Omega}\in \mathcal{H}_{\infty}</math>, (Schwartz space, rapidly decreasing smooth function)
 +
* a functional <math>\mu_{\mathbb{Z}}\in \mathcal{H}_{-\infty}</math>, where <math>\mathcal{H}_{-\infty}</math> is the space '''conjugate''' linear continuous maps from <math>\mathcal{H}_{\infty}</math> to <math>\mathbb{C}</math>
 +
* let <math>\mathbf{x}=(x_1,x_2)</math> and <math>\underline{\mathbf{x}}=\Omega x_1+x_2</math>
 +
* <math>\Theta(\underline{\mathbf{x}},\Omega)</math> appears as pairing
 +
:<math>
 +
\langle U_{(1,x)}f_{\Omega}, \mu_{\mathbb{Z}}\rangle=c\exp(\pi i ^tx_1 \underline{\mathbf{x}})\Theta(\underline{\mathbf{x}},\Omega)
 +
</math>
 +
* <math>A_i=p_i,B_i=x_i</math> in usual notation for Heisenberg algebra
 +
* <math>[X,P] = X P - P X = i \hbar</math>
  
 
==related items==
 
==related items==
22번째 줄: 247번째 줄:
 
* [[Howe duality]]
 
* [[Howe duality]]
 
* [[Heisenberg group and Heisenberg algebra]]
 
* [[Heisenberg group and Heisenberg algebra]]
−
 
  
 
==computational resource==
 
==computational resource==
30번째 줄: 254번째 줄:
 
[[분류:Lie theory]]
 
[[분류:Lie theory]]
 
[[분류:Talks and lecture notes]]
 
[[분류:Talks and lecture notes]]
 +
[[분류:migrate]]

2020년 11월 16일 (월) 05:31 기준 최신판

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

computational resource

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