Jacobi's theta function from a representation theoretic viewpoint - 편집 역사

2020년 11월 16일 (월) 12:31에 Pythagoras0님의 편집

2020-11-16T12:31:21Z

차이 보기

2020년 11월 13일 (금) 15:52에 imported>Pythagoras0님의 편집

2020-11-13T15:52:42Z

imported>Pythagoras0: /* Heisenberg group */

2015-08-01T09:31:07Z

Heisenberg group

imported>Pythagoras0: /* Riemann theta function */

2015-08-01T09:26:25Z

Riemann theta function

imported>Pythagoras0: /* overview */

2015-08-01T09:25:57Z

overview

imported>Pythagoras0: /* computing $P f_{\Omega}$ */

2015-08-01T09:24:10Z

computing $P f_{\Omega}$

imported>Pythagoras0: /* covering of the symplectic group */

2015-08-01T09:20:41Z

covering of the symplectic group

imported>Pythagoras0: /* overview */

2015-06-11T02:25:12Z

overview

imported>Pythagoras0: /* theta as matrix coefficients */

2015-06-11T01:37:17Z

theta as matrix coefficients

imported>Pythagoras0: /* theta as matrix coefficients */

2015-06-11T01:36:50Z

theta as matrix coefficients

← 이전 판		2020년 11월 13일 (금) 15:52 판
254번째 줄:		254번째 줄:
	[[분류:Lie theory]]		[[분류:Lie theory]]
	[[분류:Talks and lecture notes]]		[[분류:Talks and lecture notes]]
		+	[[분류:migrate]]

@@ 78번째 줄: / 78번째 줄: @@
 ==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
+* 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
 $$
@@ 84번째 줄: / 84번째 줄: @@
 $$
 : <math> 1 \rightarrow S^1~\rightarrow~H(2g, \mathbb{R})~\rightarrow~V \rightarrow 0</math>
 ;thm (Stone-von Neumann)
 There exists a unique irreducible unitary representation
@@ 89번째 줄: / 90번째 줄: @@
 U:H(2g,\mathbb{R})\to Aut(\mathcal{H})
 $$
-such that $U_{\lambda}=\lambda \operatorname{id}_{\mathcal{H}}$ for all $\lambda \in S^1$. In other words, if there are two representations $U^{(1)}$ and $U^{(2)}$ on $\mathcal{H}_1$ and $\mathcal{H}_2$, then there exists an isomorphism $A: \mathcal{H}_1 \to \mathcal{H}_2$ such that
+such that $U_{\lambda}=\lambda \operatorname{id}_{\mathcal{H}}$ for all $\lambda \in S^1$. In other words, if there are two such representations $U^{(1)}$ and $U^{(2)}$ on $\mathcal{H}_1$ and $\mathcal{H}_2$, then there exists an isomorphism $A: \mathcal{H}_1 \to \mathcal{H}_2$ such that
 $$
 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}
 $$
 * related to the equivalence of matrix mechanics and wave mechanics in the early days of quantum mechanics
 ===realization===
-* let $\mathcal{H}:=L^2(\mathbb{R}^g)$
+* let $\mathcal{H}_1:=L^2(\mathbb{R}^g)$
 * for $(\lambda,y_1,y_2)\in H(2g, \mathbb{R})$, $x_1\in \mathbb{R}^g$ and $\varphi\in \mathcal{H}$, define
 $$
@@ 109번째 줄: / 111번째 줄: @@
 [A_i, B_j] = \delta_{ij}C, [A_i, C] =[B_j, C] = 0
 $$
-* want to get a reprsentation $\delta U$ of $\mathfrak{h}(2g,\mathbb{R})$ on a certain dense set $\mathcal{H}_{\infty}$ of $\mathcal{H}$
+* want to get a reprsentation $\delta U$ of $\mathfrak{h}(2g,\mathbb{R})$ on a certain dense subspace $\mathcal{H}_{\infty}$ of $\mathcal{H}_1$
 * for $X\in \mathfrak{h}(2g,\mathbb{R})$, let
 $$
 \delta U_{X}f:=\lim_{t\to 0}\frac{(U_{\exp_H(tX)}f)-f}{t}
 $$
 * $A_i$ acts as $\frac{\partial f}{\partial x_i}$
 * $B_i$ acts as $2\pi i x_i f(x)$
@@ 124번째 줄: / 127번째 줄: @@
 ;prop
 There is a unique $f_{\Omega}\in \mathcal{H}_{\infty}$, unique up to scalars, such that $\delta U_{X} f_{\Omega}=0, \forall X\in W_{\Omega}$
-* Let $L=\mathbb{Z}^{2g}$ and $\sigma:L\to H(2g, \mathbb{R})$ defined by
+* Let $\sigma:\mathbb{Z}^{2g}\to H(2g, \mathbb{R})$ defined by
 $$
-\sigma(n):=((-1)^{^tn_1n_2},n),\, n\in L
+\sigma(n):=((-1)^{^tn_1n_2},n),\, n\in \mathbb{Z}^{2g}
 $$
 ;prop
@@ 142번째 줄: / 145번째 줄: @@
 ===quasi-periodicity===
-* for $n=(n_1,n_2)\in \mathbb{Z}^{2g}$,
+* for $n=(n_1,n_2)\in \mathbb{Z}^{g}\times \mathbb{Z}^{g}\mathbb{Z}^{2g}$,
 $$
 \begin{aligned}

@@ 64번째 줄: / 64번째 줄: @@
 ,\, \Omega\in \mathbb{H}_g,\mathbb{z}\in \mathbb{C}^g
 $$
-;thm (quasi-periodicity)
+* quasi-periodicity
 Let $\mathbf{a},\mathbf{b}\in \mathbb{Z}^g,\mathbf{z}\in \mathbb{C}^g,\Omega\in \mathbb{H}_g$. We have
 $$
 \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)
 $$
-;thm (modularity)
+* modularity
 Let $\gamma=\begin{pmatrix}A & B \\ C & D \\ \end{pmatrix}\in \Gamma_{1,2}$. We have
 $$

@@ 16번째 줄: / 16번째 줄: @@
 ==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
+* $V=(\mathbb{R}^{2g},A)$, 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

@@ 192번째 줄: / 192번째 줄: @@
 $$
 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 $P\mu_{\mathbb{Z}}$===

@@ 158번째 줄: / 158번째 줄: @@
 (\lambda,x)\mapsto (\lambda, \gamma x)
 $$
-* define a new action $U'$ of $H(2g,\mathbb{R})$ on $\mathcal{H}$ by
+* define a new representation $U'$ of $H(2g,\mathbb{R})$ on $\mathcal{H}$ by
 $$
 U'_{(\lambda,x)}f:=U_{(\lambda,\gamma x)}f
 $$
-* by the Stone-von Neumann theorem, there exists a unitary map $A_{\gamma}:\mathcal{H}\to \mathcal{H}$ intertwining these two
+* by the Stone-von Neumann theorem, there exists a unitary map $A_{\gamma}:\mathcal{H}\to \mathcal{H}$ intertwining $U$ and $U'$
 * let $U(\mathcal{H})$ be the group of unitary isomorphisms of $\mathcal{H}$ and define
 $$
@@ 169번째 줄: / 169번째 줄: @@
 * then for $A\in \widetilde{Mp}(2g,\mathbb{R})$, there exists $\gamma \in Sp_{2g}(\mathbb{R})$ such that
 $$
-AU_{(\lambda,x)}A^{-1}=U_{(\lambda,\gamma x)}
+AU_{(\lambda,x)}A^{-1}=U_{(\lambda,\gamma x)} \label{star}
 $$
 ;lemma
@@ 175번째 줄: / 175번째 줄: @@
 * 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 $\gamma\in Sp(2g,\mathbb{R})$ and $P\in \widetilde{Mp}(2g,\mathbb{R})$ such that $\rho(P)=\gamma$. Then
 $$
@@ 187번째 줄: / 182번째 줄: @@
 \end{aligned}
 $$
 * once we compute $P f_{\Omega}, P\mu_{\mathbb{Z}}$, the functional equation of $\Theta$ will fall out

@@ 135번째 줄: / 135번째 줄: @@
 $$
 ;thm
-Let $\Omega\in \mathbb{H}_g$ be fixed. Let $\mathcal{H}$ be a representation of $H_{2g}(\mathbb{R})$ and $f_{\Omega},\mu_{\mathbb{Z}}$ as above. For $x\in V=\mathbb{R}^{2g}$,
+Let $\Omega\in \mathbb{H}_g$ be fixed. Let $\mathcal{H}$ be a representation of $H(2g,\mathbb{R})$ and $f_{\Omega},\mu_{\mathbb{Z}}$ as above. For $x\in V=\mathbb{R}^{2g}$,
 $$
 \langle U_{(1,x)}f_{\Omega}, \mu_{\mathbb{Z}}\rangle=c\exp(\pi i ^tx_1 \underline{\mathbf{x}})\Theta(\underline{\mathbf{x}},\Omega)

@@ 121번째 줄: / 121번째 줄: @@
 * $\mathcal{H}_{\infty}$, 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 $\mathcal{h}(2g,\mathbb{R})\otimes \mathbb{C}$
+* 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 $\delta U_{X} f_{\Omega}=0, \forall X\in W_{\Omega}$