Cartan decomposition of general linear groups - 편집 역사

2021년 2월 17일 (수) 09:42에 Pythagoras0님의 편집

2021-02-17T09:42:58Z

Pythagoras0: /* 메타데이터 */ 새 문단

2020-12-28T07:53:29Z

메타데이터: 새 문단

2020년 11월 13일 (금) 13:31에 imported>Pythagoras0님의 편집

2020-11-13T13:31:04Z

2020년 9월 24일 (목) 03:35에 imported>Pythagoras0님의 편집

2020-09-24T03:35:00Z

imported>Pythagoras0: /* application to Hecke operators */

2020-02-19T08:26:58Z

application to Hecke operators

imported>Pythagoras0: /* application to Hecke operators */

2020-02-19T08:09:05Z

application to Hecke operators

imported>Pythagoras0: /* application to Hecke operators */

2020-02-19T08:00:35Z

application to Hecke operators

2020년 2월 19일 (수) 07:59에 imported>Pythagoras0님의 편집

2020-02-19T07:59:21Z

imported>Pythagoras0: 새 문서: ==computational resource== * https://drive.google.com/file/d/1LV3AvkCQAGB_ifEzpy8V-96mq-2a2amO/view

2020-02-19T07:29:32Z

새 문서: ==computational resource== * https://drive.google.com/file/d/1LV3AvkCQAGB_ifEzpy8V-96mq-2a2amO/view

새 문서

==computational resource==
* https://drive.google.com/file/d/1LV3AvkCQAGB_ifEzpy8V-96mq-2a2amO/view

@@ 59번째 줄: / 59번째 줄: @@
 [[분류:migrate]]
-== 메타데이터 ==
+==메타데이터==
 ===위키데이터===
 * ID :  [https://www.wikidata.org/wiki/Q3042798 Q3042798]

← 이전 판		2020년 12월 28일 (월) 07:53 판
58번째 줄:		58번째 줄:
	* https://drive.google.com/file/d/1LV3AvkCQAGB_ifEzpy8V-96mq-2a2amO/view		* https://drive.google.com/file/d/1LV3AvkCQAGB_ifEzpy8V-96mq-2a2amO/view
	[[분류:migrate]]		[[분류:migrate]]
		+
		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q3042798 Q3042798]

← 이전 판		2020년 11월 13일 (금) 13:31 판
57번째 줄:		57번째 줄:
	==computational resource==		==computational resource==
	* https://drive.google.com/file/d/1LV3AvkCQAGB_ifEzpy8V-96mq-2a2amO/view		* https://drive.google.com/file/d/1LV3AvkCQAGB_ifEzpy8V-96mq-2a2amO/view
		+	[[분류:migrate]]

@@ 2번째 줄: / 2번째 줄: @@
+*
+<math>
 \newcommand{\pmat}[4]{\begin{pmatrix} #1 & #2 \\ #3 & #4\end{pmatrix}}
 \def\GL#1{\mathrm{GL}_{#1}}
@@ 11번째 줄: / 11번째 줄: @@
 \newcommand{\HH}{\mathcal{H}}
 \newcommand{\fsph}{f_{\mathrm{sph}}}
+</math>
 ==application to Hecke operators==
-* Let $G = \GL2(\Qp)$ and $K = \GL2(\Zp)$
+* Let <math>G = \GL2(\Qp)</math> and <math>K = \GL2(\Zp)</math>
-* Cartan decomposition : $G = \bigcup_{(m,n)\in \Z^2 : m\geq n} K\pmat {p^m} 0 0  {p^n} K$
+* Cartan decomposition : <math>G = \bigcup_{(m,n)\in \Z^2 : m\geq n} K\pmat {p^m} 0 0  {p^n} K</math>
-* The Hecke operator $T_p\in \HH(G,K)$ is given by convolution with the characteristic function of $K\pmat p 0 0 1 K$
+* The Hecke operator <math>T_p\in \HH(G,K)</math> is given by convolution with the characteristic function of <math>K\pmat p 0 0 1 K</math>
-* Similarly, the operator $R_p$ is given by convolution with the characteristic function of $K \pmat p 0 0 p K$
+* Similarly, the operator <math>R_p</math> is given by convolution with the characteristic function of <math>K \pmat p 0 0 p K</math>
-* How $T_p$ and $R_p$ act?
+* How <math>T_p</math> and <math>R_p</math> act?
-* The double coset for $T_p$ decomposes as
+* The double coset for <math>T_p</math> decomposes as
 \[
 K \pmat p 0 0 1 K =
@@ 44번째 줄: / 44번째 줄: @@
 \]
-* The double coset for $R_p$ is the single coset $\pmat p 0 0 p K$, so
+* The double coset for <math>R_p</math> is the single coset <math>\pmat p 0 0 p K</math>, so
 \[
 \begin{aligned}

@@ 16번째 줄: / 16번째 줄: @@
 * Let $G = \GL2(\Qp)$ and $K = \GL2(\Zp)$
 * Cartan decomposition : $G = \bigcup_{(m,n)\in \Z^2 : m\geq n} K\pmat {p^m} 0 0  {p^n} K$
-* The Hecke operator $T_p\in \HH_{\GL2(\Qp)}$ is given by convolution with the characteristic function of $K\pmat p 0 0 1 K$
+* The Hecke operator $T_p\in \HH(G,K)$ is given by convolution with the characteristic function of $K\pmat p 0 0 1 K$
 * Similarly, the operator $R_p$ is given by convolution with the characteristic function of $K \pmat p 0 0 p K$
 * How $T_p$ and $R_p$ act?

@@ 15번째 줄: / 15번째 줄: @@
 ==application to Hecke operators==
 * Let $G = \GL2(\Qp)$ and $K = \GL2(\Zp)$
-* Cartan decomposition : $G = \bigcup_{\lambda} Bp^{\lambda}B$
+* Cartan decomposition : $G = \bigcup_{(m,n)\in \Z^2} K\pmat {p^m} 0 0  {p^n} K$
 * The Hecke operator $T_p\in \HH_{\GL2(\Qp)}$ is given by convolution with the characteristic function of $K\pmat p 0 0 1 K$
 * Similarly, the operator $R_p$ is given by convolution with the characteristic function of $K \pmat p 0 0 p K$
@@ 54번째 줄: / 54번째 줄: @@
 \end{aligned}
 \]
 ==computational resource==
 * https://drive.google.com/file/d/1LV3AvkCQAGB_ifEzpy8V-96mq-2a2amO/view

← 이전 판		2020년 2월 19일 (수) 07:59 판
1번째 줄:		1번째 줄:
		+	==introduction==
		+	*
		+
		+	$
		+	\newcommand{\pmat}[4]{\begin{pmatrix} #1 & #2 \\ #3 & #4\end{pmatrix}}
		+	\def\GL#1{\mathrm{GL}_{#1}}
		+	\newcommand{\Q}{\mathbb{Q}}
		+	\newcommand{\Z}{\mathbb{Z}}
		+	\newcommand{\Qp}{\Q_p}
		+	\newcommand{\Zp}{\Z_p}
		+	\newcommand{\HH}{\mathcal{H}}
		+	\newcommand{\fsph}{f_{\mathrm{sph}}}
		+	$
		+
		+	==application to Hecke operators==
		+	* Let $G = \GL2(\Qp)$ and $K = \GL2(\Zp)$
		+	* Cartan decomposition : $G = \bigcup_{\lambda} Bp^{\lambda}B$
		+	* The Hecke operator $T_p\in \HH_{\GL2(\Qp)}$ is given by convolution with the characteristic function of $K\pmat p 0 0 1 K$
		+	* Similarly, the operator $R_p$ is given by convolution with the characteristic function of $K \pmat p 0 0 p K$
		+	* How $T_p$ and $R_p$ act?
		+	* The double coset for $T_p$ decomposes as
		+	\[
		+	K \pmat p 0 0 1 K =
		+	\bigcup_{b=0}^{p-1}
		+	\pmat p b 0 1 K
		+	\bigcup
		+	\pmat 1 0 0 p K .
		+	\]
		+	* Hence
		+	\[
		+	\begin{aligned}
		+	(T_p \fsph)(1) & =
		+	\int_{K}\sum_{b}^{p-1}
		+	\fsph\left(\pmat p b 0 1 g \right)+
		+	\fsph\left(\pmat 1 0 0 p g \right)\, dg \\
		+	& =
		+	\fsph\left(\pmat p b 0 1 g \right)+
		+	\fsph\left(\pmat 1 0 0 p g \right) \\
		+	& =
		+	p \chi_1(p)\|p\|^{1/2}+p \chi_2(p)\|p\|^{-1/2} \\
		+	& =
		+	p^{1/2}(\chi_1(p)+\chi_2(p)).
		+	\end{aligned}
		+	\]
		+
		+	* The double coset for $R_p$ is the single coset $\pmat p 0 0 p K$, so
		+	\[
		+	\begin{aligned}
		+	(R_p\fsph)(1) & = \int_K \fsph\left(\pmat p 0 0 p g \right)+ dg \\
		+	& =
		+	\fsph\left(\pmat p 0 0 p g \right) \\
		+	& =
		+	\chi_1(p)\chi_2(p).
		+	\end{aligned}
		+	\]
		+
		+
	==computational resource==		==computational resource==
	* https://drive.google.com/file/d/1LV3AvkCQAGB_ifEzpy8V-96mq-2a2amO/view		* https://drive.google.com/file/d/1LV3AvkCQAGB_ifEzpy8V-96mq-2a2amO/view