"Cartan decomposition of general linear groups"의 두 판 사이의 차이

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imported>Pythagoras0
(새 문서: ==computational resource== * https://drive.google.com/file/d/1LV3AvkCQAGB_ifEzpy8V-96mq-2a2amO/view)
 
 
(사용자 2명의 중간 판 7개는 보이지 않습니다)
1번째 줄: 1번째 줄:
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==introduction==
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*
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<math>
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\newcommand{\pmat}[4]{\begin{pmatrix} #1 & #2 \\ #3 & #4\end{pmatrix}}
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\def\GL#1{\mathrm{GL}_{#1}}
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\newcommand{\Q}{\mathbb{Q}}
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\newcommand{\Z}{\mathbb{Z}}
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\newcommand{\Qp}{\Q_p}
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\newcommand{\Zp}{\Z_p}
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\newcommand{\HH}{\mathcal{H}}
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\newcommand{\fsph}{f_{\mathrm{sph}}}
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</math>
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==application to Hecke operators==
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* Let <math>G = \GL2(\Qp)</math> and <math>K = \GL2(\Zp)</math>
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* Cartan decomposition : <math>G = \bigcup_{(m,n)\in \Z^2 : m\geq n} K\pmat {p^m} 0 0  {p^n} K</math>
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* 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>
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* 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>
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* How <math>T_p</math> and <math>R_p</math> act?
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* The double coset for <math>T_p</math> decomposes as
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\[
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K \pmat p 0 0 1 K =
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\bigcup_{b=0}^{p-1}
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\pmat p b 0 1 K
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\bigcup
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\pmat 1 0 0 p K .
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\]
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* Hence
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\[
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\begin{aligned}
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(T_p \fsph)(1) & =
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\int_{K}\sum_{b}^{p-1}
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\fsph\left(\pmat p b 0 1 g \right)+
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\fsph\left(\pmat 1 0 0 p g \right)\, dg \\
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& =
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\fsph\left(\pmat p b 0 1 g \right)+
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\fsph\left(\pmat 1 0 0 p g \right) \\
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& =
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p \chi_1(p)|p|^{1/2}+p \chi_2(p)|p|^{-1/2} \\
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& =
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p^{1/2}(\chi_1(p)+\chi_2(p)).
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\end{aligned}
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\]
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* The double coset for <math>R_p</math> is the single coset <math>\pmat p 0 0 p K</math>, so
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\[
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\begin{aligned}
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(R_p\fsph)(1) & = \int_K \fsph\left(\pmat p 0 0 p g \right)+ dg \\
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& =
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\fsph\left(\pmat p 0 0 p g \right) \\
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& =
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\chi_1(p)\chi_2(p).
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\end{aligned}
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\]
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==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
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[[분류:migrate]]
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q3042798 Q3042798]
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===Spacy 패턴 목록===
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* [{'LOWER': 'cartan'}, {'LEMMA': 'decomposition'}]

2021년 2월 17일 (수) 01:42 기준 최신판

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 \[G = \bigcup_{(m,n)\in \Z^2 : m\geq n} K\pmat {p^m} 0 0 {p^n} K\]
  • 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?
  • 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

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'cartan'}, {'LEMMA': 'decomposition'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=Cartan_decomposition_of_general_linear_groups&oldid=51835"