"Cartan decomposition of general linear groups"의 두 판 사이의 차이
imported>Pythagoras0 |
imported>Pythagoras0 |
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* | * | ||
| − | + | <math> | |
\newcommand{\pmat}[4]{\begin{pmatrix} #1 & #2 \\ #3 & #4\end{pmatrix}} | \newcommand{\pmat}[4]{\begin{pmatrix} #1 & #2 \\ #3 & #4\end{pmatrix}} | ||
\def\GL#1{\mathrm{GL}_{#1}} | \def\GL#1{\mathrm{GL}_{#1}} | ||
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\newcommand{\HH}{\mathcal{H}} | \newcommand{\HH}{\mathcal{H}} | ||
\newcommand{\fsph}{f_{\mathrm{sph}}} | \newcommand{\fsph}{f_{\mathrm{sph}}} | ||
| − | + | </math> | |
==application to Hecke operators== | ==application to Hecke operators== | ||
| − | * Let | + | * Let <math>G = \GL2(\Qp)</math> and <math>K = \GL2(\Zp)</math> |
| − | * Cartan decomposition : | + | * 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 | + | * 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 | + | * 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 | + | * How <math>T_p</math> and <math>R_p</math> act? |
| − | * The double coset for | + | * The double coset for <math>T_p</math> decomposes as |
\[ | \[ | ||
K \pmat p 0 0 1 K = | K \pmat p 0 0 1 K = | ||
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\] | \] | ||
| − | * The double coset for | + | * The double coset for <math>R_p</math> is the single coset <math>\pmat p 0 0 p K</math>, so |
\[ | \[ | ||
\begin{aligned} | \begin{aligned} | ||
2020년 9월 23일 (수) 20:35 판
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
\[ \begin{aligned} (R_p\fsph)(1) & = \int_K \fsph\left(\pmat p 0 0 p g \right)+ dg \\ & = \fsph\left(\pmat p 0 0 p g \right) \\ & = \chi_1(p)\chi_2(p). \end{aligned} \]