"Cartan decomposition of general linear groups"의 두 판 사이의 차이
imported>Pythagoras0 |
imported>Pythagoras0 |
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| 16번째 줄: | 16번째 줄: | ||
* Let $G = \GL2(\Qp)$ and $K = \GL2(\Zp)$ | * Let $G = \GL2(\Qp)$ and $K = \GL2(\Zp)$ | ||
* Cartan decomposition : $G = \bigcup_{(m,n)\in \Z^2 : m\geq n} K\pmat {p^m} 0 0 {p^n} K$ | * 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 \ | + | * 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 $K \pmat p 0 0 p K$ | * 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? | * How $T_p$ and $R_p$ act? | ||
2020년 2월 19일 (수) 01:26 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
\[ \begin{aligned} (R_p\fsph)(1) & = \int_K \fsph\left(\pmat p 0 0 p g \right)+ dg \\ & = \fsph\left(\pmat p 0 0 p g \right) \\ & = \chi_1(p)\chi_2(p). \end{aligned} \]