Cartan decomposition of general linear groups

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

• https://drive.google.com/file/d/1LV3AvkCQAGB_ifEzpy8V-96mq-2a2amO/view