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 : $G = \bigcup_{(m,n)\in \Z^2 : m\geq n} K\pmat {p^m} 0 0 {p^n} K$
- 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
\[ \begin{aligned} (R_p\fsph)(1) & = \int_K \fsph\left(\pmat p 0 0 p g \right)+ dg \\ & = \fsph\left(\pmat p 0 0 p g \right) \\ & = \chi_1(p)\chi_2(p). \end{aligned} \]