I am working on the chamber homology for $SL(2,F)$, and stuck at some basic stuff on D.S. reps of $SL(2,F)$.
IfLet $J_{0}=SL(2,\mathcal{O})\cap SL(2,F)$$ I=\left( \begin{array}{cc} \mathcal{O}_{F} & \mathcal{O}_{F} \\ \varpi_{\mathbb{F}}\mathcal{O}_{F} & \mathcal{O}_{F}\\ \end{array} \right)\cap SL(2, F)$. Now, let $ w_{0}= \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \\ \end{array} \right)$ and $J_{1}=(wJ_{0}w^{-1})\cap SL(2,F)$$ w_{1}= \left( \begin{array}{cc} 0 & -\varpi^{-1}_{F} \\ \varpi_{F} & 0 \\ \end{array} \right)$, then $J_{0}=I \cup Iw_{0}I$ and $J_{1}= I \cup Iw_{1}I$ are the two max.maximal compact subgroups of $SL(2,F)$ where $ w= \left( \begin{array}{cc} 0 & 1 \\ \varpi_{\mathbb{F}} & 0 \\ \end{array} \right)$, $\varpi$$\varpi_{\mathbb{F}}$ is the uniformizer and let $I=J_{0}\cap J_{1}$ be Iwahori subgroup.
Just wondering if anybody knows how can I induce a cuspidal reps(D.S.) from a charachter belong to $J_{0}$ or/and $J_{1}$?
