Let $\beta SL(2,F)$ be a tree of $SL(2,F)$ , $\Delta$ be any edge of $\beta SL(2,F)$.The isotropy groups of the vertices of the edge are the two maximal compact subgroups $J_{0}=\left(
\begin{array}{cc}
\mathcal{O}_{\mathbb{F}} & \mathcal{O}_{\mathbb{F}} \\
\mathcal{O}_{\mathbb{F}} & \mathcal{O}_{\mathbb{F}} \\
\end{array}
\right)\cap SL(2)$ and $J_{1}=\left(
\begin{array}{cc}
\mathcal{O}_{\mathbb{F}}& \varpi_{\mathbb{F}}^{-1}\mathcal{O}_{\mathbb{F}} \\
\varpi_{\mathbb{F}} \mathcal{O}_{\mathbb{F}}& \mathcal{O}_{\mathbb{F}} \\
\end{array}
\right)\cap SL(2)$, and the isotropy group of the edge is an Iwahori
subgroup $I=J_{0}\cap J_{1}$.
Let $R(I)$, $R(J_{0})$ and $R(J_{1})$ are the representation rings, so we have this short sequence
$0 \xrightarrow{} R(I)\xrightarrow{\delta=Ind_{I}^{J_{0}}\oplus-Ind_{I}^{J_{1}}}R(J_{0})\oplus R(J_{1})\xrightarrow{}0$. Therefore, $H_{0}=\frac{R(J_{0})\bigoplus R(J_{1})}{\delta(R(I))}$, and $H_{1}=Ker\delta$.
$Q$: What are all the homologous elements in $H_{0}$?
