Let $G=SL(2,F)$ and $I=J_{0}\cap J_{1}$ be the Iwahori subgroup of $SL(2, F)$, where $J_{0}=\left( \begin{array}{cc} \mathcal{O}_{\mathbb{F}} & \mathcal{O}_{\mathbb{F}} \\ \mathcal{O}_{\mathbb{F}} & \mathcal{O}_{\mathbb{F}} \\ \end{array} \right)\cap SL(2)$ and $J_{1}=\left( \begin{array}{cc} \mathcal{O}_{\mathbb{F}}& \varpi_{\mathbb{F}}^{-1}\mathcal{O}_{\mathbb{F}} \\ \varpi_{\mathbb{F}} \mathcal{O}_{\mathbb{F}}& \mathcal{O}_{\mathbb{F}} \\ \end{array} \right)\cap SL(2)$ are the maximal compact subgroups of $SL(2, F)$. Here $F$ is a local non-Archimedean $p$-adic field, $\mathcal{O}_{F}$ is the valuation ring and $\varpi$ is the uniformizer.

Now, we know that if $\lambda^{2}= 1$, then $Ind_{I}^{G}\lambda=\lambda^{-}\oplus\lambda^{+}$. Also , we can write $Ind_{I}^{G}\lambda=Ind_{J_{0}}^{G}(Ind_{I}^{J_{0}}\lambda)$.

Does the following equality hold?

$Ind_{I}^{G}\lambda=Ind_{J_{0}}^{G}\lambda\oplus Ind_{J_{1}}^{G}\lambda$.

I do apologize for my bad English and my bad temper to create a question.