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.



The induced representations of $SL(2, F)$.
I assume that $\lambda$ is a character!? How is your $\lambda$ a representation of $J\_0$ or $J\_1$, do you mean the induction $Ind\_I^{J\_k} \lambda$ instead, but the induction is never irreducible.


On some related matters on $GL(2)$:


The question whether 
$ Ind\_{I}^{J\_i} \lambda $
splits has been answered by Silberger for $PGL(2)$ for $J\_0$ and $J\_1$ in odd residue characteristic, and by Casselman for $PGL(2)$ for $J\_0$ in all residue characteristics.


Silberger, Allan J. Irreducible representations of a maximal compact subgroup of ${\rm pgl}\_{2}$pgl2 over the $p$p-adics. Math. Ann. 229 (1977), no. 1, 1–12.


Silberger, Allan J. ${\rm PGL}\_{2}$PGL2 over the $p$p-adics: its representations, spherical functions, and Fourier analysis. Lecture Notes in Mathematics, Vol. 166


Casselman, William The restriction of a representation of ${\rm GL}\_{2}(k)$GL2(k) to ${\rm GL}\_{2}({\germ o})$GL2(o). Math. Ann. 206 (1973), 311–318.


In these cases, the restriction induction formula $Res\_{SL(2,F)} Ind^{GL(2,F)}\_{J\_i} \lambda =Ind\_{J\_i}^{SL(2,F)} \lambda$! Induction by steps imply the results, if it $ Ind\_{I}^{J\_i} \lambda $ splits. 


I am not sure, if that is necessary, but you can look at the induction restriction formula for $Res\_{J\_i} Ind^{SL(2,F)}\_{J\_i} \lambda$ and see what happens.


