Consider $G = PGL(2)$ and $PSL(2)$.
Let $F$ P$ be a non archimedean field with ring parabolic subgroup of integers $o$. Let GL(n)$ with Levi decomposition $K$ be a maximal compact in P =MN$, where $G(F)$, that N$ is the normalizer unipotent radical.
Let $NI$ \pi$ be an irreducible representation of the Iwahori subgroup $I$ or $G(o)$.
Given a standard Borel $B \subset G$ with $G(F) = B(F) K$, and a character $\chi : B(o) \rightarrow \mathbb{C}^1$.
How does $ Ind_{B(o)}^K \chi$ decompose into irreducibles?
As an intermediate step one has M(\mathbf{Z}_p)$ inflated to know:
How does $Ind_{B(o)}^I \chi$ decompose into irreducibles?
This probably easier, since we can rely on the Iwahori decomposition.
I know P(\mathbf{Z}_p)$,
how the situation behaves for $PGL(2)$ for $K = PGL(2)$ in full generality (Casselman, Silberger), and for $K=NI$ only in odd residue characteristic (Silberger), I know nothing about $SL(2)$, but $does $ Res_{K \cap PSL(2)} Ind_{B(o)}^K \chi \cong Ind_{B(o) \cap PSL(2)}^{K \cap PSL(2)} Res_{B(o) \cap PSL(2)Ind_{P(\mathbf{Z}_p)}^{GL_n(\mathbf{Z}_p)} \chi$$should give a bijection, which gives the results for $SL(2)$ from the results of $GL(2)$.
Let $\chi$ have conductor $p^k$.
Is $(Ind_{B(o)}^K \chi)^{\Gamma(p^k)}$ irreduciblepi$ decompose?
Is $(Ind_{B(o)}^K \chi)^{\Gamma(p^N)} \ominus (Ind_{B(o)}^K \chi)^{\Gamma(p^{N-1})}$ irreducible,
It would be sufficient for $N>k$?
I do not me to know what the filtration $\Gamma(p^k)$ of $NI$ should be, but for $G(o)$ consider result in the standard filtration of matricessimplest case, which are the identity modulo $p^k$.
Does the last result generalize to where $P$ parabolic in $PGL(n)$ or $PSL(n)$?is a Borel subgroup
