I can give you a partial solution for the commuting with parabolic induction.
The Iwasawa decomposition gives you
$$ Res_{G(o)} Ind_{P(F)}^{G(F)} \chi \cong Ind_{P(o)}^{G(O)} Res_{P(o)} \chi $$
Then you find that
$$ \left( Ind_{P(o)}^{G(o)} Res_{P(o)} \chi \right)^{\Gamma(p)} = Ind_{P(f)}^{G(f)} \tilde{\chi} $$
where $f$ is the residue field. Where $\tilde{\chi}$ is the irreducible sub representation of $\chi$ by the $M(p)$ invariant vectors for the Levi subgroup $M$ of $P$.