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Marc Palm
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I can give only apartial answer to the 2nd question, but it will be too long for a comment:

For the 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 $P$.

For the observation with supercuspidal, I have only a comment:

I assume that the super cuspidal is induced from an open compact group $G(o)$, thenone can use Frobenius reciprocity, i.e. consider a representation of $\chi \in G(o/p)$

$$ Hom_{G(o/p)}(\chi, \left( Res_{G(o)} Ind_{G(o)}^G \pi\right)^{\Gamma(p)}) = \bigoplus_{m \in G //G(o)} Hom_G ( \chi, Ind_{G(o/p) \cap G(o/p)^m}^{G(o/p)} \pi )$$ with the Cartan decomposition can be computed quite fast.

For $GL(n)$, you can see for which representation the induced one is actually irreducible and supercuspidal (see e.g. Kutzko and Bushnell).

For exactness:

Restriction to open compact subgroup is always an exact functor, since the representation theory of compact groups is completely irreducible.

Marc Palm
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