The cuspidal representations of $\operatorname{GL}_n(F)$ a non archimedean field $F$ with ring of integers $o$ can be classified by inducing irreducible representation from $Z\operatorname{GL}_n(o)$.
The general question:
How does the representation theory of $\operatorname{GL}_2(F)$ and the representation theory of $\operatorname{GL}_2(o)$ affect each other?
Uri Onn has shown that the irreducible representations of $\operatorname{GL}(2,o)$ will depend only upon the residue characteristic.
Does the representation theory of $\operatorname{GL}_n(F)$ only depend upon the residue characteristic? Is there a connection to the reduction steps from local field of characteristic zero to fields of characteristic $p$ (e.g. as in the context of the Fundamental lemma)?
In Stasinki "Smooth Representations of $\operatorname{GL}(2,o)$" the representations are partionated in to similarity classes of $2 \times 2$ matrices over the residue field (pg.4419) via applying Clifford theory.
What irreducible representations in Stasinski are important for describing the cuspidal representation of $\operatorname{GL}(2, F)$?
A little more general, I know that the irreducible dual of $\operatorname{GL}_n(o)$ is far from being classified, so to what extent is the irreducible dual of $\operatorname{GL}(n,F)$ described in Bushnell-Kutzo "The admissible dual of $\operatorname{GL}(N)$ via compact open subgroups".
Is the classification of the irreducible dual of $\operatorname{GL}(n,F)$ known only modulo the representation theory $\operatorname{GL}(n, o)$ or do we know all the representations of $\operatorname{GL}(n,o)$ needed for the dual of $\operatorname{GL}(n,F)$?
If the description of the dual of $\operatorname{GL}(n,F)$ is possible independently of the dual of $\operatorname{GL}_n(o)$, there is a natural last question in the context of Silberger "$\operatorname{PGL}_2$ over the $p$ adics", where he classifies the irreducibe of $\operatorname{PGL}_2(o)$ via the restricting the irreducible of $\operatorname{PGL}_2(F)$.
Can we classify the representation of $\operatorname{GL}_n(o)$ by restricting the irreducible representation of $\operatorname{GL}_n(F)$?
