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:



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> How does the representation theory of $\operatorname{GL}\_2(F)$ and the representation theory of $\operatorname{GL}\_2(o)$ affect each other?
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Uri Onn has shown that the irreducible representations of $\operatorname{GL}(2,o)$ will depend only upon the residue characteristic.



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> 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)?
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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.



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> What irreducible representations in Stasinski are important for describing the cuspidal representation of $\operatorname{GL}(2, F)$?
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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".



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> 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 $\operatorname{GL}(n,F)$?
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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 $\operatorname{PGL}\_2(F)$.



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> Can we classify the representation of $\operatorname{GL}\_n(o)$ by restricting the irreducible representation of $\operatorname{GL}\_n(F)$?
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Representations of GL(2, Q_p) and GL(2, Z_p)
Abbreviate $G={\rm GL}(n,F)$ and $K={\rm GL}(n,{\mathfrak o})$.


The general philosophy of "type theory" is the following.


When you restrict an irreducible representation of $G$ to $K$, you get a semisimple representation whose irreducible components are of two sorts:


-- irreducible representations of $K$ occuring in "a lot" of representations of $G$ by restriction. 


-- the other.


The first ones are uninteresting from the point of view of type theory. The second ones are called "typical" (say). 


To construct the smooth dual of $G$, Bushnell and Kutzko construct enough typical representations. But they are very far to have constructed the whole dual of $K$.


So here are the answers to two of your questions :


"Is the classification of the irreducible dual of $G$ known only modulo the representation theory $K$? NO


"Do we know all the representations of $K$ needed for the dual of $G$? YES


