Representations of GL(2, Q_p) and GL(2, Z_p)

Question

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 $\operatorname{GL}_n(F)$?

Answer 1

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

Answer 2

Some of your questions have several non-equivalent interpretations so it's not so easy to give precise answers, but I will try to give at least partial answers to two of your questions.

How does the representation theory of $\operatorname{GL}_2(F)$ and the representation theory of $\operatorname{GL}_2(o)$ affect each other?

One possible answer is that every supercuspidal representation of $\operatorname{GL}_2(F)$ has a unique type on $\operatorname{GL}_2(o)$ (this is due to Henniart for $\operatorname{GL}_2$ and Paskunas for $\operatorname{GL}_n$). So there is an interesting one-to-one correspondence between supercuspidals and certain representations of $\operatorname{GL}_2(o)$.

What irreducible representations of $\operatorname{GL}_2(o)$ are important for describing the cuspidal representations of $\operatorname{GL}(2,F)$?

In light of the above connection between supercuspidals and types on $\operatorname{GL}_2(o)$, at the very least the representations of $\operatorname{GL}_2(o)$ which are supercuspidal types are important. I think these are the ones which are cuspidal or which have an orbit with Eisenstein polynomial. Note that this does not say anything about non-supercuspidal representations and their types on $\operatorname{GL}_2(o)$ so is certainly not an exhaustive answer.