Let $K$ be a nonarchimedean local field with ring of integeres $R\_K$, maximal ideal $m\_K$ and finite residue field $\bf k$. Let $\pi$ be an admissible irreducible complex representation of ${\rm GL}\_2(K)$ with central character $\epsilon$. A fundamental result of Casselman says that there is a largest ideal $J\subseteq R\_K$ such that the subspace $W\_J$ of vectors in $\pi$ such that
$$
\gamma\cdot v=\epsilon(a)v\qquad
\forall\gamma=\left(\begin{array}{cc}
a & b\\
c & d 
\end{array}\right)\in{\rm GL}\_2(R\_K)
\ \text{with}\ c\in J
$$
is non-trivial and in fact $1$-dimensional. As every expert knows, this result is of paramount importance for the theory of modular forms.


Let $v\_0$ be a generator of the $1$-dimensional space $W\_J$. In some cases, it is rather easy to obtain $v\_0$ explicitly. For instance if $\pi=\pi(\mu\_1,\mu\_2)$ is a class $1$ principal series representation with trivial central character (for which $J=R\_K$) it is immediate to check that any generator of $W\_{R\_K}$ is of the form
$$
v\_0(g)=|a|^{s\_1}|d|^{s\_2}|a/d|^{1/2}v\_0(1)\quad
\text{where}\quad
g=\left(\begin{array}{cc}
a & \*\\
 & d 
\end{array}\right)r,\quad r\in{\rm GL}\_2(R\_K)
$$
and $\mu\_i=|\cdot|^{s\_i}$, $i=1$, $2$.


My question is that if a table of generators $v\_0$ has been tabulated explicitly anywhere, in particular for the supersingular representations and in other cases in which $J\subseteq m\_K^2$.



Explicit Casselman theory: reference needed
For the parabolically induced representation, I suggest to look at Casselman "Restriction of $GL(2, F)$ to to $GL(2,o)$"-paper.


For the Steinberg representations and the super cuspidal representation, I suggest to look at Bushnell-Henniart "Local Langlands conjectures for GL(2)". For the Steinberg, your ideal will be the maximal ideal $p$. For the supercuspidal stuff, you should try to understand the definition of a stratum.


The translation from stratum for super-cuspidals to what you are asking about is a time-consuming exercise. It suggest to argue with strata directly.


Also this article by Ralf Schmidt seems relevant: <http://www.math.ou.edu/~rschmidt/papers/gl2.pdf>.


Silberger has also classified representation of GL(2) in "Representations of PGL(2) over the $p$-adics" (LNM).


*I am not sure if the titles of the references are all correct.*


