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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 $J\subseteq m_K^2$.

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I deleted my answer, because you seem not to be interested in complex representations, is that right? Please specify this is in your question, although it is implicit in the term "supersingular representations". –  plusepsilon.de Jan 28 '13 at 11:57
@Marc_Palm : I'm certainly primarily interested in complex representations. I left that somewhat implicit, also the remark about modular forms should be revealing! Please repost your answer. –  Andrea Mori Jan 28 '13 at 13:48
Okay, I received a down-vote. That's why I got insecure about the coefficient field;) –  plusepsilon.de Jan 28 '13 at 13:57
Btw, your claim about $v_0(1)$ is wrong if $\mu_1|_{R^\times} \neq 1 \neq \mu_2|_{R^\times}$. –  plusepsilon.de Jan 28 '13 at 14:02
Dear Andrea Mori, I have tried to get displayed the matrices, without modify their content. –  Giuseppe Tortorella Jan 28 '13 at 18:21

1 Answer 1

up vote 3 down vote accepted

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.

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