Explicit Casselman theory: reference needed

2013-01-28T10:40:54Z

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 $J\subseteq m_K^2$.

Answer by Marc Palm for Explicit Casselman theory: reference needed

2013-01-28T11:26:29Z

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.

Explicit Casselman theory: reference needed - MathOverflow

Explicit Casselman theory: reference needed

Answer by Marc Palm for Explicit Casselman theory: reference needed