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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 ```$$
v_0(g)=|a|^{s_1}|d|^{s_2}|a/d|^{1/2}v_0(1)\quad
\text{where}\quad
g=\left(\begin{array}{cc}
a & *\\
 & d 
\end{array}\right)r,\quad r\in{\rm GL}_2(R_K).
GL}_2(R_K)
$$``` 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 $J\subseteq m_K^2$.

6 deleted 82 characters in body

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 ```$$
v_0(g)=|a|^{s_1}|d|^{s_2}|a/d|^{1/2}v_0(1)\quad
\text{where}\quad
g=\left(\begin{array}{cc}
a & *\\
 & d 
\end{array}\right)r,\quad r\in{\rm GL}_2(R_K).
$$``` 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 $J\subseteq m_K^2$.

P.S.: It seems that I can't get the matrices look right ....

5 the matricesweren't displayed rightly, so I placed backticks around the equations.

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 ```$$
\gamma\cdot v=\epsilon(a)v\qquad
\forall\gamma=\left(\begin{array}{cc}
a & bb\\
c & d 
\end{array}\right)\in{\rm GL}_2(R_K)
\ \text{with}\ c\in J
$$``` 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 ```$$
v_0(g)=|a|^{s_1}|d|^{s_2}|a/d|^{1/2}v_0(1)\quad
\text{where}\quad
g=\left(\begin{array}{cc}
a & *\
 \\
 & d 
\end{array}\right)r,\quad r\in{\rm GL}_2(R_K).
$$``` 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 $J\subseteq m_K^2$.

P.S.: It seems that I can't get the matrices look right ....

4 tag
3 edited body

Let $K$ be a nonarchimedean local field with ring of integeres $R_K$, maximal ideal $m_K$ and finite redidue 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 $$v_0(g)=|a|^{s_1}|d|^{s_2}|a/d|^{1/2}v_0(1)\quad \text{where}\quad g=\left(\begin{array}{cc} a & *\ & d \end{array}\right)r,\quad r\in{\rm GL}_2(R_K).$$ 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 $J\subseteq m_K^2$.

P.S.: It seems that I can't get the matrices look right ....

2 added 8 characters in body

Let $K$ be a nonarchimedean local field with ring of integeres $R_K$, maximal ideal $m_K$ and finite redidue 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 $$v_0(g)=|a|^{s_1}|d|^{s_2}|a/d|^{1/2}v_0(1)\quad \text{where}\quad g=\left(\begin{array}{cc} a & *\ & d \end{array}\right)r,\quad r\in{\rm GL}_2(R_K).$$ 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 $J\subseteq m_K^2$.

P.S.: It seems that I can't get the matrices look right ....

1

# Explicit Casselman theory: reference needed

Let $K$ be a nonarchimedean local field with ring of integeres $R_K$, maximal ideal $m_K$ and finite redidue field $\bf k$. Let $\pi$ be an admissible irreducible 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 $$v_0(g)=|a|^{s_1}|d|^{s_2}|a/d|^{1/2}v_0(1)\quad \text{where}\quad g=\left(\begin{array}{cc} a & *\ & d \end{array}\right)r,\quad r\in{\rm GL}_2(R_K).$$ 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 $J\subseteq m_K^2$.

P.S.: It seems that I can't get the matrices look right ....
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