Explicit Casselman theory: reference needed - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T23:40:32Z http://mathoverflow.net/feeds/question/120096 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120096/explicit-casselman-theory-reference-needed Explicit Casselman theory: reference needed Andrea Mori 2013-01-28T10:40:54Z 2013-01-29T00:36:01Z <p>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 <code>$$\gamma\cdot v=\epsilon(a)v\qquad \forall\gamma=\left(\begin{array}{cc} a &amp; b\\ c &amp; d \end{array}\right)\in{\rm GL}_2(R_K) \ \text{with}\ c\in J$$</code> is non-trivial and in fact $1$-dimensional. As every expert knows, this result is of paramount importance for the theory of modular forms.</p> <p>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 <code>$$v_0(g)=|a|^{s_1}|d|^{s_2}|a/d|^{1/2}v_0(1)\quad \text{where}\quad g=\left(\begin{array}{cc} a &amp; *\\ &amp; d \end{array}\right)r,\quad r\in{\rm GL}_2(R_K)$$</code> and $\mu_i=|\cdot|^{s_i}$, $i=1$, $2$.</p> <p>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> http://mathoverflow.net/questions/120096/explicit-casselman-theory-reference-needed/120102#120102 Answer by Marc Palm for Explicit Casselman theory: reference needed Marc Palm 2013-01-28T11:26:29Z 2013-01-28T11:34:41Z <p>For the parabolically induced representation, I suggest to look at Casselman "Restriction of $GL(2, F)$ to to $GL(2,o)$"-paper.</p> <p>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.</p> <p>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.</p> <p>Also this article by Ralf Schmidt seems relevant: <a href="http://www.math.ou.edu/~rschmidt/papers/gl2.pdf" rel="nofollow">http://www.math.ou.edu/~rschmidt/papers/gl2.pdf</a>.</p> <p>Silberger has also classified representation of GL(2) in "Representations of PGL(2) over the $p$-adics" (LNM).</p> <p><em>I am not sure if the titles of the references are all correct.</em></p>