Question

For an irreducible smooth (generic) representation $\pi$ of $G=GL_2(k)$ with central character $\omega$, where $k$ is a $p$-adic field, we define the conductor of a vector $v\in\pi$ as follows. Let $K_0({\mathfrak p}^N)$ be the subgroup of $K=GL_2({\mathfrak o})$ with lower left-hand entry congruent to $0$ modulo ${\mathfrak p}^N$. The conductor of $v$, $c(v)$, is the smallest $N$ such that $$\pi\bigg(\matrix{a&b\cr c&d}\bigg)v=\omega(a)v\ \ {\rm for\ all}\ \bigg(\matrix{a&b\cr c&d}\bigg)\in K_0({\mathfrak p}^N)$$ In other words, $v$ is fixed by $K_0({\mathfrak p}^N)$ up to the action of the center. The conductor of $\pi$, $c(\pi)$, is the smallest $N$ such that there is a $v$ with $c(v)=N$. The subspace of vectors with conductor $c(\pi)$ is one-dimensional and the unique vector $v_0$ that maps to a Whittaker function with $W_0(1)=1$ is called the new vector.

My question is: how can we write down a somewhat-explicit basis of $\pi$ in terms of the new vector $v_0$, indexed by the principal congruence subgroups $K({\mathfrak p}^M)=1_2+{\mathfrak p}^MM_2({\mathfrak o})$? In other words, how can we write a basis of $\pi^{K({\mathfrak p}^M)}$ in terms of $v_0$?

A simple nonexplicit version is this result, from Casselman's "The restriction of a representation of $GL_2(k)$ to $GL_2({\mathfrak o})$": the restriction of $\pi$ to $K$ decomposes as $$res_K^G\pi=\pi^{K({\mathfrak p}^{c(\pi)-1})}\oplus\sum_{n\ge c(\pi)}u_n(\omega)$$ where $u_n$ is the unique irreducible representation of $K$ which is trivial on $K({\mathfrak p}^n)$ but not on $K({\mathfrak p}^{n-1})$ and that contains a vector $v$ with conductor $n$ in the above sense.

Answer 1

I think that your asking how to map from one isotype to another, right?

Now, one way to map from one isotype to an other is convolution with a function $$\phi = \langle v_1, f v_2\rangle, \qquad f(k_1gk_2) =u_{n_1}(k_1) f(g) u_{n_2}(k_2),$$ where $g$ runs through a set of representatives of $G//K$, and $f$ is $Hom_{\mathbb{C}}(u_{n_2},u_{n_1})$-valued. $v_1$ and $v_2$ are vectors. This follows from Schur's orthogonality relations.

There is possibe a nicer way to express this, but this is essentially the only way! For what purpose do you need this?

Blomer and Harcos describe a basis in Section 2.6 http://arxiv.org/pdf/0904.2429.pdf.