# What is the support of the Whittaker function of a new vector on GL(2)?

Let $W$ be the normalized Whittaker function associated to a new vector in an irreducible generic representation $\pi$ of $G=GL_2(k)$, where $k$ is a $p$-adic field. Let $c$ be the conductor of $\pi$, meaning that $c$ is the smallest integer with $W(gk)=W(g)$ for all $k\in K_1(c)$ where $K_1(c)$ is the subgroup of $K=GL_2({\mathfrak o})$ with bottom row congruent to $(0,1)$ modulo $\mathfrak p^c$ (with the convention that $K_1(0)=K$).

What is the support of $W$ restricted to $K$?

Clearly, the support contains $K_0(c)$, the subgroup of upper triangular matrices modulo $\mathfrak p^c$, since $W$ won't vanish on the center of $G$. I imagine it will depend on further information about $\pi$ beyond the conductor, but I'm having trouble finding (or proving) much of anything definitive.

-
Why is W non-zero on the centre? All you know that W is non-zero on some coset $U g K_1(c)Z$. Paskunas-Stevens compute some Whittaker functions for supercuspidals, see arxiv.org/abs/math/0603051, but the fucntions there will not be new vectors in general, i think. – labirintas Dec 10 '12 at 9:49
Hi labirintas, thanks for the paper. A general (possibly nontrivial) fact, at least for GL(n), is that $W(1)\ne 0$ (this may require "newform-ness"). Since $W(z)=\omega(z)W(1)$, where $\omega$ is the central character of $\pi$, $W$ is nonzero on the center. More specifically, the "new vector" criterium can also be written as $W(gk)=\omega(d)W(g)$ for $k\in K_0(c)$, $k=\bigg(\matrix{a&b\cr c&d}\bigg)$. – B R Dec 10 '12 at 14:37
Dear BR, As you surmise, the answer depends on $\pi$. Firsly, I think it is more convenient to work in the Kirillov model than in the Whittaker model, but the two are equivalent: working in the Kirillov model means restricting to the torus $(k^{\times} \, 0 ; 0 \, 1)$. Now if $\pi$ is prinicpal series or special, the new vector has support over all of $\mathcal O_k \setminus \{0\}$, while if it is cuspidal than the support is just the char. function of $\mathcal O_K^{\times}$. In the unramified principal series case, this will be discussed in the paper(s?) of Casselman and Shalika ... – Emerton Dec 11 '12 at 12:08
... They give an explicit formula for the Whittaker function in the unramified principal series case (and for any group $G$, or at least for $GL_n$) called the Casselman--Shalika formula. I'm not sure of the a reference in the other cases, but the (possibly ramified) principal series case and the special case you can compute yourself just from the basic description in terms of parabolic induction: choose a non-zero additive character $\psi$ and just compute $\pi_{N = \psi}$ directly --- you will see that the map to the Kirillov (or Whittaker) model is given by a kind of Fourier transform. – Emerton Dec 11 '12 at 12:10
It might be that the general case (i.e. the supercuspidal case as well) is handled in another Casselman paper ("On a theorem of Atkin and Lehner", or something like that) where he discusses the theory of newforms for $GL_2$ from a representation-theoretic viewpoint. My memory is that I worked it out from that paper, combined with my own fumblings. E.g. you can describe the $U_p$-operator --- using classical Atkin-Lehner notation --- in representation theoretic terms, then consider how it acts in terms of the Kirillov model (easy, because the action of the Borel on the Kirillov model is ... – Emerton Dec 11 '12 at 12:12

## 1 Answer

You should find all ingredients needed for your calculation in the following very useful notes by Ralf Schmidt and Mahdi Asgari :

Some remarks on local newforms for GL(2). J. Ramanujan Math. Soc. 17 (2002), 115-147

It is available on line (Ralf Schmidt's webpage).

-
Hi Paul, thanks for the answer. I do agree that those notes are very useful. In fact, I was looking at it when I thought of the question! :) Perhaps I'm just being dense, but I was unable to see how to use it to get the result I'm interested in. – B R Dec 11 '12 at 19:35