Why Whittaker functions are useful?

reference help on a result of Whittaker functions of supercuspidal representations

• Let $F$ be a p-adic field, $\pi$ an irreducible supercuspidal representation of $GL(n,F)$, then it admits a unique Whittaker model $\mathcal{W}(\pi)$. For any $W\in \mathcal{W}(\pi)$, a basic result is that $W(g)$ is a compactly supported function mod $NZ$, where $N$ is the maximal unipotent subgroup and $Z$ is the center of $GL(n,F)$. Does anyone know a reference containing this result? Many thanks.

• Answer:

  I think this can be seen directly. Let's work over $G=PGL(n)$, so I don't have to keep repeating "modulo the center". Recall that supercuspidal representations can be realized as subrepresentations in $L^2(G)$ consisting of compactly supported functions. So we can define an intertwining integral from $\pi$ into the Whittaker space by $$\Big(g\rightarrow f(g)\Big)\longrightarrow \Big(g\rightarrow\int_N \bar\psi(n)f(ng)\ dn\Big)$$ ("compactly supported" gives convergence). Let $W$ denote the Whittaker function of an $f\in\pi$. If the support of $f$ is $C\subset G$, then the support of $W$ is $NC$. To see this, take $g\notin NC$. So, for each $n\in N$, $ng\notin C$. So $f(ng)=0$ for all $n$. Hence $W(g)=0$. Since $C$ is compact, $NC$ is compactly supported modulo $N$. [Added: I realized I was tacitly assuming that for some $f$, the above integral is not identically zero. Since $f$ has compact support, the integral being identically zero implies that $f$ lies in the kernel of the twisted Jacquet functor. Since the representation has a Whittaker module, the twisted Jacquet module must be non-zero, hence there is an $f$ whose integral does not vanish.] I don't know a reference off-hand for this basic fact (it also follows from the fact that the functions in the Kirillov model of a supercuspidal representation have compact support, which is in a lot of sources), though Bushnell and Henniart's http://www.sciencedirect.com/science?_ob=MiamiImageURL&_cid=272332&_user=10&_pii=S0021869398975421&_check=y&_origin=&_coverDate=01-Nov-1998&view=c&wchp=dGLzVlB-zSkWz&md5=2d152b143810609e7409f1e443889b93/1-s2.0-S0021869398975421-main.pdf gives a more, well, explicit version of it. [Added: I just realized that this is the final result of Casselman-Shalika's http://archive.numdam.org/ARCHIVE/CM/CM_1980__41_2/CM_1980__41_2_207_0/CM_1980__41_2_207_0.pdf.]