TITLE: What's the point of a Whittaker model?
QUESTION [15 upvotes]: Let $G$ be a quasi-split connected reductive group over a $p$-adic field $F$.  Let $B$ be a Borel subgroup which is defined over $F$, with $B = TU$, $T$ defined over $F$.  The choice of $T$ and $B$ gives a set of nonrestricted roots $\tilde{\Delta}$, which together with an $F$-splitting and a nontrivial unitary character $F \rightarrow S^1$ yields a unitary character $\chi$ of $U(F)$.
Let $V$ be a smooth, irreducible, admissible representation of $G(F)$.  A linear functional $\lambda: V \rightarrow \mathbb{C}$ is called a Whittaker functional for $\chi$ if for all $u \in U(F)$ and $v \in V$, we have
$$\lambda(u \cdot v) = \chi(u) \lambda(v)$$
It is a theorem that the space of Whittaker functionals for $\chi$ is at most one dimensional.  If this dimension is one, then $V$ is called $\chi$-generic.  
Fix a nonzero Whittaker functional $\lambda$, and for $v \in V$, define a function $W_v :G(F) \rightarrow \mathbb{C}$ by $W_v(g) = \lambda(g \cdot v)$.  The set $W = W_{\lambda}$ of such functions is closed under addition and scalar multiplication, and becomes a representation of $G(F)$ if we set $g \cdot W_v = W_{g\cdot v}$.  Then the representation $W$ is called a Whittaker model, and up to $G(F)$-isomorphism it does not depend on the choice of $\lambda$, since these things are all scalar multiples of each other.
From what I can see, $V \rightarrow W_{\lambda}, v \mapsto W_v$, is an isomorphism of $G(F)$-modules.  It is clearly surjective, and for injectivity, if we suppose that $0 \neq v \in V$, and $W_v(g) = 0$ for all $g \in G(F)$, then $\lambda(g \cdot v) = 0$ for all $g \in G(F)$.  This is impossible, because $g \cdot v : g \in G(F)$ spans $V$, because $V$ is irreducible.  
So my question is, what is the point of defining Whittaker models if the thing we defined is just isomorphic to our original representation? Why would it be important to regard $V$ as a set of functions $G(F) \rightarrow \mathbb{C}$?

REPLY [10 votes]: In addition to the other good answer, in the context of automorphic forms the uniqueness of local Whittaker models is specifically useful in computations of global integrals: when/if the integral can be expressed in terms of the global Fourier-Whittaker expansion of an automorphic form, then the local uniqueness expresses the thing as a product of local integrals, giving an Euler product, hence, most likely, an L-function of some kind. This is what happens in the adele-group rewrite of Mellin transforms of cuspforms, for example, as well as the adele-group rewrite of the classical Rankin-Selberg integral representation of $L(s,f\times g)$ involving Eisenstein series. These generalize to the Hecke-type integral for L-functions attached to cuspforms on $GL_n\times GL_{n-1}$, as well as the Rankin-Selberg-type integral for $GL_n\times GL_n$.