The space of Whittaker functionals is at most one-dimensional

Question

Let $\mathbf G$ be a connected, reductive group over a local field $F$, and let $(\pi,V)$ be a smooth, irreducible, admissible representation of $G = \mathbf G(F)$. Assume there exists a Borel subgroup $\mathbf B$ defined over $F$. Let $\mathbf T$ be a maximal torus of $\mathbf B$ which is defined over $F$, and let $\mathbf U = \mathscr R_u(\mathbf B)$, $U = \mathbf U(F)$. The choice of a nontrivial unitary character of $F$ and an "$F$-splitting" gives us a nontrivial unitary character $\chi: U \rightarrow S^1$. A linear functional $\lambda: V \rightarrow \mathbb{C}$ is called a $\chi$-Whittaker functional if $\lambda(\pi(u)v) = \chi(u) \lambda(v)$ for all $u \in U, v \in V$.

It is a well known result in the representation theory of reductive groups that the space of Whittaker functionals is at most one dimensional. I have seen a reference for a proof in the case $\mathbf G = \textrm{GL}_n$. Is there a reference for the general case?

Answer 1

This is proved in Shalika's multiplicity one paper:

Shalika, J. A. The multiplicity one theorem for $GL_n$. Ann. of Math. (2) 100 (1974), 171–193.

While Shalika starts off assuming $G=GL_n$ (he wants to prove the existence of Whittaker models as well), he notes in the introduction that the proof that the dimension is at most 1 works for quasi-split groups, and verifies what he needs in the appendix for quasi-split groups.

I remember parts of Shalika's paper being hard to read (I think the later parts, not where he proves dimension at most 1, but I'm no longer sure). Anyway, if you are also interested in another reference for a somewhat weaker result, Rodier extends the Gelfand-Kazhdan method for $GL_n$ to split groups over nonarchimedean fields here:

Rodier, François Whittaker models for admissible representations of reductive p-adic split groups. Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972), pp. 425–430. Amer. Math. Soc., Providence, R.I., 1973.