I am confused by this. In the original question the field is local so I didn't say that explicitly before. I don't see how you can't recover the statement I gave above by taking $G=G(F)$, $F=1$, and $\underline{\underline{G}}=G$ in the notation of the reference.

@KConrad: I am writing a book on automorphic representations with H. Hahn and I want to state this with a reference but not prove it.