Let $F$ be a local field. Is there a reference for the following fact:
No supercuspidal representation of $GL_2(F)$ has an Iwahori-fixed vector?
I have a proof, by I'd prefer a reference, because it is not enlightening.
Rough sketch of proof: We can easily see that Iwahori-fixed vector implies depth-zero and that depth-zero supercuspidal are induced from $GL_2(o)$ times the center, hence correspond modulo central characters to supercuspidal of $GL_2(o/p)$. The proof of the second conclusion is somewhat messy in my exposition.