# Supercuspidal with Iwahori fixed vector

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.

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