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.



Supercuspidal with Iwahori fixed vector
I quote from one of my papers (On Bernstein's presentation of Iwahori-Hecke algebras and representations of split reductive groups over non-Archimedean local fields, Bulletin of the Kerala Mathematics Association, Special issue on Harmonic Analysis and Quantum Groups, December 2005, also available from <http://arxiv.org/abs/math.GR/0506094>)


"Casselman, using some techniques of Jacquet, showed that under the correspondence described by Borel, the irreducible admissible representations of G(F) coming from irreducible H-modules are precisely those that occur as subquotients (or equivalently, subrepresentations) of the unramified principal series of G(F ) [Cas80, Proposition 2.6]. It is implicit in Proposition 2.5 of this article that the Jacquet module of such a representation with respect to a minimal parabolic subgroup corresponds to restriction to a certain commutative subalgebra of H (later identified by Bernstein)."


The reference is to:


[Cas80] W. Casselman. The unramified principal series of p-adic groups. I. The spherical function. Compositio Math., 40(3):387–406, 1980.


