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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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1 Answer 1

up vote 3 down vote accepted

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

"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.

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