A supercuspidal representation is an admissible representation with trivial Jacquet module for any parabolic.This implies that the matrix coeff. are compactly supported mod center.

Cartier (in Corivalis) defines super cuspidal rep as admissible rep. with compactly supported matrix coeff. mod center.

I am wondering if this (Cartier's) definition is equivalent to triviality of Jacquet module? If yes, is there a reference for such a proof?

Thanks in advance.

smoothrepresentation with trivial Jacquet module for any parabolic. Eventually, you prove that every irreducible smooth representation is admissible, by using parabolic induction and proving the result for irreducible supercuspidals. – Marty Oct 13 '11 at 16:41