Let $G(F)$ be a reductive $p$-adic group. A result of Bernstein says that we can correspond each smooth irreducible representation to a “cuspidal pair” where it is embedded, and at most finitely many smooth irreducible representations correspond to the same pair. 


My question is: Can we distinguish between them in a natural way? Specifically, I was thinking that maybe the following is true: For any two irreducible representations $V,V’$ corresponding to the same cuspidal pair, there exists some compact open $K$ such that $V$ has $K$-fixed points while $V’$ doesn’t, or the converse. Or at least that there exists some compact open K such that the dimensions $dimV^K$ and $dimV’^K$ are different. 


Does someone know a proof of the above statements somewhere? Or any result like that?


Edit: I actually realized I may have a proof of that but does it already exist?



P-adic representations corresponding to the same cuspidal pair
I don't think anything quite so simple should work in general, but something close to this should (conjecturally).


Consider the cuspidal pair in GL(2) consisting of the trivial representation of the Borel subgroup. Parabolically induce this and you get a length two indecomposable representation. Its unique sub is the trivial rep, and the quotient is the Steinberg. It's easy to check that while these both have vectors fixed by the Iwahori subgroup, only the trivial rep has vectors fixed by GL(2,O). The same kind of thing holds for GL(n), with the various representations containing the cuspidal pair (Borel,trivial) being picked out by which subgroups intermediate between the Iwahori and GL(n,O) they have fixed vectors under.


More generally, in GL(n) a cuspidal pair determines a semisimple type in the sense of Bushnell--Kutzko. This is a representation of a compact open subgroup J of G which can be taken to be contained in GL(n,O). Induce it up to GL(n,O) and you'll get a decomposable representation in general. Given a component of this induced representation, the property of containing this component upon restriction to GL(n,O) will single out a representation containing your cuspidal pair.


The same most likely holds for general groups, but this relies on lots of conjectures about being able to perform analogous constructions to the above. In general groups, there is also some uncertainty about choices which must be made.


