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?