Let $G$ be a $p$-adic linear reductive group. For an irreducible admissible smooth representation $\pi$ of $G$, there is a distribution $\Theta(\pi)$, called the character distribution, attached to $\pi$. It is given by integration against a locally integrable function which is locally constant on the regular set of $G$.

This distribution in fact determines $\pi$ up to isomorphism.

Since supercuspidal representations are rather special representations of $G$ (e.g. matrix coefficients are compactly supported mod the center), I'm wondering how to determine if a given irreducible representation $\pi$ is supercuspidal in terms of the distribution $\Theta(\pi)$? In others words, which properties of $\Theta$ make supercuspidals different from other smooth representations?

Thanks for any answer.

nowhereon the regular semisimple set, at least for $\mathrm{GL}_2(k)$.) $\endgroup$ – LSpice Apr 8 '18 at 5:59