How to show that a representation is supercuspidal?

Question

Let $G$ be a reductive group. If we know a representation $\pi$ of $G$ explicitly, how could we determine that $\pi$ is supercuspidal or not? Are there some references about this? Thank you very much.

Answer 1

I assume your $G$ is over a local field. Then what does it mean to "know explicitly" an infinite-dimensional representation of an uncountably infinite group?

If it means you can compute $\pi^K$ for any open compact-mod-centre $K$, and the action of any element of the Hecke algebra $\mathcal{H}(G, K)$ on this space, then you are in business for $G = GL_n$ and a few other examples, because in these cases the type theory of of Bushnell et al allows you to classify $\pi$'s completely in terms of the representations of compact-opens inside them. (This is more or less the approach taken in my paper with Weinstein, where we give an algorithm to compute the representations of $GL_2(\mathbf{Q}_p)$ attached to modular forms: http://arxiv.org/abs/1008.2796)