Let me take $G$ to be a simple (connected) split reductive group over a local field $K$. One way I might go about constructing a (smooth, admissible) complex representation $\sigma$ of $G$ is as follows:

- pick a parahoric subgroup $P$ of $G$, with pro-unipotent radical $U$,
- form the quotient $P/U$, which is a (connected) reductive group over a finite field,
- write down a representation $\overline{\rho} \colon P/U \to \mathrm{GL}_n(\mathbb{C})$,
- inflate this to a representation $\rho \colon P \to \mathrm{GL}_n(\mathbb{C})$,
- induce $\rho$, giving $\sigma = \mathrm{ind}_P^G(\rho)$ (compact induction).

How does this procedure work out? Excepting that I might have to take possibly smaller $U_0 \subset U$, is this expected to produce all the (smooth admissible irreducible complex) representations of $G$? When do I get something irreducible, or a supercuspidal? What is the proper formulation when $G$ is not assumed to be split over $K$?