Representations of reductive groups over local fields through parahoric induction 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 $K$?
 A: This procedure allows to construct all "level $0$" irreducible representations of G. They appear as subquotients of your compactly induced representations.  Here "level $0$ means that the representation has a non-zero fixed vector under the pro-unipotent subgroup of some parahoric. The answers to your questions are is the following paper:
Morris, Lawrence Level zero $\bf G$-types. Compositio Math. 118 (1999), no. 2, 135–157
If you use smaller groups $U_o \subset U$, then you can indeed get any irreducible representation as subquotient of a compactly induced representation. However when the compactly induced representation is irreducible it is automatically supercuspidal (see e.g. Bushnell-Henniart for a proof of that). All explicitely known supercuspidal representations
are indeed obtained by compact induction. But it is still conjectural that they all are. 
In general a compactly induced representation from an irreducible representation of a compact open subgroup splits in two part. An admissible part which is a finite sum of supercuspidal representations and a non admissible part which contains non-supercuspidal as irreducible subquotients. 
To describe the non supercuspidal representations by compact open data, a good point of view is that  of "types". You may read Bushnell and Kutzko's papers on that subject. 
