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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 $K$?

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Bushnell&Henniart "Local Langlands for GL(2)$ us a good place to start. I am not sure about the terminology you use, but I guess you ask something like mathoverflow.net/questions/101067/…? –  plusepsilon.de Oct 15 '12 at 9:59

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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.

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Thanks, I'm going to be reading that minutiously. What about taking smaller $U_0$ inside $U$, does this give higher level representations, and are those expected to be all of them? –  Will Oct 16 '12 at 10:20

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