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What is the current status of representations of $GL_n(F)$ (and other algebraic groups)?

When $F$ is a local field, the representations of $GL_n(F)$ are classified by Bernstein and Zelevinsky in terms of cuspidal representations. What about other types of groups (and when $F$ is some other field)? Thank you very much.

Edit: I am interested in the representations which relates to Langlands program.

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    $\begingroup$ I think this question is way too broad, and thus does not really allow for any sort of answer. What other fields might you be interested in? And what other algebraic groups? And what sort of problems in their representation theory? $\endgroup$
    – Tobias Kildetoft
    Apr 11, 2014 at 8:16
  • $\begingroup$ Even in the Bernstein--Zelevinsky setting their classification isn't the end of the story: one still has to classify the cuspidals (which is much harder), cf. the book of Bushnell and Kutzko. $\endgroup$
    – David Loeffler
    Apr 11, 2014 at 8:29
  • $\begingroup$ @DavidLoeffler, thank you very much. What is the name of the book of Bushnell and Kutzko? $\endgroup$
    – Jianrong Li
    Apr 11, 2014 at 8:39
  • $\begingroup$ @TobiasKildetoft, thank you very much. I have edited the post. $\endgroup$
    – Jianrong Li
    Apr 11, 2014 at 8:42
  • $\begingroup$ When $F$ is local non-archimedean field,... $\endgroup$
    – Marc Palm
    Apr 11, 2014 at 8:57

1 Answer 1

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Let $G$ be reductive over the field $F$.

When $F$ is finite, there is Deligne-Lusztig theory.

When $F$ is archimedean, there is Langlands classification. (Knapp's book "Representation theory of semisimple groups tries to develop it).

When $F$ is non-archimedean and $G$ is $GL(n)$, $SL(n)$, then Bushnell and Kutzko have constructed the unitary dual explicitly.

For general $G$ and non-archimedean $F$, there is no construction of all supercuspidals known: see e.g. here for further references.

For general $F$ without trivial topology, you get discrete groups. To classify their unitary representation up to equivalence is hopeless. So you rather want to consider algebraic representations here. The smooth admissible/unitary representation theory of $G(F)$ for a local field $F$ is automatically algebraic in some sense.

Last but not least, a theory of zeta integrals would require Fourier-selfdual additive group of the field. This property characterizes local and finite fields among all fields.

I can't say much about the geometric Langland's program. Uses different fields and seems pretty algebraic to me, somebody else?

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    $\begingroup$ I think when $F$ is finite you mean Deligne--Lusztig theory (although Kazhdan--Lusztig theory also plays a role). For $\mathrm{GL}_n(q)$ there is also the formidable work of Sandy Green ("The characters of the finite general linear groups", 1955, Trans. Amer. Math. Soc.). $\endgroup$
    – Jay Taylor
    Apr 11, 2014 at 10:42
  • $\begingroup$ There is also the work of Zelevinsky which uses PSH algebras to get a much clearer picture than in Green's paper, although in the end it is the same classifying set. $\endgroup$
    – Adam Gal
    Apr 13, 2014 at 16:18

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