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](http://en.wikipedia.org/wiki/Bernstein%E2%80%93Zelevinsky_classification) 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.



What is the current status of representations of $GL_n(F)$ (and other algebraic groups)?
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](https://mathoverflow.net/questions/101067/are-all-irreducible-supercuspidal-representation-induced-from-compact-mod-center) 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.



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


