Does anyone know the classification of irreducible admissible representations of GL(n) (over real,complex and p-adic fields), or some references?
Sorry if this question is not appropriate here.
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$\begingroup$ There are two fields involved here. A representation is a map GL(n)(K) --> GL(N)(L) for some fields K and L. Which pairs (K,L) are you interested in? Also, do you care about infinite dimensional representations? I think the question is appropriate, but I am trying to pair down the number of cases I, or someone else, will have to write about. $\endgroup$– David E SpeyerCommented Jan 4, 2010 at 16:05
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$\begingroup$ @David Speyer: I believe the K in GL(n,K) is a locally compact field of characteristic 0, and the vector space is infinite dimensional over the complex numbers (otherwise, why "admissible"?). This is the most standard setup, as far as I recall. (This is already much of what I remember from a whole course on p-adic representation theory I took in grad school.) Agreed that the question would be better if it were more detailed and focused... $\endgroup$– Pete L. ClarkCommented Jan 4, 2010 at 16:49
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2$\begingroup$ My advice to the OP is to pick up any book on automorphic representations that has survey articles in (for example the "Motives" volumes) and simply to read the articles on GL_n; there will typically be two, one on the case of R,C and one on the non-arch case. Here there will be precise statements of the local Langlands conjectures, all of which are theorems in the case of GL_n. $\endgroup$– Kevin BuzzardCommented Jan 4, 2010 at 18:50
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2 Answers
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There's a classification of irreducible admissible representations of real and complex reductive algebraic groups (in particular, GL(n)) due to Langlands, which is the basis of the local Langlands conjecture. A reference for this material is Knapp's "Representation theory of semisimple groups: an overview based on examples".
For GL(n) over a nonarchimedean local field, the papers "Induced reprsentations of reductive p-adic groups I,II" of Bernstein and Zelevinsky provide one step of the classification. It leaves the classification of the supercuspidal representations undertermined. Bushnell and Kutzko's "The admissible dual of GL(N) via compact open subgroups" determines the full set of irreducible admissible representations.
Update: I second Buzzard's comment above, especially with regards to the "Motives" proceedings. Kudla's and Knapp's articles in Motives II are quite nice, and contain several references including the ones I've mentioned.
answered Jan 4, 2010 at 17:41
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Let me focus on the case when $K$ is non-archimedean; the archimedean case is somewhat easier.
There is a coarse classification, valid for any reductive group, into supercuspidals, and all the others --- the others are ones that can be parabolically induced from supercuspidals of proper Levi's, and so in principle are understood by induction, while the supercuspidals are the basic building blocks. There is a subtley that certain parablic inductions of irreducibles are not themselves irreducible, which leads to so-called special representations (EDIT: and in the general, ie. non-$GL_n$ case, so-called packets), but at least in the $GL_n$ case these are well-understood too. (EDIT: In particular the packets are actually just singletons).
So everything comes down to the supercuspidals. (This is explained in the introduction to Harris and Taylor's book, among many other places.) This coarse classification is also compatible in a natural way with the local Langlands correspondence.
For $GL_n(K)$ the supercuspidals are completely classified. (This is the difference between $GL_n$ and most other groups.) In fact there are two forms of the classification.
(1) Via the local Langlands correspondence (a theorem of Harris and Taylor).
(2) Via the theory of types (a theorem of Bushnell and Kutzko).
The first classification relates them to local $n$-dimensional Galois reps. The second relates more directly to the internal group theoretic structure of the representations.
As far as I know, the two classifications are not reconciled in general (say for large $n$, where large might be $n > 3,$ or something of that magnitude), and this is an ongoing topic of investigation by experts in the area. (Any updates/corrections to this statement would be welcome!)
The difference in the archimedean case is that there are no supercuspidals, so everything comes down to inducing characters of tori, and understanding the reducibility of these parabolic inductions.
