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3 added 134 characters in body

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 other 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.

2 edited body

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 other 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, but at least in the $GL_n$ case there these are well-understood too.

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.

1

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 other 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, but at least in the $GL_n$ case there are well-understood too.

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.