My knowledge about representation theory of locally compact groups is rather scattered. As I got more interested with this subject, I would like to know some good references, where I could learn the basic facts of this theory. So I'm looking for the book (or books, if there is no one souce which can cover the material which I want to learn) which will contain:
1. The result that each irreducible representation of compact group is finite dimensional.
2. The theorem which states that for any locally compact group irreducible representations separate the points.
3. Discussion about how the topology on the set of $Irr G$ (classes of) irreducible representation is defined.
4. The answer to the question: if $G$ is compact then $Irr G$ is discrete and if $G$ is discrete then $Irr G$ is compact.
5. Definition of the Plancherel measure.
6. The theorem which states that for any compact group $G$ we have:
a) every representation of $G$ splits into direct sum of irreducible ones
b) every irreducible representation is contained in left regular representation.
I will be very grateful for any help.
[Math] Representation theory of locally compact groups
representation-theorytopological-groups
Best Answer
A lot of this would fall under the subject of harmonic analysis; I learned a large part of what I know of it from Anton Deitmar's Principles of Harmonic Analysis. This covers the compact case completely (here, everything is essentially contained in the Peter--Weyl theorem), and roughly sketches the general Plancherel theorem. This is a pretty difficult topic, and the main reference for most of the proofs seems to still be Jacques Dixmier's $C^\ast$-algebras and Von Neumann Algebras. Unfortunately, I can speak from experience when I say that these are pretty difficult books to read, and that his treatment of the Plancherel theorem relies on huge amounts of material from both of them.
As far as I'm aware, in the non-compact case you can't really put a topology on $Irr\ G$ in a sensible way. This is where $C^*$-algebras come in; you instead look at the dual of the group's associated $C^*$-algebra which gives you something to work with. Then, in the compact case I've never personally seen anyone put a topology on this set -- that's not to say that it isn't done at all. Certainly though, in my experience I guess that we essentially put the discrete topology on it, which speculatively answers question 4. It's never occurred to me to try and work out how this fits in with the $C^*$-algebra approach until now, and it's now long enough since I read about this stuff that I doubt I could say anything sensible.
Let me quickly give a few brief answers to some of your simpler questions. There's a discussion on means of showing that irreducible reps of compact groups are finite dimensional here, including some ways of doing it without the Peter--Weyl theorem. The theorem that irreducible representations separate points is the Gelfand--Raikov theorem. I don't have my copy to hand, but I suspect that a proof is in Deitmar's book. If not, it's a very well-known result, so a reference shouldn't be too hard to turn up. Semisimplicity of representations should also be established during the proof of the Peter--Weyl theorem. That every irreducible representation is contained in the regular representation is Frobenius reciprocty (the regular representation is isomorphic the the representation obtained by inducing the trivial representation of the trivial subgroup to the entire subgroup).
That just leaves that definition of the Plancherel measure. As I mentioned above, this is in general extremely non-trivial. Here's an answer in the form of the Plancherel theorem, which at least gives a non-constructive definition: