# Representation theory of locally compact groups

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.

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:

Let $$G$$ be locally compact, second countable, unimodular and type 1 (a condition on the $$C^*$$-algebra of $$G$$), equipped with a choice $$\mu$$ of Haar measure. Then there exists a unique measure $$\hat{\mu}$$ on the $$Irr\ G$$, such that one has a direct integral decomposition $$L^2(G)\simeq\int_{V\in Irr\ G}^{\oplus}HS(V)\ d\hat{\mu}(V),$$ where $$HS(V)$$ denotes the Hilbert--Schmidt space associated to V.

• As far as I remeber the topology on unitary dual is called Fell topology. I found somewhere on mathoverflow its definition but still I would like to read further about this. Moreover, apparently there must be some topology on the unitary dual since I found for example the following equivalent definition of so called property T "trivial representation is an isolated point in the unitary dual" Jul 12, 2015 at 20:19
• Yes, the Fell topology is defined on the dual of the group's $C^*$-algebra. There's then a natural bijection between this algebra and the unitary dual of $G$, so I suppose you can define a topology on the unitary dual in this way. As far as I know, this is the only way in which the dual is usually topologized. However, to actually be able to say much at all about this (apart from in very simple examples, such as when $G$ is abelian) I think that you'd really need to learn about what's going on on the $C^*$-algebra side of things.
– PL.
Jul 12, 2015 at 20:25
• Ok, so it appears that this is just the same definition as for $C^*$-algebras. I have some familiarity with the notion of group $C^*$-algebra. If you have a discrete group $\Gamma$ then you can form two $C^*$-algebras $C^*_u(\Gamma)$ and $C^*_r(\Gamma)$ called universal and reduced (resp.) C*-algebras of group $\Gamma$. Both of them are unital therefore their spectra are compact (not necessarily Hausdorff) by a general theorem which can be found in Dixmier's book. The construction of $C^*$-algebra of group can be performed also for nondiscrete groups so the question whether the ... Jul 12, 2015 at 20:40
• ... obtained spectrum for a compact group $G$ will be discrete is still meaningful and (at least for me) interesting. Note that in this more general setting $C^*$-algebras of the underlying group no longer have to be unital. Jul 12, 2015 at 20:42
• By the way, his name (second line) is Anton Deitmar, not "Dietmar". Mar 8, 2016 at 13:26

Gerald B. Folland, a course in abstract harmonic analysis, is the best source and contains the answers to your questins.