# Regular representation of a compact group (like SU(2) for example)

In the case of finite groups, according to Peter–Weyl theorem, the regular representation of the group $$G$$ on $$L^2(G)$$ decomposes into a direct sum of all irreducible unitary representations. For each irrep, the representation is multiplied by the identity representation that has the same dimension as as the irrep, so it should look something like this : $$R = \oplus_{q=0}^{n} \mathcal{\pi}_{\text{irrep}}^{q} \otimes \mathcal{I}_{dim(irrep)}$$

Is there an equivalent for the case of compact groups ? (I'm specifically searching for SU(2) and SO(3)). My first idea was to write the same expression (for $$n=\infty$$) but I'm sure that the Peter–Weyl theorem is only valid for finite groups.

Context : I'm reading Quantum Theory, Groups and Representations: An Introduction of Peter Woit.

• Commented Feb 23, 2023 at 14:34
• The finite group version, which you are quoting and called "peter weyl," is older than PW (which IS for compact groups!), and is due to Frobenius - I think? Maybe? Perhaps someone knowledgeable might correct me. Commented Feb 23, 2023 at 14:52
• Or are we misunderstanding your question? Commented Feb 23, 2023 at 14:53
• You might be right, I quoted the finite version of Peter Weyl, I will go back to my notes and see the two presentations that were posted in the only answer so far. I will update my question if necessary. Commented Feb 23, 2023 at 15:01

As peter a g essentially said in the comments, the Peter-Weyl Theorem is not limited to finite groups, but covers compact groups. The best and most efficient introduction that I know is the set of lecture notes by Joel Feldman on the Haar measure. It is especially suited for folks in mathematical physics, and he proves everything except, if I remember correctly, one tricky point in the proof of Peter-Weyl, which is that matrix elements of finite dimensional representations separate points. This is needed in order to bring down the Stone-Weierstrass hammer which gives density in the entire $$L^2$$ space of the group. This is only tricky if one wants extremely general compacts groups. For groups like $$SU(N)$$, this is trivial since the fundamental representation already can distinguish group elements.