What is the *real* representation theory of $SO(3)$?

Question

It seems like all the resources I find on representation theory are done for representations over the complex numbers. For those, I know how to classify all representations of $SO(3)$: they are direct sums of irreducible ones, and the irreducible representations are isomorphic to the action of $SO(3)$ on the spherical harmonics. But applying these results to real representations seems to go wrong, since $\mathbb{C}$ being algebraically closed pops up in a bunch of places.

I think the action on the real spherical harmonics continues to be irreducible, but does that classify all of the irreducible real representations? And does the classic result that representations of compact groups are direct sums of irreducible ones hold for real representations? Any resources on representation theory that discusses real representations as well as complex ones would be helpful too!

Answer 1

Partial answer to the second part of your question. Yes, that is enough. The spherical harmonics form a basis for the $2 \ell + 1$ dimensional representations of SO(3). Here is a nice reference. The author calculates the action of rotation about the z axis applied to the $Y_{l,m}$ and from this the character table for $SO(3)$.

Answer 2

The result that a representation of a compact group is a direct sum of irreducibles holds over the reals too. If you know the proof for complex representations, the same proof works for real ones (if you have a representation on a vector space $V$ and $U$ is an invariant subspace, pick $\pi':V\rightarrow U$ a projection of vector spaces, and then use integration over the group to produce $\pi:V\rightarrow U$ a projection which is also a map of representation spaces).

Answer 3

Since I had a similar question which I think I found the answer to I might as well post it again here.

The universal (double) cover $SU(2, \mathbb{C})$ has irreducible real representations in all dimensions which are odd or divisible by $4$, see e.g. here. As in the complex case, the irreducible real representations in odd dimensions descend to $SO(3, \mathbb{R})$ and are isomorphic to the real vector space of spherical harmonics of degree $\ell$ on the $2$-sphere.

Some more detail regarding $SU(2, \mathbb{C})$, $SO(3, \mathbb{R})$ and real representations is given in chapter II sections 5 and 6 of Representations of Compact Lie Groups by Bröcker and Dieck.