rt.representation theory - irreducible representations of O(2) - reference?

I just see now, that the issue is appearently real representations. I consider complex representations. I not experienced with real representations and whether my strategy works there as well.

You can induce from $SO(2)$. Define on $SO(2)$ the rep $epsilon_n: theta mapsto e^{i theta n}$. Let $rho_n$ be the induced one, then $rho_n$ is irreducible if $n neq 0$. You have $rho_n cong rho_{-n}$ and $rho_{0} = 1 oplus det$. These are up to isomorphism all irreducible representations.

Reference: Traces of Hecke operator by Knightly and Li.

A proof also is in my thesis: http://ediss.uni-goettingen.de/bitstream/handle/11858/00-1735-0000-000D-F074-7/palm.pdf?sequence=1 on pg 101.