I am looking for a list of the irreducible representations of O(2). Could someone please provide a reference?
EDIT: I am particularly interested in the representations on IR^2 (irreducible or not)
I am looking for a list of the irreducible representations of O(2). Could someone please provide a reference? EDIT: I am particularly interested in the representations on IR^2 (irreducible or not) 


You can look into this section of the book Group theory in physics by WuKi Tung to begin with. EDIT: Theorem 11.3 from this book works for complex irreps only (see Jose's comments to the answer and to the question). 


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.unigoettingen.de/bitstream/handle/11858/0017350000000DF0747/palm.pdf?sequence=1 on pg 101. 

