*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.