Irreducible decomposition for local representations of quantum Teichmüller space

We give an irreducible decomposition of the so-called local representations \cite{math/0407086} of the quantum Teichmüller space $\mathcal{T}_q(\Sigma)$ where $\Sigma$ is a punctured surface of genus $g>0$ and $q$ is a $N$-th root of unity with $N$ odd.