I'm trying to understand Kitaev's quantum double by following Shawn X. Cui's notes (Topological Quantum Computation). Let $L(s_0,s_1)$ denote the subspace of excited states at sites $s_0,s_1$ and that $D(s_0),D(s_1)$ are representations of the quantum double acting on $L(s_0,s_1)$ which commute. Then he claims that based on "general representation theory of quantum doubles", the space can be decomposed as $$ L(s_0,s_1)=\bigoplus_{(C,\chi)\in\text{irrep}(DG)} V^*_{(C,\chi)}\otimes V_{(C,\chi)} $$ where $D(s_0)$ acts on $V_{(C,\chi)}^*$ and $D(s_1)$ acts on $V_{(C,\chi)}$.
Now the paper actually gives the explicit formula of basis states and how $D(s_0),D(s_1)$ acts on them, so in theory I can definitely go do the calculations and check that the decomposition is correct. However, it seems that there should be a more abstract proof, which may utilize the fact that $D(s_0),D(s_1)$ commute.
EDIT. I think I (mostly) understand now. Notice that $L(s_0,s_1)$ is isomorphic to the group algebra of the quantum double $DG$, i.e., $\cong \mathbb{C}[DG]$, i.e., equation (60) shows that $L(s_0,s_1)$ has an o-basis which we can denote by $|h,g\rangle$, whereas $DG$ is the algebra with basis $D(h,g),h,g,\in G$, and that left multiplication is defined as $D(s_0)$ acting on $|h,g\rangle$ and right multiplication is defined as $D(s_1)$ acting on the state. It should then be noted that if $H$ is a finite group, then $$ \mathbb{C}[H] \cong \bigoplus_{\chi\in\text{irrep}(H)} V_\chi^*\otimes V_\chi $$ This can be seen by using the isomorphism $\oplus_\chi \chi$ and the fact that $V_\chi^* \otimes V_\chi \cong \text{End}(V_\chi)$. The statement then follows, albeit some details in which $DG$ is different from a group.