TITLE: Classification of $\operatorname{Rep} D(G)$
QUESTION [8 upvotes]: Let $G$ be a finite group and $D(G)$ its quantum double. Its finite dimensional complex representations are classified in this Dijkgraaf et al. Quasi-Quantum Groups Related To Orbifold Models. However, in the paper, the authors claimed the examples obtained by natural inductions are complete, but without a proof. My impression is that the representation theory for a Hopf algebra is not completely known yet in general. How would one classify the representations of $D(G)$?
My understanding is still at the level of representation of a finite group $G$. In that case, the classification can be done due to

$\mathbb{C}[G]$ is semi-simple
$\lvert\operatorname{Irrep}(G)\rvert =\lvert\operatorname{Conj}(G)\rvert$.

Are there similar statements for $D(G)$ (better with proofs)? Pointers towards anything relevant will be appreciated. Thank you!
EDIT: While there's only one chosen answer, the others might be valuable for you. Here is a table of answers below so far.

Konstantinos Kanakoglou pointed out several papers that directly answered my question.
I wrote up a note spelling out Konstantinos Kanakoglou's wonderful answer. I am willing to discuss the detail of the proof.
zibadawa timmy's higher categorical view point.

REPLY [3 votes]: There's a higher way to come at this. I'll be a little light on the rigorous details here, but everything I mention can be found in the book "Tensor Categories" by Etingof, Gelaki, Nikshych, and Ostrik.  The book's a very good starting point for moving from the Hopf algebras perspective to the tensor category perspective, which is where a lot of current research is being done.
If one considers a semisimple Hopf algebra $H$, and take $\mathcal{C}=\operatorname{Rep}(H)$ to be the category of finite dimensional left (or right) modules of $H$, then there is a braided tensor equivalence $\operatorname{Rep}(D(H))\cong \mathcal{Z}(\operatorname{Rep}(H))$, where $\mathcal{Z}(\mathcal{C})$ denotes the categorical center of the category $\mathcal{C}$. This center construction works not just for the particular choice here, but any tensor (aka monoidal) category with sufficiently similar properties.  The objects of the center are the pairs $(V,\gamma_V)$ where $V$ is an object of $\mathcal{C}$ and $\gamma_V$ is a natural family of isomorphisms called a "half-braiding" (because they piece together into a braiding on the entire category).
In the case of $H=\mathbb{C}G$ with $G$ a finite group, we can go one step better.  There is a Morita-equivalence between $\operatorname{Rep}(G)$ and $\text{Vec}_G$, where the latter is the space of $G$-graded finite-dimensional vector spaces (over $\mathbb{C}$).  This is equivalent to saying these categories have the same centers, up to braided tensor equivalence, so we could just as well compute $\mathcal{Z}(\text{Vec}_G)$ instead.  Once you actually write down what the half-braiding conditions are, this center becomes very easy to determine: it's $\text{Vec}_G^G$ (sometimes denoted ${}^G_G\mathcal{M}$, or some variation thereof depending on the use of left/right (co)actions), the category of finite dimensional $G$-graded, $G$-equivariant vector spaces.  At this point it's easy to decide the isomorphism classes of the irreducibles, and you find that they are parameterized by pairs $(g,\chi)$ where $g$ is an element in a complete set of representatives of the conjugacy classes of $G$, and $\chi$ is an element in a complete set of representatives for the irreducible representations (or characters) of $C_G(g)$.  So the isomorphism type of the module depends only on the conjugacy class of $g$ and the isomorphism class of $\chi$.
When you understand the objects of $\text{Vec}_G^G$ it becomes readily apparent that the irreducible objects are just induced representations from $C_G(g)$ to $G$, but where the implicit grading of this induction via cosets of $C_G(g)$ is relevant to deciding the full action of $D(G)$.
And if you want to go even further than that, you can change the associativity morphism of $\text{Vec}_G$ via a normalized 3-cocycle $\omega$ to obtain the category $\text{Vec}_G^\omega$, and then we have $\mathcal{Z}(\text{Vec}_G^\omega)\cong\operatorname{Rep}(D^\omega(G))$, where $D^\omega(G)$ is the twisted Drinfeld double, and is in general a quasi-Hopf algebra and not a Hopf algebra. These objects are also quickly described in the paper you mention. The description of the irreducibles is similar, except now we're using irreducible projective representations for particular 2-cocycles of $C_G(g)$ obtained from $\omega$.  
This category, as a braided tensor category, will only depend on the cohomology class of $\omega$, while $D^\omega(G)$ can have wildly different structures even for representatives of the same cohomology class.  Since those structures are also quite nightmarish to deal with directly for any non-trivial 3-cocycle, most people end up gravitating towards dealing with them through their representation categories, instead.