Let $G$ be an affine group scheme over a field $k$ of characteristic zero.

I understand that when $G$ is algebraic there is an equivalence between the category $\text{Rep}(G)$ of (rational) representations of $G$ and the category $\text{QCoh}(BG)$ of quasicoherent sheaves on the fppq classifying stack $BG$.

I have three questions:

- Which subcategory, if any, of $\text{QCoh}(BG)$ corresponds to the category $\text{Rep}_{f}(G)$ of finite dimensional representations of $G$? Does it makes sense to talk about coherent sheaves on $BG$?
- Does an equivalence of this form hold when $G$ is not assumed to be algebraic?
- How about at the level of bounded derived categories?