Let $H$ and $K$ be affine group schemes over a field $k$ of characteristic zero. Let $\varphi:H\to Aut(K)$ be a group action. Then we can form the semi-direct product $G = K\ltimes H$.
Problem: Describe the tannakian category $Rep_k(G)$ in terms of $Rep_k(K)$, $Rep_k(H)$ and $\varphi$.
If $\varphi$ is the trivial action, $G$ is the direct product $K\times H$. In this case, $Rep_k(G) = Rep_k(K) \boxtimes Rep_k(G)$ is just the Deligne tensor product of the tannakian categories.
Question: What happens in the opposite scenario where the action $\varphi$ is faithful? Can one characterize this situation in terms of Ext groups of simple objects in $Rep_k(K)$ and $Rep_k(H)$?
I'm mostly insterested in the case where $K$ and $H$ are extensions of $\mathbb{G}_m$ by a pro-unipotent group but I have no idea where to start.
Best Answer
The algebraic stack BG is the quotient of BK by the action of H, induced by its action on K. So we can describe coherent sheaves on it (aka reps of G) via descent from BK - i.e. reps of G are H-equivariant sheaves on BK, or H-equivariant objects in Rep K. Now this is not yet the answer you want, since it involves Rep K and H, rather than Rep H.
One indirect (and maybe slightly imprecise) answer is to use the Morita equivalence between Rep H-module categories and categories with H action: the Rep H-module category Rep G corresponds under this equivalence to Rep K as a category with H action. One direction takes a category with H action to its H-equivariant objects, which carry a natural Rep H action. The other direction takes a Rep H category to the category of its eigenobjects. Here an eigenobject is an object $M\in C$ with a functorial identification $$V\ast M \simeq \underline{V}\otimes M$$ of the module action of $V\in Rep K$ with the naive tensor product by the underlying vector space of $V$. (see e.g. this paper) -- again this is just playing with monadic formalism.. So I would then characterize Rep G as the Rep H-module (via induction of representations) whose eigenobject category is Rep K as an H-category..
This Morita equivalence for finite groups appears in papers of Mueger and Ostrik cited in here. For affine group schemes I proved a derived version of this result with Nadler and Francis, but it's not available sadly. I haven't thought this through in the usual Tannakian setting, so maybe I'm missing something obvious, but it should be a fairly straightforward (though 2-categorical!) application of Barr-Beck I would think: we're simply claiming that categories over BH are described via descent from a point, and the descent data is an action of H.