For an algebraic group $G$ over a field $k$, the abelian category $ Rep_k(G)$ of $k$-linear locally finite representations of $G$ can be identified $QCoh(BG_{lis-et})$. Suppose now I replace $G$ with a group stack $ \mathfrak{G}$, i.e. a group object in the $\infty$-category of $1$-Artin stacks. Then I can consider the abelian category $D_{qc}(B\mathfrak{G}_{lis-et})^\heartsuit$. Here I am thinking of $ B\mathfrak{G} $ as $ [ \bullet / \mathfrak{G}]$ which I suppose will be a $2$-Artin stack.

Question: Is there a way to relate $D_{qc}(B\mathfrak{G}_{lis-et})^\heartsuit$ to some category of "representations" of the group stack $\mathfrak{G}$? (Im not even sure what a representation should be maybe something like a functor $ B\mathfrak{G} \rightarrow D^{[0 ,1]}(Vect_k)$?)

  • 1
    $\begingroup$ Do you want to identify representations with descent data for a vector space along the covering map from Spec $k$ to $BG$? $\endgroup$ – S. Carnahan Aug 30 '18 at 0:02
  • $\begingroup$ I'm not qualified, but I would guess that the right notion of representation is a representation of the connected components of $\mathfrak G$. $\endgroup$ – Phil Tosteson Aug 30 '18 at 1:26
  • $\begingroup$ @S.Carnahan Yes I think this is what I want to do. $\endgroup$ – Anette Aug 30 '18 at 5:35
  • 2
    $\begingroup$ I wouldn't expect there to be a lot of interesting reps since they (at least the finite dimensional ones to be safe) will factor through the affinization of your group-stack. Group-stacks do tend to have interesting representations on categories though.. $\endgroup$ – David Ben-Zvi Aug 30 '18 at 10:50
  • $\begingroup$ Ahh so maybe it would be better to consider a group stack acting on objects of the form $ [V_0 / V_1]$ which are obtained by taking the stack quotient of two vector spaces $ V_0, V_1$ which are related by a linear map $\varphi: V_1 \rightarrow V_0$? By stack quotient I mean that $\varphi$ defines an action of $V_1$ on $ V_0$ by $ (v_1, v_0) \mapsto \varphi(v_1) + v_0$. Sorry if that was gibberish. $\endgroup$ – Anette Aug 30 '18 at 11:28

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.