For an algebraic group $G$ there should be an equivalence $\operatorname{Rep}(G) \simeq \operatorname{IndCoh}(BG)$. I'm trying to understand what the dualizing sheaf (or complex) of $BG$ is. Here's what confuses me:
On one hand $\omega_{BG}$ should be the monoidal unit, hence correspond to the trivial representation. That is also what I get if I write sheaves on $BG$ as colimit over the Cech nerve, and what gives group homology under push forward to a point.
On the other hand, at least for simple groups the category of representation admits self duality with the unit map sending the base field to the regular representation. For ind coherent sheaves there's also self duality determined by Vardier duality with the unit map being the (push forward along the diagonal of) the dualizing sheaf. I'm assuming these should agree, so the dualizing sheaf corresponds to the regular representation.
What am I missing here?
Edit: Well, my mistake was in the parenthesis, $\Delta_* \omega_{BG}$ is not the same as $\omega_{BG}$. It is exactly the operation of taking an induced representation $Ind_G^{G\times G} k$ which gives the regular representation. I'll still want to know how to compute this object using stacks, and in particular how to find it's cohomological shift.
