# Koszul duality, and coherent sheaves on $pt/G \times_{\mathfrak{g}/G} pt/G$

My questions are the following (from this paper of Arinkin-Gaitsgory):

Q1 Let $P \subset G$ be algebraic groups (in my case, $P$ being a parabolic subgroup of a reductive group $G$, but the following should hold in more generality). The natural map (of algebraic stacks) induces a map $\alpha: pt/P \times_{\mathfrak{p}/P} pt/P \rightarrow pt/G \times_{\mathfrak{g}/G} pt/G$. Let $\Delta: pt/G \rightarrow pt/G \times_{\mathfrak{g}/G} pt/G$ be the diagonal map (and abusing notation, denote also by $\Delta$ the corresponding map for $P$ instead of $G$). Let $\Delta_G = \Delta_* \mathcal{\omega}_{pt/G}$, and define $\Delta_P$ similarly (where $\omega$ denotes the canonical sheaf).

Is it true that $\alpha^{!} \Delta_G = \Delta_P$, and if so, why?

Q2 It is stated on pg $97$ of that paper that $End(\Delta_G)=Sym_{\mathcal{O}_{pt/G}}(\mathfrak{g}/G[-2])$. How is this statement proven? (It is used to formulate a Koszul duality-type result; I was having trouble unwinding the explanation given in that paper.)

Consider the same statement where one gets rid of the $G$-equivariance (i.e. replacing $pt/G$ by $pt$ and $\mathfrak{g}/G$ by $\mathfrak{g}$). If it's easier to answer the above question in this setting, I would also be happy with that. I understand this statement is similar to Koszul duality between the exterior and symmetric algebras.

Q3 On the same page, it is also stated that $\mathcal{F} \rightarrow Hom(\Delta_G, \mathcal{F})$ is an equivalence between $IndCoh(pt/G \times_{\mathfrak{g}/G} pt/G) \rightarrow Mod(End(\Delta_G))$. Why is this true? Again, I was having trouble unwinding the explanation given on that page and in $5.3$.

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If I'm not mistaken, one of the authors is just down the road from you. Have you tried asking him or one of his students? –  S. Carnahan Dec 11 '12 at 5:52
I can also try doing that (though it would take longer). If you think this question is inappropriate for MO, let me know and I'll delete it. –  Vinoth Dec 11 '12 at 16:14
I haven't checked it carefully but if true Q1 should just follow from base change for $\alpha$ and the two $\Delta$'s.. –  David Ben-Zvi Dec 11 '12 at 16:42
and Q3 looks to me like Barr-Beck-Lurie (but then again most things do!) –  David Ben-Zvi Dec 11 '12 at 16:43
Q2 should also follow from base change/projection formula I would guess. but sorry I haven't looked at this part of the paper or thought about this carefully. –  David Ben-Zvi Dec 11 '12 at 16:46