*I'll be using homological grading throughout this question.*


Let $G$ be a compact connected lie group. The following isomorphisms are classical and can be proven using several methods:


$$H^{\bullet}(BG;\mathbb{R}) \cong Sym^{\bullet}(\mathfrak{g}^\*[2])^G$$


$$H^{\bullet}(G; \mathbb{R}) \cong Sym^{\bullet}(\mathfrak{g}^\*[1]
)^G$$


Where on the RHS we take invariants w.r.t. the adjoint action of $G$ on the lie algebra. Replacing $G$ with its complexification (which is the complex points $\mathbb{G} (\mathbb{C})$ of a connected semi-simple complex algebraic group) doesn't change the cohomology so let us do this. Next let's interpret the graded rings above as (derived if you like) prestacks (functors $CommAlg\_{\mathbb{C}} \to \infty$-groupoids). Translating to this language we get the following (which is just a different way to notate the above):


$$SpecH^{\bullet}(B\mathbb{G}(\mathbb{C});\mathbb{C}) \cong \mathfrak{g}\_{\mathbb{C}}[-2] \text{ //}G \cong \mathbb{T}\_{B\mathbb{G}}[-1] \text{ //} \mathbb{G}$$


$$Spec H^{\bullet}(\mathbb{G}(\mathbb{C}); \mathbb{C}) \cong \mathfrak{g}\_{\mathbb{C}}[-1] \text{ //}G \cong \mathbb{T}\_{B\mathbb{G}} \text{ //} \mathbb{G}$$


I find it weird that both sides of the equation naturally take as input a connected semisimple complex algebraic group $\mathbb{G}$ but the isomorphism itself is of a transcendental nature (by passing through the real points of the compact real form as a topological group). This mysterious relationship does not manifest itself at the level of the proofs of these statement (at least in the proofs I know). It would be nice to know whether these interpretations can be turned to proofs (at least giving one of the results given the other one):



> 
> **Question 1:** Is their a **higher mechanism** at work here? Phrased differently: Are these isomorphisms a formal consequences of the formalism of DAG together with some basic facts about groups?
> 
> 
> 



Interpretation of the cohomology of compact lie groups and their classifying spaces in DAG?
I don't know of a DAG mechanism that implies these statements, any more formally than the standard proofs -- i.e., we reduce to a maximal torus, where the statement follows from the shape of the cohomology of the circle.


There are higher mechanisms though of which this statement is a hint. On one side you have topology (of BG), on the other side (derived) algebraic geometry -- suggesting that rather than something formal this is a shadow of mirror symmetry. Indeed the statement is [essentially equivalent to] the mirror symmetry between the G-equivariant A-model on a point and the B-model into a shift of the Cartan mod Weyl group. That is probably a natural setting for this question.


Or if you prefer, the statement can be seen (at the risk of tremendous overkill) as a feature of geometric Langlands (roughly the case of ${\mathbf P}^1$), specifically of the wonderful derived Geometric Satake theorem of Bezrukavnikov-Finkelberg <https://arxiv.org/abs/0707.3799> (though of course the basic statements above are used in the proof!)
Namely there's an equivalence of monoidal dg categories


$$D\_{LG\_+}(LG/LG\_+) \simeq D\_{coh}({\mathfrak g}^{\vee,\ast}[2]/G^\vee)$$


between equivariant sheaves on the affine Grassmannian for $G$ and equivariant coherent sheaves on the coadjoint representation of the Langlands dual. Your statement is an "affinization" of this -- describing modules for the endomorphisms of the unit (or natural coefficient rings) on both sides -- on the left there's the equivariant category


$$D\_{LG\_+}(pt)=D\_G(pt)=D(BG)$$


and on the right there's the affinization of the coadjoint quotient,


$${\frak g}^{\vee,\ast}[2]//G^\vee \simeq {\frak h}^{\vee,\ast}[2]//W
\simeq {\frak h}[2]//W\simeq {\frak g}[2]//G.$$


