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Warning: non-specialist writing, some rubbish possible.

The formula $h^*(BG)\cong h^*(BT)^W$ valid for complex oriented cohomology of the classifying space of a compact Lie group $G$ with maximal torus $T$ and Weyl group $W$ suggests that it might come from some sort of equivalence like $G\sim T///W$ where "///" stands for some hypothetical homotopy quotient, like homotopy colimit but not in spaces but rather some sort of equivariant homotopy colimit in the category of topological groups or maybe rather in the category of homotopy associative H-spaces and homotopy coherent homomorphisms, or maybe some sort of stacky quotient instead. Is there such a thing?

(Slightly later: comments by Ben Wieland show that what I wrote is not clear enough. Let me try to explain at least approximately: by $T$///$W$ I mean a universal object among topological group homomorphisms $f:T\to G$ together with homotopies between $f$ and $fw$ for each $w\in W$, where $w:T\to T$ is the action map. Presumably these homotopies must satisfy coherence conditions, and maybe I should require them to induce homomorphisms $T\to G^{[0,1]}$ or something similar, here I am not sure.)

My motivation (not counting that it might be useful to know such thing if it is true) comes from the fascinating paper "Root systems and elliptic curves" by Looijenga (Invent. Math. 38 (1976), pp. 17-32) where he constructs a "third invariant theory" using the Weyl group action on, in place of the Cartan subalgebra or maximal torus that are used for the "first", resp. "second" invariant theory, the abelian variety $A:=Q^\vee\otimes E$ where $Q^\vee$ is the lattice generated by the dual root system and $E$ is an elliptic curve. (Invariants are with respect to the $W$-action on sections of the line bundle over $A$ corresponding to the divisor formed by the fixed hypertori of $W$-reflections, and are expressed using certain theta functions.)

I wonder if some similar "homotopy quotient group" $A///W$ can be formed to yield some new kind of group-like object.

However I must admit that before this irresponsible "jump from second to third" it would be more appropriate to first understand the "first" case -- whether the corresponding Lie algebra $\mathfrak g$ might be realized as some sort of "homotopy quotient Lie algebra" $\mathfrak h///W$ of the Cartan subalgebra by the Weyl group...

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    $\begingroup$ I love this question. So deliciously irresponsible! $\endgroup$ – André Henriques Apr 8 '18 at 21:12
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    $\begingroup$ When $h^*=H^*(-,\mathbb Z)$, the formula $h^*(BG)\cong h^*(BT)^W$ holds for $*\le 4$, but it fails in higher degrees when $G$ is an arbitrary compact simply-connected Lie group. See §3 of my paper arxiv.org/pdf/1602.02968.pdf. I don't really know what happens for other complex oriented cohomology theories, but I've heard that things are much nicer for $K$-theory. Maybe it has to do with the question of whether $h^*$ is Landweber exact. $\endgroup$ – André Henriques Apr 8 '18 at 21:42
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    $\begingroup$ @მამუკაჯიბლაძე For any complex oriented $E$ and Lie group $G$ with $H^*(BG;Z)$ torsion-free, we have $E^*(BG)=E^*(BT)^W$ $\endgroup$ – user43326 Apr 9 '18 at 11:24
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    $\begingroup$ Sorry to be a plodding literalist, but ... $BT//W=BNT$. I believe that $BG$ stably splits off of $BNT$, a refinement of the usual statements about invariants, which may be adequate for some purposes... If you're in a setting that allows disconnected groups (eg, $O(n)$), then you can't get around $BNT$. But maybe it is possible to force (some) spaces to be simply connected to slide along the map $BNT\to BG$. $\endgroup$ – Ben Wieland Apr 10 '18 at 17:37
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    $\begingroup$ @მამუკაჯიბლაძე Assuming $G$ is connected, are you familiar with the fact that after localizing (bousfield localization of simply connected spaces) away from the primes dividing $|W|$ (the cardinality of the weyl group) one can show (just by following carefully the classical computation of rational cohomology) that the map $BT \to BG$ induces a homotopy equivalence $BT // W \cong BG$? where by $//$ i mean the homotopy orbits of the action of $W$ on $BT$. $\endgroup$ – Saal Hardali Apr 11 '18 at 13:27
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Here's the "answer" that I started writing, then put away for a while. The short answer is: although "T//W" is not the same as G, they "look sort of the same" from the point of view of certain generalized cohomology theories. If you replace the role of the maximal torus with "arbitrary abelian subgroups", this "looking the same" can be given homotopical meaning. Finally, this data that your generalized cohomology (such as elliptic cohomology) "sees" can be encoded (according to Lurie) in terms of objects of derived algebraic geometry, which for elliptic cohomology you can think of as the "A///W" that you want to have.

...

As pointed out in the comments, $h^*(BG)\neq h^*(BT)^W$ for many compact simply connected $G$ and generalized cohomology theories (though equality does hold for $G=U(n)$ or $SU(n)$ and $h^*$ complex oriented).

If you try to naively make a "homotopical" version of this, by talking about classifying spaces, it gets even worse: in general $BG \neq BT_{hW}$, where the right-hand side is the homotopy invariant quotient of $BT$ by the action of the Weyl group (in fact, $BT_{hW}= B(NT)$, the classifying space of the normalizer.)

What is true is if the appropriate set of primes is inverted in your cohomology theory, then you get an equivalence: e.g., if $|W|$ is inverted in $h^*$, then $h^*(BG)=h^*(NT)$ for connected $G$ (as Saal Hardali points out in the comments). But it's not true integrally.

(It is true rationally. Which you can then reinterpret as a statement of Chern-Weil theory: $H^*(BG;\mathbb{R}) = (\operatorname{Sym} \mathfrak{g}^*)^G$; i.e., seen by $\mathbb{R}$-cohomology, $BG$ "looks like" $\mathfrak{g}//G$.)

Another thing to do is to generalize from "maximal torus" to "arbitrary abelian subgroup". So let $\mathcal{A}(G)$ be the category of "abelian subgroups of $G$": really, the category whose objects are $G/A$, the $G$-orbits with abelian isotropy group $A$, and whose maps are the space of $G$-equivariant maps between such orbits. Note that for connected $G$ this category contains the object $G/T$ with $\mathrm{Aut}(G/T)=W$, so it really is a generalization of "$W$ acting on $T$". Then we obtain a "better" approximation map: $$ \mathrm{hcolim}_{G/A\in \mathcal{A}(G)} BA = \mathrm{hcolim}_{G/A\in \mathcal{A}(G)} (G/A)_{hG}\to BG,$$ which actually makes sense for all compact Lie group $G$ (including finite groups). This approximation actually turns out to be an equivalence (though not for deep reasons). Note: $\mathrm{hcolim}$ is homotopy colimit, and $(-)_{hG}$ is homotopy orbits.

Side note: the above "formula" implies a spectral sequence for any cohomology theory $h^*$ for computing $h^*(BG)$. This spectral sequence has as its "edge" map: $$ h^*(BG) \to \operatorname{lim}_{G/A\in \mathcal{A}(G)} h^*(BA). $$ Hopkins, Kuhn, and Ravenel proved that for a finite $G$, and a complex oriented $h^*$, this edge map becomes an isomorphism after inverting $|G|$. This is one version of their famous "character theorem".

There are analogues of this that apply equivariant cohomology theories other than "Borel equivariant cohomology", in which case things are more subtle. For instance, consider equivariant $K$-theory. If for a $G$-space $X$ I let $K_G(X)$ denote the spectrum whose homotopy groups are the $G$-equivariant $K$-theory of $X$, then there is supposed to be an equivalence $$ K_G(*) \approx \mathrm{hlim}_{G/A\in \mathcal{A}(G)} K_G(G/A) =\mathrm{hlim}_{G/A\in \mathcal{A}(G)} K_A(*). $$ Here $G$ is any compact Lie group, including finite ones (in which case it is a kind of Artin induction theorem). (This is basically asserted in Lurie's "Survey of Elliptic Cohomology", though I haven't seen a proof written out in full anywhere. $\mathrm{hlim}$ is homotopy limit.)

In other words, equivariant $K$-theory "sees" $G$ (more precisely, $(*//G)$) as indistinguishable from a derived colimit of abelian subgroups $A$ (more precisely, $(*//A)$). This is reminiscent of and related to (but distinct from and more complicated than) the observation that $RG=RT^G$ for representation rings of compact connected groups (because $\pi_0 K_G(*)=RG$).

Another equivariant cohomology theory you might try is elliptic cohomology. If you had a "naturally occuring" example of an equivariant elliptic cohomology, you would conjecture that it satisfies an analogue of the above equivalence. Or, if you're Jacob Lurie, you construct equivariant elliptic cohomology for arbitrary abelian compact Lie groups, then extend it to arbitrary compact Lie groups by imposing an equivalence as above.

Thus, Lurie associates to each abelian compact Lie group $A$ an object $M_A$ in derived algebraic geometry, so that:

  • $M_{U(1)}$ is a "derived elliptic curve" $E$,

  • $M_{A\times B}=M_A\times M_B$, so in particular

  • $M_{U(1)\otimes Q} = E\otimes Q$ for a lattice $Q$, and

  • $M_{Z/n}= $ kernel of $[n]:E\to E$.

Then for an arbitrary compact Lie group $G$ you should set $M_G$ to be some sort of colimit of a functor on $\mathcal{A}(G)$ that you can construct which sends $G/A\mapsto M_A$. (Apparently it's delicate to know exactly what sort of geometric object $M_G$ will be, but for the purposes of extracting an equivariant cohomology theory out of this business it's probably not too important. I'll pretend it is some kind of geometric object for the purposes of this discussion.)

For $G$ connected, you should think of $M_G$ as a derived and souped-up version of $E\otimes Q^{\vee}//W$ (souped-up because we use all abelian subgroups of $G$ and not just the torus with its Weyl group action).

As a bonus, Lurie can extend this set-up to Lie 2-groups $\widetilde{G}$ which are extensions of the form $1\to \mathbb{B}U(1)\to \widetilde{G}\to G\to 1$, associated to some cohomology class $c\in H^4(BG;\mathbb{Z})$. Then $M_{\widetilde{G}}$ is (I think, and modulo things that I haven't said quite correctely) supposed to be the principal space of a line bundle over $M_G$, whose sections are to be (presumably) described by the theta-functions that Looijenga writes down in that paper you mention. (As an aside, I have no idea what motivates Looijenga to do whatever it is he does in that paper.)

Unfortunately, I don't know much more, since most of this is only described in Lurie's "Survey" mentioned above, which is only an announcement. (Versions of this story were perceived much earlier, most notably by Grojnowski in "Delocalized equivariant elliptic cohomology".)

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