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Let $G$ be a compact connected Lie group with maximal torus $T$ and Weyl group $W$. Recall the following two isomorphisms.

Isomorphism 1: $R(G) \cong R(T)^W$, where $R(-)$ denotes the representation ring.

Isomorphism 2: $H^{\bullet}(BG, \mathbb{Q}) \cong H^{\bullet}(BT, \mathbb{Q})^W$.

What relationship is there, if any, between these two isomorphisms? Can I see them both as special cases of some more fundamental fact?

It's tempting to conjecture a relationship involving K-theory but I don't know how to make this work out. In particular, any finite-dimensional representation $V$ of $G$ determines a vector bundle on $BG$ and hence a class in $K^{\bullet}(BG)$, and I hoped that I could apply the Chern character to this class, but for a space like $BG$ with infinite cohomological dimension the Chern character lands in the direct product, rather than the direct sum, of the cohomology groups. There is some story here involving equivariant K-theory and the Atiyah-Segal completion theorem but I am not sure it is enough to, say, deduce either of Isomorphism 1 or 2 from the other.

Here is another even vaguer attempt. By Peter-Weyl, some version of Isomorphism 1 should follow from a suitable isomorphism of the form $G/G \cong T/W$, where on the LHS I am taking the quotient with respect to conjugation. (I'm not sure exactly what category these quotients should be happening in.) On the other hand, Isomorphism 2 should follow from something like the observation that the fiber sequence $G/T \to BT \to BG$ gives a fiber sequence $(G/T)/W \to (BT)/W \to BG$, where $(G/T)/W$ has the rational homotopy type of a point, at least if I'm reading Allen Knutson's answer correctly here, and by the quotient $(BT)/W$ I mean the homotopy quotient. So $(BT)/W \to BG$ is a rational homotopy equivalence, and I might be able to go from this map $(BT)/W \to BG$ to a map $T/W \to G/G$ by taking some version of the free loop space. Maybe? (I might need to work with stacks instead of spaces.)

Edit: I think the two isomorphisms above can be restated in terms of equivariant K-theory and equivariant cohomology respectively as follows:

Isomorphism 1.1: $K_G^{\bullet}(\text{pt}) \cong K_T^{\bullet}(\text{pt})^W$

Isomorphism 2.1: $H_G^{\bullet}(\text{pt}, \mathbb{Q}) \cong H_T^{\bullet}(\text{pt}, \mathbb{Q})^W$

One way to probe the relationship between these two isomorphisms further, then, is to ask:

What happens when I replace $\text{pt}$ above with some other $G$-space? In what generality do the two isomorphisms above continue to hold? What other equivariant cohomology theories can I use here?

In particular, if we find that after replacing $\text{pt}$ by more general spaces one of the isomorphisms holds for different spaces than the other, that would be evidence against a more fundamental fact having them both as corollaries.

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    $\begingroup$ I don't know how to answer your question, but let me try to clarify one thing that I probably said hastily in a talk some time. Of course, there is no equivalence (or even a map) of stacks $T/W \to G/G$ - the stabilizers are all wrong, even if the orbit spaces agree. Let us write $N=N_G(T)$. You do have a map of stacks $BN \to BG$, which gives a map on loop spaces $N/N \to G/G$ ($BN$ is what you write as $BT/W$). Inside $N/N$ you have a copy of $T/N = (T/T)/W$ which also maps to $G/G$. Not sure what else to say yet. $\endgroup$
    – Sam Gunningham
    Commented May 20, 2014 at 16:19
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    $\begingroup$ Isn't $G/N$ rationally contractible ($N$ the normalizer of a maximal torus)? That would imply a rational equivalence $BN\to BG$ and that any rational cohomology theory can be computed on $BN$. $\endgroup$
    – Matthias Wendt
    Commented May 20, 2014 at 18:43
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    $\begingroup$ In addition to $H_G^*(X, {\Bbb Q}) = H_T^*(X, {\Bbb Q})^W$ (any space $X$, essentially due to Borel), these isomorphisms also hold for equivariant Chow groups and (higher) algebraic $K$-theory (of coherent sheaves). I think it's also true for (algebraic/topological) cobordism, but don't know much about that subject. The unifying theme is a version of the splitting principle; the proofs I know in these different contexts use different techniques, though it's certainly reasonable to ask for a mother-of-all proof. $\endgroup$
    – Dave Anderson
    Commented May 20, 2014 at 19:08
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    $\begingroup$ @DavidSpeyer: yes, for finite CW-complexes. It means that in rational homotopy, $G/N$ is equivalent to the point and thus all rational cohomology theories like singular cohomology, K-theory, cobordism (all with rational coefficients) are trivial for $G/N$. I thought a bit more about this, and the rational contractibility of $G/N$ also implies that $X\times^N EG\to X\times^GEG$ also induces isomorphisms in all rational cohomology theories for $X$ any finite $G$-CW-complex. I think this explains the equivariant rational isomorphisms in the question. $\endgroup$
    – Matthias Wendt
    Commented May 20, 2014 at 19:57
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    $\begingroup$ @Matthias: I think Dave's point is that your argument works for Borel-equivariant K-theory but it's not clear that one can go from this to the same statement for what is usually called equivariant K-theory. Atiyah-Segal compares the former to the completion of the latter, but as Dave says it's unclear whether this completion can be undone. $\endgroup$
    – Qiaochu Yuan
    Commented May 20, 2014 at 20:13

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Some googling has led me to an answer to the second question, at least: $H_G^{\bullet}(X, \mathbb{Q}) \cong H_T^{\bullet}(X, \mathbb{Q})^W$ holds for any $G$-space $X$ and the argument is essentially the same, and I learned from a paper of Harada, Landweber, and Sjamaar that $K_G^{\bullet}(X) \otimes \mathbb{Q} \cong (K_T^{\bullet}(X) \otimes \mathbb{Q})^W$ holds at least for $X$ compact (actually we only need to invert $|W|$). So this is encouraging, although I don't know how similar the proofs can be made.

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