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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.

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.

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.

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.

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.)

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.