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Reference: M. Atiyah, R. Bott, "The moment map and equivariant cohomology," Topology 23 (1984) (Page 5)

Let $G= T$ be a torus of dimension $l$. You may assume $G=(\mathbb C^*)^{l}$. Then the equivariant cohomology ($\mathbb C$-coefficient) is $$H^*_{G}:=H^*(BG)=\mathbb C[u_1,\cdots,u_l]$$

The paper (page 5) claims that $u_i$'s can be viewed naturally as coordinates on the Lie algebra $\mathfrak g$ or its complexification $\mathfrak g^\mathbb C$. But how to understand this precisely? Thanks

In what follows, the support $S$ of a module over $H^*_G$ should be contained in $Spec (H^*_G) \equiv Spec(\mathbb C[u_1,\cdots,u_l])$. Why can $S$ be naturally considered a subset of $\mathfrak g^\mathbb C$ as well? And how to understand it?

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    $\begingroup$ For the first part, the keyword is Chern-Weil theory. This provides an isomorphism from the ad(G)-polynomials on the Lie algebra to the cohomology of the classifying space. Once $H_G^\ast$ is identified as polynomials on $\mathfrak{g}$ we can naturally view generators of the polynomial ring as coordinates on the Lie algebra. Algebraic geometry then states that the support of a module over a ring $R$ is a closed subset of $Spec(R)$. The spectrum of the polynomial ring is an affine space, which can be identified with the Lie algebra (via the Chern-Weil iso). $\endgroup$
    – Matthias Wendt
    Commented Dec 19, 2016 at 20:59

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