Let us first precise the question : let $T$ be a torus, $\alpha : T \to \mathbb{C}$ be an irreducible character. I am interested in the $T$-equivariant Euler class of the ($T$-equivariant) bundle $\xi\_\alpha:\mathbb{C}\to pt$ where $T$ acts on $\mathbb{C}$ via $\alpha$. $\alpha$ corresponds to an integral form $\alpha^\ast$ on the dual of the Lie algebra of $T$, and in their magnificent paper "The Moment Map and Equivariant Cohomology", Atiyah and Bott say that the equivariant Euler class of the bundle $\xi\_\alpha$ is exactly $\alpha^\ast$ (seen as an equivariant cohomology class of $pt$).


My question then is : why is this true ?



Why is the equivariant Euler class a character ?
Well, there's going to be *some* map $T^\* \to H^2\_T(pt)$ taking a weight $\lambda$ to the equivariant Euler class of the corresponding line bundle over a point. If you add weights, that tensors the line bundles, so adds the first Chern classes; hence this is an additive map between these two free abelian groups. Since it's natural it's going to be $Aut(T)$-equivariant. Already we see that Atiyah and Bott have to be right, up to scale.


