Let $T$ be a complex torus and $X$ a smooth quasiprojective $T$variety with finitely many fixed points. Denote by $\varphi:H_{T}(X)\rightarrow H_{T}(X^T)$ the map on equivariant cohomology induced by the inclusion $X^T\hookrightarrow X$ of the fixed points. If $X$ were compact, then we could explicitly describe $\varphi$ by computing the weights of the isotropy representations on the tangent spaces to fixed points. Still, I am wondering about what can be said if one does not assume $X$ to be compact. In particular, should one invoke essentially the same procedure to describe $\varphi$? I would appreciate references and/or suggestions.
