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Weights on equivariant cohomology?

Although the words "equivariant cohomology" don't appear there, this goes back to Deligne's Hodge III, since he defines the mixed Hodge structure on a simplicial variety and you can define the Borel construction … connected, then there is a spectral sequence $$H^p(BG) \otimes H^q(X) \implies H^{p+q}_G(X).$$ If $X$ is smooth and projective this must therefore degenerate immediately for weight reasons and so the equivariant
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