On the localization theorem in equivariant cohomology

We present a simple proof of a precise version of the localization theorem in equivariant cohomology. As an application, we describe the cohomology algebra of any compact symplectic variety with a multiplicity-free action of a compact Lie group. This applies in particular to smooth, projective spherical varieties. 1 A precise version of the localization theorem Let X be a topological space with an action of a compact torus T . Let H T (X) be the equivariant cohomology algebra of X with coefficients in the field Q of rational numbers. The equivariant cohomology algebra of the point is denoted by ST ; then H ∗ T (X) is a ST -algebra. Any weight of T defines an element of degree 2 of ST ; this identifies ST with the symmetric algebra over Q of the group of weights of T . Let Γ ⊂ T be a subtorus, let XΓ ⊂ X be its fixed point set and let iΓ : X Γ → X, iT,Γ : X T → X be the inclusion maps. They define homomorphisms of ST -algebras i∗Γ : H ∗ T (X) → H ∗ T (X ), i∗T,Γ : H ∗ T (X ) → H T (X T ). Recall that the ST -algebra H ∗ T (X Γ) is isomorphic to ST ⊗ST/Γ H ∗ T/Γ(X Γ). In particular, the ST -module H ∗ T (X T ) = ST ⊗Q H ∗(XT ) is free. The following statement is a variant of a result of Chang and Skjelbred (see [4] §2 and also [8] p. 63).