Abelianization for hyperkähler quotients

We study an integration theory in circle equivariant cohomology in order to prove a theorem relating the cohomology ring of a hyperkähler quotient to the cohomology ring of the quotient by a maximal abelian subgroup, analogous to a theorem of Martin for symplectic quotients. We discuss applications of this theorem to quiver varieties, and compute as an example the ordinary and equivariant cohomology rings of a hyperpolygon space. Let X be a symplectic manifold equipped with a hamiltonian action of a compact Lie group G. Let T ⊆ G be a maximal torus, let ∆ ⊂ t∗ be the set of roots of G, and let W = N(T )/T be the Weyl group. If the symplectic quotients X/G and X/T are both compact, Martin’s theorem [M, Theorem A] relates the cohomology1 of X/G to the cohomology of X/T . Specifically, it says that H(X/G) ∼= H∗(X/T )W ann(e0) ,