Localization for nonabelian group actions

Suppose X is a compact symplectic manifold acted on by a compact Lie group K (which may be nonabelian) in a Hamiltonian fashion, with moment map µ : X → Lie(K) ∗ and Marsden-Weinstein reduction MX = µ −1 (0)/K. There is then a natural surjective map �0 from the equivariant cohomology H ∗ K (X) of X to the cohomology H ∗ (MX). In this paper we prove a formula (Theorem 8.1, the residue formula) for the evaluation on the fundamental class of MX of any �0 ∈ H ∗ (MX) whose degree is the dimension of MX, provided that 0 is a regular value of the moment map µ on X. This formula is given in terms of any class � ∈ H ∗ (X) for which �0(�) = �0, and involves ∗

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