Localization for nonabelian group actions

Abstract

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 $μ: X \to {\rm Lie}(K)^*$ and Marsden-Weinstein reduction $\xred = μ^{-1}(0)/K$. There is then a natural surjective map $κ_0$ from the equivariant cohomology $H^*_K(X) $ of $X$ to the cohomology $H^*(\xred)$. In this paper we prove a formula (Theorem 8.1, the residue formula) for the evaluation on the fundamental class of $\xred$ of any $η_0 \in H^*(\xred)$ whose degree is the dimension of $\xred$, provided that $0$ is a regular value of the moment map $μ$ on $X$. This formula is given in terms of any class $η\in H^*_K(X)$ for which $κ_0(η) = η_0$, and involves the restriction of $η$ to $K$-orbits $KF$ of components $F \subset X$ of the fixed point set of a chosen maximal torus $T \subset K$. Since $κ_0$ is