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 $\mu :X\to \mathrm{Lie}(K{)}^{\ast }$ and Marsden-Weinstein reduction $\xred = μ^{-1}(0)/K$. There is then a natural surjective map ${\kappa }_0$ from the equivariant cohomology $H_K^{\ast }(X)$ 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 $\mu$ on $X$. This formula is given in terms of any class $\eta \in H_K^{\ast }(X)$ for which ${\kappa }_0(\eta )={\eta }_0$, and involves the restriction of $\eta$ to $K$-orbits $KF$ of components $F\subset X$ of the fixed point set of a chosen maximal torus $T\subset K$. Since ${\kappa }_0$ is

42 pages, LaTex version no. 2.09, Introduction and Section 8 have been rewritten in revised version

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