Symplectic quotients by a nonabelian group and by its maximal torus

This paper examines the relationship between the symplectic quotientX//G of a Hamiltonian G-manifold X , and the associated symplectic quotient X//T , where T ⊂ G is a maximal torus, in the case in which X//G is a compact manifold or orbifold. The three main results are: a formula expressing the rational cohomology ring of X//G in terms of the rational cohomology ring of X//T ; an ‘integration’ formula, which expresses cohomology pairings on X//G in terms of cohomology pairings on X//T ; and an index formula, which expresses the indices of elliptic operators on X//G in terms of indices on X//T . The results of this paper are complemented by the results in a companion paper [15], in which different techniques are used to derive formulæ for cohomology pairings on symplectic quotients X//T , where T is a torus, in terms of the T -fixed points of X . That paper also gives some applications of the formulæ proved here. In order to state the main results of this paper, we introduce some notation. The symplectic quotient X//G is defined to be the topological quotient μ G (0)/G, where μG : X → Lie(G) is a moment map for the G-action on X . A choice of maximal torus T ⊂ G induces a natural projection Lie(G) ։ Lie(T ), and composing with μG gives a moment map μT : X → Lie(T ) for the T -action, with X//T := μ T (0)/T . In most of this paper we make some additional simplifying assumptions: we assume that both μ G (0) and μ −1 T (0) are compact manifolds, on which the respective Gand T -actions are free. It follows that X//G and X//T are compact symplectic manifolds. In section 6 we show how to modify the main results when various of these assumptions are dropped. For every weight α of T there is a characteristic class e(α) ∈ H(X//T ) naturally associated to the principal T -bundle μ T (0) → X//T (for a precise definition see section 2).