Symplectic quotients by a nonabelian group and by its maximal torus

TL;DR

This paper establishes formulas relating the symplectic quotient of a Hamiltonian G-manifold to that of its maximal torus T, including cohomology, pairing, and index formulas, for cases where the quotient is a compact manifold or orbifold.

Contribution

It provides new explicit formulas connecting the cohomology, pairings, and indices of symplectic quotients by a nonabelian group and its maximal torus.

Findings

Cohomology ring of X//G expressed via X//T

Cohomology pairings on X//G in terms of X//T

Indices of elliptic operators on X//G related to those on X//T

Abstract

This paper examines the relationship between the symplectic quotient X//G of a Hamiltonian G-manifold X, and the associated symplectic quotient X//T, where T 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, in which different techniques are used to derive formulae 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 formulae proved here.)