The Biinvariant Diagonal Class for Hamiltonian Torus Actions

Suppose that an algebraic torus G acts algebraically on a projective manifold X with generically trivial stabilizers. Then the Zariski closure of the set of pairs {(x, y) ∈ X × X | y = gx for some g ∈ G} defines a nonzero equivariant cohomology class [∆G] ∈ H G×G(X×X). We give an analogue of this construction in the case where X is a compact symplectic manifold endowed with a hamiltonian action of a torus, whose complexification plays the role of G. We also prove that the Kirwan map sends the class [∆G] to the class of the diagonal in each symplectic quotient. This allows to define a canonical right inverse of the Kirwan map.