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Let $M$ be a smooth manifold and $G$ a compact connected Lie group acting on$M$ by isometries. In this paper, we study the equivariant cohomology of ${\bfX}=T^\ast M$, and relate it to the cohomology of the Marsden-Weinstein reducedspace via certain residue formulae. In case that $\bf X$ is a compactsymplectic manifold with a Hamiltonian $G$-action, similar residue formulaewere derived by Jeffrey, Kirwan et al.

Singular equivariant asymptotics and the momentum map. Residue formulae in equivariant cohomology