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First, since $\overline N$ is contractible its equivariant cohomology is the same as for $pt$. The Poincare pairing is uniquely determined by $\int_{\overline N} 1$ (since it is linear with respect to $H^*_G(pt)$-linear). H^*_G(pt)$). More precisely, any cohomology class of$\overline N$has the form$\alpha\cdot 1$where$\alpha$is an equivariant cohomology class of$pt$and$1$denotes the unit cohomology class in$\overline N$and we have $$\langle \alpha\cdot 1,\beta\cdot 1\rangle =\alpha\beta\int_{\overline N} 1.$$ I don't know a good way to compute this$\int_{\overline N} 1$for arbitrary$N$- other than replacing$\overline N$by a resolution and using fixed point localization. By the way, if$\overline N$is the minimal orbit, then${\mathbb C}^2\times {\overline N}$is the same as the Uhlenbeck space of$\mathbb C^2$of second Chern class 1 - that should give you another way to compute that integral (is it obvious that you get the same answer?) 1 First, since$\overline N$is contractible its equivariant cohomology is the same as for$pt$. The Poincare pairing is uniquely determined by$\int_{\overline N} 1$(since it is$H^*_G(pt)$-linear). I don't know a good way to compute this for arbitrary$N$- other than replacing$\overline N$by a resolution and using fixed point localization. By the way, if$\overline N$is the minimal orbit, then${\mathbb C}^2\times {\overline N}$is the same as the Uhlenbeck space of$\mathbb C^2\$ of second Chern class 1 - that should give you another way to compute that integral (is it obvious that you get the same answer?)