Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$, and let $N$ be a nilpotent orbit of $\mathfrak{g}$. What is the equivariant cohomology of its closure, $H^*_G(\overline{N})$, with respect to the group $G$ for $\mathfrak{g}$?

Also, $H^*_G(\overline{N})$ has a natural "inner product" which takes value in the quotient field $S$ of $H^*_G(pt)$, defined via the equivariant integration. It would be nice to know this structure too.

I would be happy if I know the answer for the minimal nilpotent orbit for the simply-laced $\mathfrak{g}$.

Let me give the background to my question. Let $H^*_G(pt)=\mathbb{C}[t_1,\ldots,t_r]$ so that $t_1$ has degree 2, ..., $t_r$ has degree $h^\vee$ ($r$ is the rank of $\mathfrak{g}$. As $\mathrm{Spec} H^*_G(pt)= \mathfrak{h}/W$, the standard flat metric on $\mathfrak{h}$ determines a metric on $\mathfrak{h}/W$, which we denote by $\langle .,. \rangle$. Note that the vector field $\partial/\partial t_r$ is unique up to a scalar multiplication. So, $\langle \partial/\partial t_r,\partial/\partial t_r\rangle$ determines a rational function on $\mathfrak{h}/W$, i.e. an element of $S$ (unique up to a scalar multiplication).

Let $N$ be the minimal nilpotent orbit of a simply-laced $\mathfrak{g}$. My collaborators and I calculated $\int_{\overline{N}} 1$. And it equaled $\langle \partial/\partial t_r,\partial/\partial t_r\rangle$.

This suggests that $H^*_G(\overline{N})$ has a natural basis corresponding to $\partial/\partial t_i$, and the inner product given by the equivariant integral equals $\langle.,.\rangle$.

Is this something known in the literature?