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 $\mathfrak{g}$.




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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?



Equivariant cohomology of nilpotent orbits
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 
$H^\*\_G(pt)$). 


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 $\int\_{\overline N} 1$ 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?)


