Let $G=Gl\_n(\mathbb{C})$ and $\mathcal{N}$ be the nilpotent cone associated to it i.e nilpotent matrices inside $\mathfrak{g}=\mathfrak{gl}\_n(\mathbb{C})$.


We have the variety $\tilde{\mathcal{N}}$ with the Springer resolution $p:\tilde{\mathcal{N}} \to \mathcal{N}$ with the Springer sheaf $p\_{!}(\mathbb{Q}[\dim \mathcal{N}])$ with its action of the Weyl group $S\_n$. In this case we know it decomposes as $$\bigoplus\_{\lambda \in \mathcal{P}\_n}\operatorname{IC}(O\_{\lambda},\mathbb{Q})\otimes V\_{\lambda} $$ where $O\_{\lambda}$ is the nilpotent orbit associated to the partition $\lambda$ and $V\_{\lambda}$ is the $S\_n$ irreducible module indexed by $\lambda$.


I'm looking for an explicit description of $H\_c(\mathcal{N},p\_{!}(\mathbb{Q}[\dim \mathcal{N}]))^{S\_{\mu}}$ where $\mu$ is a partition of $n$ and $S\_{\mu}=S\_{\mu\_1} \times \dotsb \times S\_{\mu\_k}$. I think this should be related to the cohomology $H(T^\*(G/P\_{\mu}),\mathbb{Q})$ where $P\_{\mu}$ is the parabolic subgroup associated to $\mu$.


The only thing I was able to notice is that $$H\_c(\mathcal{N},p\_{!}(\mathbb{Q}[\dim \mathcal{N}]))^{S\_{\mu}}=\bigoplus\_{\lambda \in \mathcal{P}\_n}H\_{c}(\operatorname{IC}(O\_{\lambda},\mathbb{Q}))\otimes V\_{\lambda}^{S\_{\mu}} $$ and $V\_{\lambda}^{S\_{\mu}} \neq 0$ only if $\lambda \geq \mu$. This happens exactly when $O\_{\lambda} \subseteq \mathfrak{u}\_{\mu}$ where $\mathfrak{u}\_{\mu}$ is the Lie algebra of the unipotent radical of $P\_{\mu}$.



Invariants of cohomology of Springer sheaf
You want to look at the partial Grothendieck-Springer resolution, i.e. the variety of pairs $ (g \in G/ P\_\mu, v \in g \mathfrak p\_\mu g^{-1})$.


The partial Grothendieck-Springer resolution is smooth, so the shifted constant sheaf on it is perverse, and its pushforward to $\mathfrak g$ is pure.


The partial Grothendieck-Springer resolution is covered by the Grothendieck-Springer resolution, so the pushforward of the shifted constant sheaf is a summand of the pushforward of the shifted constant sheaf of the Grothendieck-Springer resolution.


In particular, the pushforward from the partial Grothendieck-Springer resolution is a middle extension from the subset of regular semisimple elements.


On a regular semisimple element, the fiber of the map from this variety to $\mathfrak g$ is $S\_n/S\_\mu$. It follows from this and the middle extension property that the pushforward from he partial Grothendieck-Springer resolution, restricted to the nilpotent cone, is the $S\_\mu$-invariants of the Springer sheaf.


So, by the Leray spectral sequence with compact support, the cohomology you want is the compactly supported cohomology of the space of pairs $g \in G/ P\_\mu, v \in g \mathfrak p\_\mu g^{-1}$ with $v$ nilpotent.


The fiber of the map to $G/P\_\mu$ is the product of the nilpotent cone of the associated Levi with a vector space. I think we know that the compactly supported cohomology of the nilpotent cone is supported in a single degree, as is the compactly supported cohomology of a vector space, so in the end you get a shift of the cohomology of $G/P\_\mu$.


