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 $P_{\mu}$.
Best Answer
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 $G/P_\mu$.