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 $P_{\mu}$.