# Invariants of cohomology of Springer sheaf

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 $$P_{\mu}$$.

• Do you really mean for $\mathfrak u_\mu$ to be the nilpotent cone, or rather the Lie algebra of the unipotent radical? Sep 8, 2022 at 16:35
• Sorry the Lie lagebra of the unipotent radical. I'm editing! Sep 9, 2022 at 7:27

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 $$G/P_\mu$$.