Let $\pi:\tilde{\mathfrak{g}}\rightarrow\mathfrak{g}$ the Grothendieck-Springer resolution of a semisimple Lie algebra $\mathfrak{g}$, over $\mathbb{C}$.

We know it's a small map, and that $\pi_{*}\mathbb{Q}_{\ell}$ is a perverse with an action of the Weyl group and is the intermediate extension of what happens on the regular semisimple locus.

If we take $W$-invariants of this sheaf, then we obtain the constant sheaf, as it will be the IC extension of the constant sheaf on something smooth.

In particular, it means that for each $\gamma\in \mathfrak{g}$, the Weyl group invariants on $R\Gamma(X_{\gamma},\mathbb{Q}_{\ell})$ is just $\mathbb{Q}_{\ell}$, where $X_{\gamma}$ is the Springer fiber at $\gamma$. How can we see it directly, without using perversity?


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