I am confused about the following: can one describe the action of the Weyl group on the cohomology of each fiber of the Grothendieck-Springer resolution? I only need the case of ${\mathfrak sl}\_n$. The Grothendieck-Springer resolution for ${\mathfrak s \mathfrak l}\_n = {\mathfrak s \mathfrak l}(V)$, where $V$ is an $(n+1)$-dimensional complex vector space, is the variety $\widetilde {\mathfrak g}$ consisting of pairs of an element $g$ in ${\mathfrak s \mathfrak l}(V)$ and a full flag $F$ of subspaces in $V$ such that $g$ preserves $F$. The natural projection map $\pi: {\widetilde {\mathfrak g}} \to {\mathfrak g}$ is called the Grothendieck-Springer resolution. Over the open regular semisimple locus in $\mathfrak g$ it is a regular covering with deck group equal to the Weyl group $W=S\_n$ in the case of ${\mathfrak s \mathfrak l}\_n$. My question is, is it possible to describe this action, given the action of the Weyl group on the cohomology of the fibers of the Springer sheaf? Could one perhaps describe the action of the Weyl group on the fibers of the Grothendieck-Springer sheaf as some kind of induced representation (since I think that the action of the Weyl group on the cohomology of the Springer fiber is given by an induced representation of the Weyl group by a result of de Concini, Lusztig, and Procesi)?



Is it possible to describe the action of the Weyl group on the cohomology of the fibers of the Grothendieck-Springer resolution?
As a $W$ module, the cohomology of a fiber of the Grothendieck-Springer resolution over $g$ is isomorphic to the cohomology of the fiber of the Springer resolution over its nilpotent part $g\_n$. Note, this is not true as a graded representation: for example for a regular semi-simple element, we get $\# W$ points with a regular $W$ action (so a regular $W$ module all in degree 0) whereas for the Springer fiber over 0, we get $G/B$. This follows from localization in equivariant cohomology applied to a torus in which the semi-simple part $g\_s$ is generic: we get a flat family of $W$-modules over the Lie algebra of the torus with generic fiber the cohomology of the Grothendieck-Springer, and special fiber the cohomology of the Springer fiber.


