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)?
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$\begingroup$ I don't understand what "this action" means. The action of deck transformations? $\endgroup$– Ben Webster ♦Commented Sep 24, 2015 at 11:55
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$\begingroup$ @BenWebster Thank you very much for your help! I think that there is an action of the Weyl group on the Grothendieck-Springer sheaf by endomorphisms, so should not this be the same as an action on the Weyl group on the cohomology of each fiber of the Grothendieck-Springer sheaf? Or am I confused? $\endgroup$– Yellow PigCommented Sep 24, 2015 at 11:58
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$\begingroup$ @BenWebster In other words, like you said, the group $W=S_n$ of deck transformations acts on the restriction of the Grothendieck-Springer sheaf to the regular semisimple part, hence $W$ also acts on the Grothendieck-Springer sheaf, which is the IC extension of its restriction to the regular semisimple part. $\endgroup$– Yellow PigCommented Sep 24, 2015 at 12:05
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$\begingroup$ Right, so you just described the action. Since you clearly knew what the action was, I wasn't so sure what you were asking. $\endgroup$– Ben Webster ♦Commented Sep 24, 2015 at 12:13
1 Answer
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.
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$\begingroup$ Thank you very much! I would be very grateful if you could also give me a hint, a suggestion, or a reference to help me find (in a computable way) the grading on the induced representation $Ind_{W_L}^W$ (where $L$ is the Levi subgroup defining the type of the nilpotent $g_n$) which is the cohomology of the Grothendieck-Springer fiber over $g$. I need this grading because I need to find (say in the example of ${\mathfrak s \mathfrak l}_4$) the dimensions of the cohomology groups of the stalks of each simple perverse sheaf which is a summand of the Grothendieck-Springer sheaf in type A. $\endgroup$ Commented Sep 24, 2015 at 17:51