Lusztig defined (in Sec. 5, also Sage) a Springer action of the affine Weyl group on the homology of affine Springer fibers (Iwahori one, i.e. in an affine flag variety). In the regular semisimple unramified case, Goresky, Kottiwtz and MacPherson had a formula (14.4) for this action assuming the homology is pure, for example in the equal valuation case, where they said
It can be shown that this action coincides with the Springer action defined by Lusztig [Lu] and (using a statement from [KL1] whose proof does not appear in the literature) by Sage [S1].
May I bother asking for hint how one can show it coincides with Lusztig's action? Thanks!
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Just to add a failed attempt for mine for top homology in the $SL(2)$ case (must have done something stupid...). The way I understand Lusztig's action in the $SL(2)$ case is the following: The Springer sheaf of $SL(2)$ is the direct sum of (shifted) constant sheaf on nilpotent cone and a skyscraper sheaf at zero, while the Weyl group acts trivially on the first and by $-1$ on the second. Consequently, suppose $\dot{X}_{\gamma}$ is the ASF in the affine flag variety with $\dot{X}_{\gamma}\rightarrow X_{\gamma}$ where the latter is the ASF in $Gr$. The fiber of this map is either a point or $\mathbb{P}^1$ (like the Springer resolution). The generator of the affine Weyl group that corresponds to this choice of $Gr$ (there are two choices) acts by $-1$ on the $H^2$ of this $\mathbb{P}^1$ and acts trivially otherwise, as for the Springer sheaf. In other words, for the dual action on the top homology, it acts trivially on those non-contracted components in $Gr$ and acts as $-1$ on those contracted. For another generator of the affine Weyl group, we then use the projection to the other $Gr$. This does not look like a very interesting action - the two generators of the affine Weyl group have commuting actions - and is different from the action that Goresky-Kottiwtz-MacPherson give.
