As usual, by Springer fiber, I mean the fixed points $X^u$ of a unipotent element $u$ of the group $G$ on the flag variety $X=G/B$. It's a lovely theorem that when $G=SL_n$, the induced map on cohomology $H^*(X)\to H^*(X^u)$ is surjective.

However, I've been told many times that for other $G$, this isn't the case. Can anyone point me to a good reference for finding these, or a handy source of examples?

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    $\begingroup$ Do you really mean SL_n or GL_n? For GL_n this was proven by Hotta and Springer in "A Specialisation Theorem for ...". However I thought this map is already not surjective for all unipotent elements of SL_n. $\endgroup$ – Jay Taylor Aug 1 '13 at 5:18
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    $\begingroup$ One easy example is provided by the subregular orbits in non simply laced type. They are isomorphic to subregular Springer fibres in much larger (roughly double) simply laced types ("folding"), and hence $H^2$ cannot be surjective. In general, this restriction map must land in the $A_G(x)$ invariants, so any non-trivial $A_G(x)$ action gives you counter-examples. ($A_G(x)$ is the component group of the centralizer.) $\endgroup$ – Geordie Williamson Aug 1 '13 at 7:44

I think the standard early reference is the paper by Hotta and Springer here, in which they work with $\ell$-adic cohomology (but the results seem to carry over to other settings). What they show is that the map in cohomology is surjective on the top degree in the target space when mapping into the subspace fixed by the component group, while the precise image in other degrees might be more elusive when there is a nontrivial component group for the unipotent class in question. (This is essentially why all goes well for type $A$, where component groups are trivial.)

Examples in rank 2 for types $B_2, G_2$ show already what goes wrong when the component group is nontrivial (which happens there for the subregular orbit, as Jay observes). But as far as I know, the problem remains quite difficult to study beyond what is done by Hotta-Springer.

[I gave a short survey of Springer theory in Chapter 9 of my 1995 AMS book on conjugacy classes, but unfortunately overstated the Hotta-Springer result in 9.6(4). Corrections are posted on my homepage.]


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