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The cohomology of Nakajima quiver varieties is a quite interesting object. It's equipped with some natural classes given by the Chern classes of the tautological bundles associated to the spaces in the quiver representation. We say that a quiver variety satisfies Kirwan surjectivity if these classes generate the cohomology ring.

I know that in type A, Nakajima quiver varieties are homotopic to Spaltenstein varieties, and I can see surjectivity using a presentation of that cohomology.

If all the entries in the dimension vector are 1, then the quiver variety is a hypertoric variety, and we know Kirwan surjectivity for those. There are also hyperpolygon spaces.

Are there any other large classes of examples where this is proven?

I'm particularly interested in what's in the literature from the finite case. It seems that this case is "obvious" based on work of Kodera and Naoi, but no one seems to have actually written it down (at least not anyone citing their paper).

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As an update, Kevin McGerty and Thomas Nevins prove Kirwan surjectivity for Nakajima quiver varieties in this recent paper.

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