On the Kirwan map for moduli of Higgs bundles

Let C be a smooth complex projective curve and G a connected complex reductive group. We prove that if the center Z (G) of G is disconnected, then the Kirwan map H* (Bun(G,C), Q) -> H* (MTHiggsss (G, C), Q) from the cohomology of the moduli stack of all G-bundles to the moduli stack of semistable G-Higgs bundles, cannot be surjective. We establish similar results for intersection cohomology and for the cohomology of the coarse moduli spaces. The proof uses a Borel-Quillen-style localization result for equivariant cohomology of stacks to reduce to an explicit calculation.