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 $H^\big(\operatorname{Bun}(G,C),$\mathbb{Q}$\big)\rightarrow H^\big($\mathcal{M}${\operatorname{Higgs}}^{\operatorname{ss}},$\mathbb{Q}$\big)$fromthecohomologyofthemodulistackof$G$-bundles to the moduli stack of semistable $G$-Higgs bundles, fails to be surjective: more precisely, the “variant cohomology” (and variant intersection cohomology) of the stack ${\mathcal{M}}_{\mathrm{Higgs}}^{}$ of semistable $G$-Higgs bundles, is always nontrivial. We also show that the image of the pullback map $H^*\big(M{\operatorname{Higgs}}^{\operatorname{ss}},$\mathbb{Q}$\big)\rightarrow H^*\big($\mathcal{M}$_{\operatorname{Higgs}}^{\operatorname{ss}},$\mathbb{Q}$\big)$, from the cohomology of the moduli space of semistable $G$-Higgs bundles to the stack of semistable $G$-Higgs bundles, cannot be contained in the image of the Kirwan map. The proof uses a Borel-Quillen–style localization result for equivariant cohomology of stacks to reduce to an explicit construction and calculation.