On the Kirwan map for moduli of Higgs bundles

Let $C$ be a smooth complex projective curve and $G$ connected reductive group. We prove that if the center $Z(G)$ of is disconnected, then Kirwan map $H^*\big(\operatorname{Bun}(G,C),\mathbb{Q}\big)\rightarrow H^*\big(\mathcal{M}_{\operatorname{Higgs}}^{\operatorname{ss}},\mathbb{Q}\big)$ from cohomology moduli stack $G$-bundles to semistable $G$-Higgs bundles, fails surjective: more precisely, variant (and intersection cohomology) $\mathcal{M}_{\operatorname{Higgs}}^{\operatorname{ss}}$ always nontrivial. also show image pullback $H^*\big(M_{\operatorname{Higgs}}^{\operatorname{ss}},\mathbb{Q}\big)\rightarrow H^*\big(\mathcal{M}_{\operatorname{Higgs}}^{\operatorname{ss}},\mathbb{Q}\big)$, space bundles cannot contained in map. The proof uses Borel-Quillen--style localization result for equivariant stacks reduce an explicit construction calculation.