To a reductive group $G$ and a smooth projective complex curve $C$, one associates the moduli stack of $G$-Higgs bundles on $C$. There is a natural map from the cohomology of the moduli stack of $G$-Higgs bundles to the cohomology of the moduli stack of semistable $G$-Higgs bundles, known as the Kirwan map. In the case of $G=GL(n)$, Markman proved that the Kirwan map is surjective for the component of Higgs bundles of degree $d$ coprime to $n$. By contrast, I explain recent work in which we show (generalizing Hitchin for $SL(n)$) that the Kirwan map for moduli of $G$-Higgs bundles fails to be surjective whenever G has disconnected centre. This is accomplished via the study of a group action on the moduli stack and the induced action on its cohomology. We also establish similar results for the cohomology of the moduli space (rather than stack), and for intersection cohomology. This talk is based on joint work with Thomas Nevins and Shiyu Shen.