Semiprojectivity of the moduli of principal $G$-bundles with $\lambda$-connections

Let $X$ be a compact connected Riemann surface of genus $g\ge 2$ and $G$ a connected reductive affine algebraic group over $\mathbb{C}$. We prove the semiprojectivity of the moduli spaces of semistable $G$-Higgs bundles and $G$-bundles with $\lambda$-connections of fixed topological type $d\in {\pi }_1(G)$. As an application, in the smooth case we describe the resulting Bialynicki - Birula decomposition and derive cohomological and motivic consequences.