Let $X$ be a compact connected Riemann surface of genus $g \geq 2$ and $G$ a nontrivial connected reductive affine algebraic group over $\mathbb{C}$. We consider the moduli spaces of regularly stable $G$-Higgs bundles and holomorphic $G$-connections of fixed topological type $d\in \pi_1(G)$. We show that these two moduli spaces have the same Grothendieck motives and their $E$-polynomials are equal. Also, we show that their Hodge structures are pure and isomorphic.PDF: Motivic invariance for principal $G$-bundles with $\lambda$-connections.pdf
