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Let $X$ be a compact connected Riemann surface of genus $g \geq 2$ and $G$ anontrivial connected reductive affine algebraic group over $\mathbb{C}$. Weconsider the moduli spaces of regularly stable $G$-Higgs bundles andholomorphic $G$-connections of fixed topological type $d\in \pi_1(G)$. We showthat these two moduli spaces have the same Grothendieck motives and their$E$-polynomials are equal. Also, we show that their Hodge structures are pureand isomorphic.

Motivic invariance for principal $G$-bundles with $\lambda$-connections