Motivic realization of rigid G-local systems on curves and tamely ramified geometric Langlands

For a reductive group $G$, we prove that complex irreducible rigid $G$-local systems with quasi-unipotent monodromies and finite order abelianization on a smooth curve are motivic, generalizing a theorem of Katz for $GL_n$. We do so by showing that the Hecke eigensheaf corresponding to such a local system is itself motivic. Unlike other works in the subject, we work entirely over the complex numbers. In the setting of de Rham geometric Langlands, we prove the existence of Hecke eigensheaves associated to any irreducible $G$-local system with regular singularities. We also provide a spectral decomposition of a naturally defined automorphic category over the stack of regular singular local systems with prescribed eigenvalues of the local monodromies at infinity. Finally, we establish a relationship between rigidity for complex local systems and automorphic rigidity, answering a conjecture of Yun in the tame complex setting.