  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.
