Generalized Kuga-Satake theory and rigid local systems, II: rigid Hecke eigensheaves

This paper uses rigid Hecke eigensheaves, building on Yun’s work on the construction of motives with exceptional Galois groups, to produce the first robust examples of generalized Kuga-Satake theory' outside the Tannakian category of motives generated by abelian varieties. To strengthen our description of the motivic’ nature of Kuga-Satake lifts, we digress to establish a result that should be of independent interest: for any quasi-projective variety over a (finitely-generated) characteristic zero field, the associated weight-graded of its intersection cohomology arises from a motivated motive in the sense of Andr'{e}, and in particular from a classical homological motive if one assumes the Standard Conjectures. This extends work of de Cataldo and Migliorini.