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.