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

Abstract

We use 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 quasiprojective variety over a (finitely generated) characteristic-zero field, the associated graded of the weight filtration on its intersection cohomology arises from a motivated motive in the sense of André, and in particular from a classical homological motive if one assumes the standard conjectures. This extends work of de Cataldo and Migliorini.