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.
Keywords
Galois representations, rigid local systems, Kuga–Satake
construction, geometric Langlands
