This paper is the substantially revised version of paragraph 5 of Polylogarithmic Extensions on Mixed Shimura varieties. We study the functor "canonical construction", which associates to a representation of the group underlying mixed Shimura data a mixed system of sheaves on the Shimura variety, consisting of an admissible variation of Hodge structure for each embedding of the reflex field into C, a mixed lisse l-adic sheaf, a bifiltered flat vector bundle and comparison isomorphisms. Due to work of Milne, the situation is rather well understood in the case of pure Shimura data. Making full use of the results of Mixed structures on fundamental groups, we extend the construction to the mixed setting.
This paper is now available in Springer Lecture Notes in Mathematics, number 1650.