Functoriality of motivic lifts of the canonical construction

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Let (G,{mathfrak {X}}) be a Shimura datum and K a neat open compact subgroup of G(mathbb {A}_f). Under mild hypothesis on (G,{mathfrak {X}}), the canonical construction associates a variation of Hodge structure on text {Sh}_K(G,{mathfrak {X}})(mathbb {C}) to a representation of G. It is conjectured that this should be of motivic origin. Specifically, there should be a lift of the canonical construction which takes values in relative Chow motives over text {Sh}_K(G,{mathfrak {X}}) and is functorial in (G,{mathfrak {X}}). Using the formalism of mixed Shimura varieties, we show that such a motivic lift exists on the full subcategory of representations of Hodge type {(-1,0),(0,-1)}. If (G,{mathfrak {X}}) is equipped with a choice of PEL-datum, Ancona has defined a motivic lift for all representations of G. We show that this is independent of the choice of PEL-datum and give criteria for it to be compatible with base change. Additionally, we provide a classification of Shimura data of PEL-type and demonstrate that the canonical construction is applicable in this context.