Let (G,X) be a Shimura datum and K a neat open compact subgroup of $G(\mathbb{A}_f)$. Under mild hypothesis on (G,X), the canonical construction associates a variation of Hodge structure on $\textrm{Sh}_K(G,X)(\mathbb{C})$ 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 $\textrm{Sh}_K(G,X)$ and is functorial in (G,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,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.

中文翻译：

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规范结构的动机提升的函数性

设 (G,X) 是 Shimura 数据，K 是 $G(\mathbb{A}_f)$ 的整齐开紧子群。在 (G,X) 的温和假设下，规范构造将 $\textrm{Sh}_K(G,X)(\mathbb{C})$ 上的 Hodge 结构变化与 G 的表示相关联。据推测，这应该是出于动机。具体来说，应该有一个规范构造的提升，它在 $\textrm{Sh}_K(G,X)$ 上的相对 Chow 动机中取值并且在 (G,X) 中是函子。使用混合 Shimura 变体的形式主义，我们表明这种动机提升存在于 Hodge 类型 {(-1,0),(0,-1)} 表示的完整子类别中。如果 (G,X) 配备了 PEL 数据的选择，则 Ancona 已为 G 的所有表示定义了一个动机提升。