I am trying to understand the theory of integral model of Shimura variety of Hodge type, like for example in Kisin's paper "Integral models for Shimura varieties of abelian type".

I understand that everything is done by a careful study of the tensors $\{s_\alpha\} \subset V^\otimes$ giving the group $G$ as a subgroup of $\mathrm{GSp}(V)$, that come from the embedding $X \to X_{\mathrm{Siegel}}$, where $X$ is the Shimura variety of Hodge type and $X_{\mathrm{Siegel}}$ is Siegel variety. So far so good. Note that the various automorphic objects obtained by $V$ by the standard machinery are those associated to restriction of to $X$ of the universal abelian scheme over $X_{\mathrm{Siegel}}$ (we get the étale cohomology, the de Rham cohomology etc...).

I am also reading the paper "Integral canonical models for Spin Shimura varieties" by Madapusi Pera, that deals with a particular case. Here, there is only one tensor $\pi \in V^{\otimes 2} \otimes (V^\vee)^{\otimes 2} = \mathrm{End}(\mathrm{End}(V))$, that is moreover an idempotent. Madapusi Pera uses a lot $L = \mathrm{Im}(\pi) \subset V \otimes V^\vee$, that is a $G$-representation: for example he considers the automorphic object associated to $L$.

Is there an analogue of the $G$ representation $L$ for a general Shimura variety of Hodge type?

Clearly $\pi$ is the analogue of the tensors $\{s_\alpha\}$ (probably in the $\mathrm{Gspin}$ case it is canonical, while the $\{s_\alpha\}$ in general is not), but I do not see what should I consider for $L$.

Thank you!


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