Shimura varieties of Hodge type

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!