Compatibility of canonical $\mathrm{\ell }$-adic local systems on Shimura varieties, II

Let $(G,X)$ be a Shimura datum. In previous work with Christian Klevdal, we showed that the canonical $G({\mathbb{Q}}_{\mathrm{\ell }})$-valued local systems on Shimura varieties for $G$ form compatible systems after projection to the adjoint group of $G$. In this note, we strengthen this result to prove compatibility for the $G({\mathbb{Q}}_{\mathrm{\ell }})$-local systems themselves. We also include the crystalline compatibility, extending the adjoint case established in our joint work Jake Huryn, Kiran Kedlaya, and Christian Klevdal.