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

Abstract

Let $(G,X)$ be a Shimura datum. In previous work with Christian Klevdal, we showed that the canonical $G({\mathbb{Q}}_{\ell })$-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 $G({\mathbb{Q}}_{\ell })$-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.

Comments welcome. This update: (1) fixes some cross-referencing with our joint work with Huryn, Kedlaya, and Klevdal; (2) gives more details on companions with reductive monodromy group; (3) changes the title to avoid confusion with our previous work with Klevdal